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Now let's think about what the inter, whether the interquartile range should change. And I don't think it will, because remember, everything shifts. And even though the first quartile is gonna be five more, but the third quartile is gonna be five more as well. So the difference shouldn't change. And indeed, look, the d... | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
So the difference shouldn't change. And indeed, look, the distance does not change, or the difference does not change. But similarly, if we scale everything up, if we were to scale up the first quartile and the third quartile by five, well then their difference should scale up by five. And we see that right over there.... | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
Which of the following are conditions for this type of interval? So before I even look at these choices, and they say choose all answers that apply, so it might be more than one, let's just think about what the conditions for inference for this type of interval actually are. So we've done this many times in many differ... | Conditions for inference for difference of means AP Statistics Khan Academy.mp3 |
And that's the idea that each of our samples are random, or we are conducting some type of an experiment where we randomly assign folks to one, or eggs in this case, to one of two groups. In this case, we are taking two samples, and we would hope that they are truly random samples. The second is the normal condition. A... | Conditions for inference for difference of means AP Statistics Khan Academy.mp3 |
And the normal condition is a little bit different depending on whether we're talking about means or whether we're talking about proportions. The random condition is essentially the same. The normal condition when we're talking about means, remember, they're looking at the difference between mean weights of eggs, is yo... | Conditions for inference for difference of means AP Statistics Khan Academy.mp3 |
One is if the underlying distribution is normal. The second way is if your sample sizes for each of your samples are greater than or equal to 30. So if your first sample size is greater than or equal to 30 and your second sample size is greater than or equal to 30, or even if the underlying data, you don't know whether... | Conditions for inference for difference of means AP Statistics Khan Academy.mp3 |
When we're dealing with means. And then the third condition, and we see this whether we're dealing with means or proportions or differences of means or differences of proportions, is the independence condition. And this is the idea that either your individual observations are done with replacement in both of your sampl... | Conditions for inference for difference of means AP Statistics Khan Academy.mp3 |
So with that little bit of a review, let's see which of these apply. They observe at least 10 heavy eggs and 10 light eggs in each sample. So this actually is the normal condition when we are dealing with proportions, not for means. So I would rule this out. It's a good distractor choice. The eggs in each sample are ra... | Conditions for inference for difference of means AP Statistics Khan Academy.mp3 |
So I would rule this out. It's a good distractor choice. The eggs in each sample are randomly selected from their population. Yep, that's the random condition right over there, so I would select that. They sample an equal number of each type of egg. So this is a common misconception that whether we're dealing with mean... | Conditions for inference for difference of means AP Statistics Khan Academy.mp3 |
Yep, that's the random condition right over there, so I would select that. They sample an equal number of each type of egg. So this is a common misconception that whether we're dealing with means or proportions, when we're thinking about the difference between, say, means or the difference between proportions, that som... | Conditions for inference for difference of means AP Statistics Khan Academy.mp3 |
The amount in the sample had a mean of 503 milliliters and a standard deviation of five milliliters. They want to test if this is convincing evidence that the mean amount for bottles in this batch is different than the target value of 500 milliliters. Let mu be the mean amount of liquid in each bottle in the batch. Wri... | Writing hypotheses for a significance test about a mean AP Statistics Khan Academy.mp3 |
Write an appropriate set of hypotheses for their significance test, for the significance test that the quality-control expert is running. So pause this video and see if you can do that. All right, now let's do this together. So first, you're going to have two hypotheses. You're gonna have your null hypothesis and your ... | Writing hypotheses for a significance test about a mean AP Statistics Khan Academy.mp3 |
So first, you're going to have two hypotheses. You're gonna have your null hypothesis and your alternative hypothesis. Your null hypothesis is going to be a hypothesis about the population parameter that you care about, and it's going to assume kind of the status quo, no news here. And so the parameter that we care abo... | Writing hypotheses for a significance test about a mean AP Statistics Khan Academy.mp3 |
And so the parameter that we care about is the mean amount of liquid in the bottles in the batch. So that's mu right over there. And what would be the assumption that that would be, the no news here? Well, it would be 500 milliliters. That's the target value. So it's reasonable to say, well, you know, the null is it's ... | Writing hypotheses for a significance test about a mean AP Statistics Khan Academy.mp3 |
Well, it would be 500 milliliters. That's the target value. So it's reasonable to say, well, you know, the null is it's doing what it's supposed to, that where the actual mean for the batch is actually what the target needs to be, is actually 500 milliliters. Some of you might have said, hey, wait, didn't they say the ... | Writing hypotheses for a significance test about a mean AP Statistics Khan Academy.mp3 |
