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And then you have the results that are less than three standard deviations below the mean. This tail right there. So that tells us that less than three standard deviations below the mean and more than three standard deviations above the mean combined have to be the rest. Well, the rest, it's only 0.3% for the rest. And... | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
Well, the rest, it's only 0.3% for the rest. And these two things are symmetrical. They're going to be equal. So this right here has to be half of this, or 0.15%. And this right here is going to be 0.15%. So the probability of having a one-year-old baby girl in the US that is more than 12.8 kilograms, if you assume a p... | ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3 |
And what we're gonna do in this video is think about how to classify them, or use the words that people typically use to classify distributions. So let's first look at this distribution right over here. That's the distribution of the lengths of houseflies. So someone went out there and measured a bunch of houseflies, a... | Classifying shapes of distributions AP Statistics Khan Academy.mp3 |
So someone went out there and measured a bunch of houseflies, and then said, hey, look, there's many houseflies that are between 6 1⁄10 of a centimeter and 6 1⁄2 1⁄10 of a centimeter. Looks like there's about 40 houseflies there. And then if you say between 6 1⁄2 and 7 1⁄10, there's about 30 houseflies. And if you were... | Classifying shapes of distributions AP Statistics Khan Academy.mp3 |
And if you were to say between 5 1⁄2 1⁄10 and 6 1⁄10, it looks like it's about the same amount. This type of distribution is usually described as being symmetric. Why is it called that? Because if you were to draw a line down the middle of this distribution, both sides look like mirror images of each other. This one lo... | Classifying shapes of distributions AP Statistics Khan Academy.mp3 |
Because if you were to draw a line down the middle of this distribution, both sides look like mirror images of each other. This one looks pretty exactly symmetric, but more typically, when you're collecting data, you'll see roughly symmetric distributions. Now here we have a distribution that gives us the dates on penn... | Classifying shapes of distributions AP Statistics Khan Academy.mp3 |
So someone went out there, observed a bunch of pennies, looked at the dates on them. They saw many pennies, looks like a little bit more than 55 pennies, had a date between 2010 and 2020, while very few pennies had a date older than 1980 on them. And this type of distribution, when you have a tail to the left, you can ... | Classifying shapes of distributions AP Statistics Khan Academy.mp3 |
Now in future videos, we'll come up with more technical definitions of what makes it left-skewed, but the way that you can recognize it is you have the high points of your distribution on the right, but then you have this long tail that skews it to the left. Now the other side of a left-skewed, you might say, well, tha... | Classifying shapes of distributions AP Statistics Khan Academy.mp3 |
It looks like it's a little over 35. None of them actually have zero. They all have at least one representative, but they would fall into this bucket, while very few have more than 50 representatives. So here, where the bulk of our distribution is to the left, but we have this tail that skews us to the right, this is k... | Classifying shapes of distributions AP Statistics Khan Academy.mp3 |
So here, where the bulk of our distribution is to the left, but we have this tail that skews us to the right, this is known as a right-skewed distribution. Now if we look at this next distribution, what would this be? Pause this video and think about it. Well, this could be a distribution of maybe someone went around t... | Classifying shapes of distributions AP Statistics Khan Academy.mp3 |
Well, this could be a distribution of maybe someone went around the office and surveyed how many cups of coffee each person drank, and if they found someone who drank one cup of coffee per day, maybe this would be them, and then they found another person who drinks one cup of coffee, that's them. Then they found three ... | Classifying shapes of distributions AP Statistics Khan Academy.mp3 |
A large amount of our data fell into this right bucket of three cups of coffee, but then we had this tail to the left. So this would be left-skewed. Now these right two distributions are interesting. One could argue that this distribution here, which is telling us the number of days that we had different high temperatu... | Classifying shapes of distributions AP Statistics Khan Academy.mp3 |
One could argue that this distribution here, which is telling us the number of days that we had different high temperatures, that this looks roughly symmetric, or actually even looks exactly symmetric, because if you did that little exercise of drawing a dotted line down the middle, it looks like the two sides are mirr... | Classifying shapes of distributions AP Statistics Khan Academy.mp3 |
And so what we could do is we could set up some buckets of time studied and some buckets of percent correct, and then we could survey the students and or look at the data from the scores on the test. And then we can place students in these buckets. So what you see right over here, this is a two-way table, and you can a... | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