Some of you might have said, hey, wait, didn't they say the amounts in the sample had a mean of 503 milliliters? Why isn't this 503? Remember, your hypothesis is going to be about the population parameter, your assumption about the population parameter. This 503 milliliters right over here, this is a sample statistic. ... | Writing hypotheses for a significance test about a mean AP Statistics Khan Academy.mp3 |
What we're going to do in this video is get some practice classifying whether a random variable is a binomial variable, and we're gonna do it by looking at a few exercises from Khan Academy. So this says a manager oversees 11 female employees and nine male employees. They need to pick three of these employees to go on ... | Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3 |
Let x equal the number of female employees chosen. So they're going to do three trials, and on each of those trials, you could say success is if they pick a female employee, and then the random variable x is the number of females out of those three. Is x a binomial variable? Why or why not? So pause this video and see ... | Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3 |
Why or why not? So pause this video and see if you can work through this on your own. All right, now let's go through each choice. Choice A says each trial isn't being classified as a success or failure, so x is not a binomial variable. I disagree with this. Each trial is being classified as a success or failure. It's ... | Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3 |
Choice A says each trial isn't being classified as a success or failure, so x is not a binomial variable. I disagree with this. Each trial is being classified as a success or failure. It's either going to be female or not, and since we're counting the number of female employees, if in a trial we pick a female, that wou... | Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3 |
It's either going to be female or not, and since we're counting the number of female employees, if in a trial we pick a female, that would be a success, so each trial is being classified as a success or failure. So this one over here isn't true. There is no fixed number of trials, so x is not a binomial variable. There... | Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3 |
There is a fixed number of trials. They're doing three trials. They're picking three hats, three names out of a hat. The trials are not independent, so x is not a binomial variable. So this is interesting. So for example, trial one, what's the probability of success? Well, there are 20 employees, 20 names in the hat, a... | Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3 |
The trials are not independent, so x is not a binomial variable. So this is interesting. So for example, trial one, what's the probability of success? Well, there are 20 employees, 20 names in the hat, and 11 of the outcomes would be success, so you have 11 20th probability of success, but in trial two, what's the prob... | Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3 |
Well, there are 20 employees, 20 names in the hat, and 11 of the outcomes would be success, so you have 11 20th probability of success, but in trial two, what's the probability of success given success in, I'll say, trial one, T1? Well, if you succeeded in trial one, that means that there's now only 10 female names in ... | Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3 |
The trials are not independent, so that violates that condition for being a binomial variable. In order to be a binomial variable, all your trials have to be independent of each other, and so we'd rule this last one out because this last one says that x has a binomial distribution, or it is or does meet all the conditi... | Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3 |
So here we have different scenarios, and I have the conditions for binomial variable written right over here, and so once again, pause the video and look at each of these scenarios for random variables, and look at these conditions, and think about whether these random variables are binomial or not. So let's look at th... | Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3 |
Let y be equal to the number of aces drawn. Well, in our introductory video to binomial variables, we talked about if we're doing without replacement, your probability of getting an ace on a given trial, where trial is you're taking a card out of the deck, it's going to be dependent on whether you got aces in previous ... | Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3 |
Not independent, not independent trials. Now, on the other hand, if on every trial you looked at whatever card you got and put it back in the deck, then they would be independent trials. The probability of getting an ace on each trial would be the same, but not when you have without replacement. So this is not binomial... | Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3 |
So this is not binomial right over here, because you don't have independent trials. The second scenario. 60% of a certain species of tomato live after transplanting from pot to garden. Eli transplants 16 of these tomato plants. Assume that the plants live independently of each other. So whether one plant lives isn't de... | Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3 |
Eli transplants 16 of these tomato plants. Assume that the plants live independently of each other. So whether one plant lives isn't dependent on whether another plant lives. Let T equal the number of tomato plants that live. All right, so let's look at the conditions. The outcome of each trial can be classified as eit... | Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3 |
Let T equal the number of tomato plants that live. All right, so let's look at the conditions. The outcome of each trial can be classified as either success or failure. So each trial over here is one of the tomato plants, and we have 16 of those trials, and success is if the tomato plant lives, and failure is if it die... | Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3 |
So each trial over here is one of the tomato plants, and we have 16 of those trials, and success is if the tomato plant lives, and failure is if it dies. So we have either success or failure. Each trial is independent of the others. They tell us the plants live independently of each other. So whether or not a neighbori... | Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3 |