So one way to read this is that 20 out of the 200 total students got between 60 and 79% on the test and studied between 21 and 40 minutes. So there's all sorts of interesting things that we could try to glean from this, but what we're going to focus on this video is two more types of distributions other than the joint ... | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
And a marginal distribution is just focusing on one of these dimensions. And one way to think about it is you can determine it by looking at the margin. So for example, if you wanted to figure out the marginal distribution of the percent correct, what you could do is look at the total of these rows. So these counts rig... | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
So these counts right over here give you the marginal distribution of the percent correct. 40 out of the 200 got between 80 and 100. 60 out of the 200 got between 60 and 79, so on and so forth. Now a marginal distribution could be represented as counts or as percentages. So if you represent it as percentages, you would... | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
Now a marginal distribution could be represented as counts or as percentages. So if you represent it as percentages, you would divide each of these counts by the total, which is 200. So 40 over 200, that would be 20%. 60 out of 200, that'd be 30%. 70 out of 200, that would be 35%. 20 out of 200 is 10%. And 10 out of 20... | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
60 out of 200, that'd be 30%. 70 out of 200, that would be 35%. 20 out of 200 is 10%. And 10 out of 200 is 5%. So this right over here in terms of percentages gives you the marginal distribution of the percent correct based on these buckets. So you could say 10% got between a 20 and a 39. Now you could also think about... | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
And 10 out of 200 is 5%. So this right over here in terms of percentages gives you the marginal distribution of the percent correct based on these buckets. So you could say 10% got between a 20 and a 39. Now you could also think about marginal distributions the other way. You could think about the marginal distribution... | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
Now you could also think about marginal distributions the other way. You could think about the marginal distribution for the time studied in the class. And so then you would look at these counts right over here. You'd say a total of 14 students studied between zero and 20 minutes. You're not thinking about the percent ... | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
You'd say a total of 14 students studied between zero and 20 minutes. You're not thinking about the percent correct anymore. A total of 30 studied between 21 and 40 minutes. And likewise, you could write these as percentages. This would be 7%. This would be 15%. This would be 43%. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
And likewise, you could write these as percentages. This would be 7%. This would be 15%. This would be 43%. And this would be 35% right over there. Now another idea that you might sometimes see when people are trying to interpret a joint distribution like this or get more information or more realizations from it is to ... | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
This would be 43%. And this would be 35% right over there. Now another idea that you might sometimes see when people are trying to interpret a joint distribution like this or get more information or more realizations from it is to think about something known as a conditional distribution. Conditional distribution. And ... | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
Conditional distribution. And this is the distribution of one variable given something true about the other variable. So for example, an example of a conditional distribution would be the distribution, distribution of percent correct, correct, given that students, students, students study between, let's say, 41 and 60 ... | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
Between 41 and 60 minutes. Well, to think about that, you would first look at your condition. Okay, let's look at the students who have studied between 41 and 60 minutes. And so that would be this column right over here. And then that column, the information in it, can give you your conditional distribution. Now an imp... | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
And so that would be this column right over here. And then that column, the information in it, can give you your conditional distribution. Now an important thing to realize is a marginal distribution can be represented as counts for the various buckets or percentages, while the standard practice for conditional distrib... | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
So the conditional distribution of the percent correct given that students study between 41 and 60 minutes, it would look something like this. Let me get a little bit more space. So if we set up the various categories, 80 to 100, 60 to 79, 40 to 59, continue it over here, 20 to 39, and zero to 19, what we'd wanna do is... | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
We are told a restaurant owner installed a new automated drink machine. The machine is designed to dispense 530 milliliters of liquid on the medium-sized setting. The owner suspects that the machine may be dispensing too much in medium drinks. They decide to take a sample of 30 medium drinks to see if the average amoun... | Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3 |
They decide to take a sample of 30 medium drinks to see if the average amount is significantly greater than 500 milliliters. What are appropriate hypotheses for their significance test? And they actually give us four choices here. I'll scroll down a little bit so that you can see all of the choices. So like always, pau... | Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3 |