They tell us the plants live independently of each other. So whether or not a neighboring plant lives or dies doesn't affect whether the plant next to it lives or dies. So each trial is independent of the others. There is a fixed number of trials. Yes, we have 16 right over there. The probability P of success on each t... | Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3 |
There is a fixed number of trials. Yes, we have 16 right over there. The probability P of success on each trial remains constant. Well, yeah, according to at least the scenario, they're saying that we have a 60% chance for each tomato plant, which is each trial. So it meets all of the conditions right over here. So thi... | Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3 |
Well, yeah, according to at least the scenario, they're saying that we have a 60% chance for each tomato plant, which is each trial. So it meets all of the conditions right over here. So this one is binomial. Now let's look at this third scenario. In a game of luck, a turn consists of a player continuing to roll a pair... | Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3 |
Now let's look at this third scenario. In a game of luck, a turn consists of a player continuing to roll a pair of six-sided die until they roll a double, two of the same face values. Let x equal the number of rolls in one turn. So you're gonna keep rolling until they roll a double. Well, the thing that jumps out at me... | Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3 |
So you're gonna keep rolling until they roll a double. Well, the thing that jumps out at me is that you don't have a fixed number of trials, not fixed number of trials. You could say each trial, each roll is a trial. Success is getting a double, which has a fixed probability. Whether or not you get a double on each tri... | Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3 |
Let's say that your school has a population of 80 students in it. Maybe it's not your whole school, maybe it's just your grade. So there's 80 students in your population, and you want to get an estimate of the average height in your population. And you think it's too hard for you to go and measure the height of all 80 ... | Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3 |
And you think it's too hard for you to go and measure the height of all 80 students, so you decide to find a simple, or take a simple random sample. You think it's reasonable for you to measure the heights of 30 of these students. And so what you want to do is randomly sample 30 of the 80 students, and take their avera... | Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3 |
So once you decide to do this, you say, well, how do I select those 30 students, and how do I select it so that I feel good that it is actually random? And there's several ways that you could approach this. One way to do it is associate every person in your school with a piece of paper, and put them all in a bowl, and ... | Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3 |
So let's do that. So let's say this is alphabetically the first person in the school, they're on a slip of paper, then the next slip of paper gets the next person, and you're gonna go all the way down, so you're gonna have 80 pieces of paper. They all should be the same size. And then you throw them all, you throw them... | Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3 |
And then you throw them all, you throw them all into a bowl of some kind. And this seems like a very basic way of doing it, but it's actually a pretty effective way of getting a simple, of getting a simple random sample. So I'll try to draw a little, I don't know, it looks like a little fishbowl or something. All right... | Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3 |
All right, so that's our bowl. And so all the pieces of paper go in there. And then you put a blindfold on someone, and they can't feel what names are there, and so they should pick out the first 30 without replacing them, because you obviously don't wanna pick the same, you don't wanna pick out the same name twice. An... | Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3 |
And those 30 names that you pick, that would be your simple random sample, and then you could measure their heights to estimate the average height for the population. This would be a completely legitimate way of doing it. Other ways that you could do it, if you have a computer or a calculator, you could use a random nu... | Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3 |
And the random functions on computer programming languages or on your calculator, they tend to be something, you know, some place you'll see something like a math.rand, short for random. You might see something like random. You might see, you might see something like random. Without anything passed into it, it might gi... | Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3 |
Without anything passed into it, it might give you a number between zero and one, or zero and 100. And you have to be very careful on how you use this to make sure that you have an even chance of picking certain numbers. But what you would do in this situation, if you had access to some random number generator, and it ... | Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3 |
And so the first student alphabetically, assign the number zero, one. And you could just say one if you're using a random number generator, but I'll use two digits for it just because it'll be useful and consistent. And in a little bit, we'll use another technique where it's gonna be nice to be consistent with our numb... | Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3 |
And so the next one, zero, two, and you go all the way to 79, and all the way to 80. And then you use your random number generator to keep generating numbers from one to 80. And as long as you don't get repeats, you pick the first 30 to be your actual random sample. Another related technique, which is a little bit more... | Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3 |
Another related technique, which is a little bit more old school, but is definitely the way that it has been done in the past and even done now sometimes, is to use a random digit table. You still start with these number associations with each student in the class, and then you use a randomly generated list of numbers.... | Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3 |