I'll scroll down a little bit so that you can see all of the choices. So like always, pause this video and see if you can have a go at it. Okay, now let's do this together. So let's just remind ourselves what a null hypothesis is and what an alternative hypothesis is. One way to view a null hypothesis, this is the hypo... | Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3 |
So let's just remind ourselves what a null hypothesis is and what an alternative hypothesis is. One way to view a null hypothesis, this is the hypothesis where things are happening as expected. Sometimes people will describe this as the no difference hypothesis. It'll often have a statement of equality where the popula... | Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3 |
It'll often have a statement of equality where the population parameter is equal to the value where the value is what people were kind of assuming all along. The alternative hypothesis, this is a claim where if you have evidence to back up that claim, that would be new news. You are saying, hey, there's something inter... | Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3 |
There is a difference. And so in this context, the no difference, we would say the null hypothesis would be we would care about the population parameter, and here we care about the average amount of drink dispensed in the medium setting. So the population parameter there would be the mean, and that the mean would be eq... | Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3 |
And then the alternative hypothesis, this is what the owner fears, is that the mean actually might be larger than that, larger than 530 milliliters. And so let's see, which of these choices is this? Well, these first two choices are talking about proportion, but it's really the average amount that we're talking about. ... | Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3 |
We see it up here. They decided to take a sample of 30 medium drinks to see if the average amount, they're not talking about proportions here, they're talking about averages, and in this case, we're talking about estimating the population parameter, the population mean for how much drink is dispensed on that setting. A... | Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3 |
Only these two are even dealing with the mean. And the difference between this one and this one is this says the mean is greater than 530 milliliters, and that indeed is the owner's fear. And this over here, this alternative hypothesis is that it's dispensing, on average, less than 530 milliliters, but that's not what ... | Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3 |
So I would definitely pick choice C. Let's do another example. The National Sleep Foundation recommends that teenagers aged 14 to 17 years old get at least eight hours of sleep per night for proper health and wellness. A statistics class at a large high school suspects that students at their school are getting less tha... | Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3 |
To test their theory, they randomly sample 42 of these students and ask them how many hours of sleep they get per night. The mean from this sample, the mean from the sample, is 7.5 hours. Here's their alternative hypothesis. The average amount of sleep students at their school get per night is, what is an appropriate e... | Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3 |
The average amount of sleep students at their school get per night is, what is an appropriate ending to their alternative hypothesis? So pause this video and see if you can think about that. So let's just first think about a good null hypothesis. So the null hypothesis is, hey, there's actually no news here that everyt... | Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3 |
So the null hypothesis is, hey, there's actually no news here that everything is what people were always assuming. And so the null hypothesis here is that, no, the students are getting at least eight hours of sleep per night. And so that would be that, and remember, we care about the population of students, and we care... | Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3 |
And a good clue for the alternative hypothesis is when you see something like this, where they say a statistics class at a large high school suspects, so they suspect that things might be different than what people have always been assuming or actually what's good for students. And so they suspect that students at thei... | Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3 |
And so if you wanted to write this out in words, the average amount of sleep students at their school get per night is less than eight hours. Now, one thing to watch out for is, one, you wanna make sure you're getting the right parameter. Sometimes it's often a population mean, sometimes it's a population proportion. B... | Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3 |
So they all agree to put in their salaries into a computer, and so these are their salaries, they're measured in thousands, so one makes 35,000, 50,000, 50,000, 50,000, 56,000, two make 60,000, one make 75,000, and one makes 250,000, so she's doing very well for herself. And the computer spits out a bunch of parameters... | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
The mean is roughly 76.2, the computer would calculate it by adding up all of these numbers, these nine numbers, and then dividing by nine. And the median is 56. And median is quite easy to calculate, you just order the numbers and you take the middle number here, which is 56. Now what I want you to do is pause this vi... | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
Now what I want you to do is pause this video and think about for this data set, for this population of salaries, which measure of central tendency is a better measure? All right, so let's think about this a little bit. I'm gonna plot it on a line here, I'm gonna plot my data so we get a better sense, so we just don't ... | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
So let's say this is zero, let's say this is, let's see, one, two, three, four, five, so this would be 250, this is 50, 100, 150, 200, 200, and let's see, let's say if this is 50, then this would be roughly 40 right here, and I just wanna get rough, so this would be about 60, 70, 80, 90, close enough. I could draw this... | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
Okay, that's pretty good. So let's plot this data. So one student makes 35,000, so that is right over there. Two make 50, or three make 50,000, so one, two, and three, I'll put it like that. One makes 56,000, which would put them right over here. One makes 60,000, or actually two make 60,000, so it's like that. One mak... | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
Two make 50, or three make 50,000, so one, two, and three, I'll put it like that. One makes 56,000, which would put them right over here. One makes 60,000, or actually two make 60,000, so it's like that. One makes 75,000, so that's 60, 70, 75,000, this one's gonna be right around there, and then one makes 250,000, so o... | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
One makes 75,000, so that's 60, 70, 75,000, this one's gonna be right around there, and then one makes 250,000, so one's salary is all the way around there. And then when we calculate the mean, as 76.2 is our measure of central tendency, 76.2 is right over there. So is this a good measure of central tendency? Well, to ... | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
Well, to me, it doesn't feel that good, because our measure of central tendency is higher than all of the data points except for one. And the reason is is that you have this one, that our data is skewed significantly by this data point at $250,000. It is so far from the rest of the distribution, from the rest of the da... | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
And this is something that you see in general. If you have data that is skewed, and especially things like salary data, where someone might make, most people are making 50, 60, $70,000, but someone might make $2 million, and so that will skew the average, or skew the mean, I should say, when you add them all up and div... | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
The median at 56 sits right over here, which seems to be much more indicative for central tendency. And think about it. Even if you made this, instead of 250,000, if you made this 250,000,000, which would be $250 million, which is a ginormous amount of money to make, it would skew the mean incredibly, but it actually w... | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
This could be a trillion dollars. This could be a quadrillion dollars. The median is going to stay the same. So the median is much more robust if you have a skewed data set. Mean makes a little bit more sense if you have a symmetric data set, or if you have things that are, where things are roughly above and below the ... | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
So the median is much more robust if you have a skewed data set. Mean makes a little bit more sense if you have a symmetric data set, or if you have things that are, where things are roughly above and below the mean, or things aren't skewed incredibly in one direction, especially by a handful of data points like we hav... | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
And so what about spread? Well, you might immediately say, well, Sal, you already told us that the mean is not so good, and the standard deviation is based on the mean. You take each of these data points, find their distance from the mean, square that number, add up those squared distances, divide by the number of data... | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
And so since this is based on the mean, which isn't a good measure of central tendency in this situation, and this is also going to skew that standard deviation, this is going to be, this is a lot larger than if you look at the actual, you want an indication of the spread. Yes, you have this one data point that's way f... | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
How do we calculate the interquartile range? Well, you take the median, and then you take the bottom group of numbers and calculate the median of those. So that's 50 right over here. And then you take the top group of numbers, the upper group of numbers, and the median there, 60 and 75, is 67.5. If this looks unfamilia... | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
And then you take the top group of numbers, the upper group of numbers, and the median there, 60 and 75, is 67.5. If this looks unfamiliar, we have many videos on interquartile range and calculating standard deviation and median and mean. This is just a little bit of a review. And then the difference between these two ... | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