And you start at the beginning. And you say, okay, we're interested in getting 30 two-digit numbers from one to 80, including one in 80. So one technique that you could use is you start it right at the beginning, and you could say, all right, this is a randomly generated list of numbers. So the first number here is 59.... | Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3 |
So the first number here is 59. Is 59 between one and 80? Sure is. As long as we, you know, if this was a zero one, that would have worked. If this was an eight zero, that would have worked. If this was a zero zero, it wouldn't have worked. If this was an eight one, it wouldn't have worked. | Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3 |
As long as we, you know, if this was a zero one, that would have worked. If this was an eight zero, that would have worked. If this was a zero zero, it wouldn't have worked. If this was an eight one, it wouldn't have worked. But this would be our, this right over here, that would be our first name that we, you could im... | Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3 |
If this was an eight one, it wouldn't have worked. But this would be our, this right over here, that would be our first name that we, you could imagine the same as picking that first name out of the hat, whoever's associated with number 59. Now, you would move on. You get the next two digits. The next two digits are 83... | Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3 |
You get the next two digits. The next two digits are 83. They don't fall into our range from one to 80, so we're not going to use it. Then you look at the next two digits. So we get a five and a nine. Well, that fits in our range, but we already picked 59. We already picked person 59, so we're not gonna pick 59 again. | Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3 |
Then you look at the next two digits. So we get a five and a nine. Well, that fits in our range, but we already picked 59. We already picked person 59, so we're not gonna pick 59 again. So we keep moving on. Then we get a 37. Well, that's in our range. | Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3 |
We already picked person 59, so we're not gonna pick 59 again. So we keep moving on. Then we get a 37. Well, that's in our range. We haven't picked that yet. We do that. Then we get a zero zero. | Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3 |
Well, that's in our range. We haven't picked that yet. We do that. Then we get a zero zero. Once again, not in our range. I think you see where this is going. 91, not in our range. | Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3 |
Then we get a zero zero. Once again, not in our range. I think you see where this is going. 91, not in our range. 23, it's in our range, and we haven't picked it yet. So we're gonna pick the 23. I think you see where this is going. | Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3 |
91, not in our range. 23, it's in our range, and we haven't picked it yet. So we're gonna pick the 23. I think you see where this is going. We're gonna keep going down this list in the way that I've just described until we get 30 of these. We've just gotten three. We just have to keep on going. | Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3 |
I think you see where this is going. We're gonna keep going down this list in the way that I've just described until we get 30 of these. We've just gotten three. We just have to keep on going. And this isn't an exhaustive list of all of the different ways that you can get random numbers, but it starts to give you some ... | Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3 |
What we're going to do in this video is think about under which conditions does the sampling distribution of the sample proportions? In which situations does it look roughly normal? And under which situations does it look skewed right? So does it look something like this? And under which situations does it look skewed ... | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
So does it look something like this? And under which situations does it look skewed left? Maybe something like that. And the conditions that we're going to talk about, and this is a rough rule of thumb, that if we take our sample size and we multiply it by the population proportion that we care about, and that is great... | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
And the conditions that we're going to talk about, and this is a rough rule of thumb, that if we take our sample size and we multiply it by the population proportion that we care about, and that is greater than or equal to 10, and if we take the sample size and we multiply it times one minus the population proportion, ... | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
According to the supplier, approximately 12% of the population of these tangerines is overripe. Suppose that Emiliana calculates the daily proportion of overripe tangerines in her sample of 50. We can assume the supplier's claim is true and that the tangerines each day represent a random sample. What will be the shape ... | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
What will be the shape of the sampling distribution, what will be the shape of the sampling distribution of the daily proportions of overripe tangerines? Pause this video, think about what we just talked about, and see if you can answer this. All right, so right over here, we're getting daily samples of 50 tangerines. ... | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
So for this particular example, our n is equal to 50, and our population proportion, the proportion that is overripe is 12%, so p is 0.12. So if we take n times p, what do we get? np is equal to 50 times 0.12. Well, 100 times this would be 12, so 50 times this is going to be equal to six. And this is less than or equal... | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
Well, 100 times this would be 12, so 50 times this is going to be equal to six. And this is less than or equal to 10. So this immediately violates this first condition, and so we know that we're not going to be dealing with a normal distribution. And so the question is, how is it going to be skewed? And the key realiza... | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
And so the question is, how is it going to be skewed? And the key realization is, remember, the mean of the sample proportions, of the sampling distribution of the sample proportions, or the mean of the sampling distribution of the daily proportions, that that's going to be the same thing as our population proportion. ... | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
So if I were to draw it, let me see if I were to draw it right over here, where this is 50%, and this is 100%, our mean is gonna be right over here at 12%. And so you're gonna have it really high over there, and then it's gonna be skewed to the right. You're gonna have a big, long tail. So this is going to be skewed to... | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
So this is going to be skewed to the right. Let's do another example. So here we're told, according to a Nielsen survey, radio reaches 88% of children each week. Suppose we took weekly random samples of N equals 125 children from this population, and computed the proportion of children in each sample whom radio reaches... | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
Suppose we took weekly random samples of N equals 125 children from this population, and computed the proportion of children in each sample whom radio reaches. What will be the shape of the sampling distribution of the proportions of children the radio reaches? Once again, pause this video and see if you can figure it ... | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
All right, well, let's just figure out what N and P are. Our sample size here, N, is equal to 125, and our population proportion of the proportion of children that are reached each week by radio is 88%. So P is 0.88. So now let's calculate NP. So N is 125 times P is 0.88. And is this going to be greater than or equal t... | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
So now let's calculate NP. So N is 125 times P is 0.88. And is this going to be greater than or equal to 10? Well, we don't even have to calculate this exactly. This is almost 90% of 125. This is actually going to be over 100. So it for sure is going to be greater than 10. | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
Well, we don't even have to calculate this exactly. This is almost 90% of 125. This is actually going to be over 100. So it for sure is going to be greater than 10. So we meet this first condition. But what about the second condition? We could take N, 125, times one minus P. So this is times 0.12. | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
So it for sure is going to be greater than 10. So we meet this first condition. But what about the second condition? We could take N, 125, times one minus P. So this is times 0.12. So this is 12% of 125. Well, even 10% of 125 would be 12.5. So 12% is for sure going to be greater than that. | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
We could take N, 125, times one minus P. So this is times 0.12. So this is 12% of 125. Well, even 10% of 125 would be 12.5. So 12% is for sure going to be greater than that. So this too is going to be greater than 10. I didn't even have to calculate it. I could just estimate it. | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
So 12% is for sure going to be greater than that. So this too is going to be greater than 10. I didn't even have to calculate it. I could just estimate it. And so we meet that second condition. So even though our population proportion is quite high, it's quite close to one here, because our sample size is so large, it ... | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
I could just estimate it. And so we meet that second condition. So even though our population proportion is quite high, it's quite close to one here, because our sample size is so large, it still will be roughly normal. And one way to get the intuition for that is, so this is a proportion of zero. Let's say this is 50%... | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
And one way to get the intuition for that is, so this is a proportion of zero. Let's say this is 50%, and this is 100%. So our mean right over here is going to be 0.88 for our sampling distribution of the sample proportions. If we had a low sample size, then our standard deviation would be quite large. And so then you ... | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
If we had a low sample size, then our standard deviation would be quite large. And so then you would end up with a left skewed distribution. But we saw before, the higher your sample size, the smaller your standard deviation for the sampling distribution. And so what that does is it tightens up, it tightens up the stan... | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
And so what that does is it tightens up, it tightens up the standard deviation. And so it's going to look more normal. It's gonna look closer, it's gonna look closer to being normal. So we'll say approximately normal, because it met our conditions for this rule of thumb. Is it gonna be perfectly normal? No. In fact, if... | Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3 |
Let's say that I run a website that currently has this off-white color for its background, and I know the mean amount of time that people spend on my website. Let's say it is 20 minutes, and I'm interested in making a change that will make people spend more time on my website. My idea is to make the background color of... | P-values and significance tests AP Statistics Khan Academy.mp3 |
But after making that change, how do I feel good about this actually having the intended consequence? Well, that's where significance tests come into play. What I would do is first set up some hypotheses, a null hypothesis and an alternative hypothesis. The null hypothesis tends to be a statement that, hey, your change... | P-values and significance tests AP Statistics Khan Academy.mp3 |
The null hypothesis tends to be a statement that, hey, your change actually had no effect. There's no news here. And so this would be that your mean is still equal to 20 minutes, is still equal to 20 minutes after, after the change to yellow, in this case, for our background. And we would also have an alternative hypot... | P-values and significance tests AP Statistics Khan Academy.mp3 |
And we would also have an alternative hypothesis. Our alternative hypothesis is actually that our mean is now greater because of the change, that people are spending more time on my site. So our mean is greater than 20 minutes after, after the change. Now, the next thing we do is we set up a threshold known as the sign... | P-values and significance tests AP Statistics Khan Academy.mp3 |
Now, the next thing we do is we set up a threshold known as the significance level, and you will see how this comes into play in a second. So your significance level, significance level, is usually denoted by the Greek letter alpha, and you tend to see significant levels like 1 1 100th or 5 1 100th or 1 10th or 1%, 5% ... | P-values and significance tests AP Statistics Khan Academy.mp3 |
Let's just say it's going to be 0.05. And what we're going to now do is we're going to take a sample of people visiting this new yellow background website, and we're gonna calculate statistics, the sample mean, the sample standard deviation, and we're gonna say, hey, if we assume that the null hypothesis is true, what ... | P-values and significance tests AP Statistics Khan Academy.mp3 |
However, if the probability of getting the statistics for that sample are at the significance level or higher, then we say, hey, we can't reject the null hypothesis, and we aren't able to have evidence for the alternative. So what we would then do, I will call this step three. In step three, we would take a sample, tak... | P-values and significance tests AP Statistics Khan Academy.mp3 |
We measure the sample mean here. Let's say that for that sample, the mean is 25, 25 minutes. We are also likely to, if we don't know what the actual population standard deviation is, which we typically don't know, we would also calculate the sample standard deviation. Then the next step is we calculate a p-value, and t... | P-values and significance tests AP Statistics Khan Academy.mp3 |
Then the next step is we calculate a p-value, and the p-value, which stands for probability value, is the probability of getting a statistic at least this far away from the mean if we were to assume that the null hypothesis is true. So one way to think about it, it is a conditional probability. It is the probability th... | P-values and significance tests AP Statistics Khan Academy.mp3 |
And in other videos, we have talked about how to do this. If we assume that the sampling distribution of the sample means is roughly normal, we can use the sample mean, we can use our sample size, we can use our sample standard deviation, perhaps we use a t-statistic to figure out roughly what this probability is going... | P-values and significance tests AP Statistics Khan Academy.mp3 |
So let me call that step five. So step five, there are two situations. If my p-value, if my p-value, if it is less than alpha, then I reject my null hypothesis, reject, reject my null hypothesis, and say that I have evidence for my alternative hypothesis. Now, if we have the other situation, if my p-value is greater th... | P-values and significance tests AP Statistics Khan Academy.mp3 |
Now, if we have the other situation, if my p-value is greater than or equal to, in this case, 0.05, so if it's greater than or equal to my significance level, then I cannot reject the null hypothesis. I wouldn't say that I accept the null hypothesis. I would just say that we do not, do not reject, reject the null hypot... | P-values and significance tests AP Statistics Khan Academy.mp3 |
And so let's say when I do all of these calculations, I get a p-value, which would put me in this scenario right over here, let's say that I get a p-value of 0.03. 0.03 is indeed less than 0.05, so I would reject the null hypothesis and say that I have evidence for the alternative. And this should hopefully make logica... | P-values and significance tests AP Statistics Khan Academy.mp3 |
It's 3 100ths, and so since that probability is less than our probability threshold here, we'll reject it and say we have evidence for the alternative. On the other hand, there might have been a scenario where we do all of the calculations here and we figure out a p-value, a p-value that we get is equal to 0.5, which y... | P-values and significance tests AP Statistics Khan Academy.mp3 |
A world where the null hypothesis is true and I get this result, well, you know, it seems reasonably likely. And so this is the basis for significant tests generally, and as you will see, it is applicable in almost every field you'll find yourself in. Now, there's one last point of clarification that I wanna make very,... | P-values and significance tests AP Statistics Khan Academy.mp3 |
What we're going to do in this video is calculate by hand the correlation coefficient for a set of bivariate data. And when I say bivariate, it's just a fancy way of saying for each x data point, there is a corresponding y data point. Now before I calculate the correlation coefficient, let's just make sure we understan... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
So we assume that these are samples of the x and the corresponding y from a broader population. And so we have the sample mean for x and the sample standard deviation for x. The sample mean for x is quite straightforward to calculate. It would just be one plus two plus two plus three over four, and this is eight over f... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
It would just be one plus two plus two plus three over four, and this is eight over four, which is indeed equal to two. The sample standard deviation for x, we've also seen this before. This should be a little bit of review. It's gonna be the square root of the distance from each of these points to the sample mean squa... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
It's gonna be the square root of the distance from each of these points to the sample mean squared. So one minus two squared plus two minus two squared plus two minus two squared plus three minus two squared. All of that over, since we're talking about sample standard deviation, we have four data points, so one less th... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
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