And then the difference between these two is 17.5. And notice, this distance between these two, this 17.5, this isn't going to change even if this is $250 billion. So once again, it is both of these measures are more robust when you have a skewed data set. So the big takeaway here is mean and standard deviation, they'r... | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
So the big takeaway here is mean and standard deviation, they're not bad. If you have a roughly symmetric data set, if you don't have any significant outliers, things that really skew the data set, mean and standard deviation can be quite solid. But if you're looking at something that could get really skewed by a handf... | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
Let's say it's a bunch of balls, each of them have a number written on it. For that population, we could calculate parameters. So a parameter you could view as a truth about that population. We've covered this in other videos. So for example, you could have the population mean, the mean of the numbers written on top of... | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
We've covered this in other videos. So for example, you could have the population mean, the mean of the numbers written on top of that ball. You could have the population standard deviation. You could have the proportion of balls that are even, whatever, these are all population parameters. Now, we know from many other... | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
You could have the proportion of balls that are even, whatever, these are all population parameters. Now, we know from many other videos that you might not know the population parameter or it might not even be easy to find. And so the way that we try to estimate a population parameter is by taking a sample. So this rig... | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
So this right over here is a sample of size n, sample of size n. And then we can calculate a statistic from that sample, based on that sample, maybe we picked n balls from there. And so from that, we can calculate a statistic that is used to estimate this parameter. But we know that this is a random sample right over h... | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
So every time we take a sample, the statistic that we calculate for that sample is not necessarily going to be the same as the population parameter. In fact, if we were to take a random sample of size n again and then we were to calculate the statistic again, we could very well get a different value. So these are all g... | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
And so an interesting question is, is what is the distribution of the values that I could get for these statistics? What is the frequency with which I could get different values for the statistic that is trying to estimate this parameter? And that distribution is what a sampling distribution is. So let's make this even... | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
So let's make this even a little bit more concrete. Let's imagine where our population, I'm gonna make this a very simple example. Let's say our population has three balls in it, one, two, three, and they're numbered one, two, and three. And it's very easy to calculate, let's say the parameter that we care about right ... | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
And it's very easy to calculate, let's say the parameter that we care about right over here is the population mean. And that of course is going to be one plus two plus three, all of that over three, which is six divided by three, which is two. So that is our population parameter. But let's say that we wanted to take sa... | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
But let's say that we wanted to take samples, let's say samples of two balls at a time, and every time we take a ball, we'll replace it. So each ball we take, it is an independent pick. And we're gonna use those samples of two balls at a time in order to estimate the population mean. So for example, this could be our f... | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
So for example, this could be our first sample of size two. And let's say in that first sample, I pick a one and let's say I pick a two. Well, then I can calculate the sample statistic here. In this case, it would be the sample mean, which is used to estimate the population mean. And in this, for this sample of two, it... | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
In this case, it would be the sample mean, which is used to estimate the population mean. And in this, for this sample of two, it's going to be 1.5. Then I can do it again. And let's say I get a one and I get a three. Well, now when I calculate the sample mean, the average of one and three or the mean of one and three ... | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
And let's say I get a one and I get a three. Well, now when I calculate the sample mean, the average of one and three or the mean of one and three is going to be equal to two. Let's think about all of the different scenarios of samples we can get and what the associated sample means are going to be. And then we can get... | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
And then we can get see the frequency of getting those sample means. And so let me draw a little bit of a table here. So make a table right over here. And let's see, these are the numbers that we pick. And remember, when we pick one ball, we'll record that number, then we'll put it back in, and then we'll pick another ... | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
And let's see, these are the numbers that we pick. And remember, when we pick one ball, we'll record that number, then we'll put it back in, and then we'll pick another ball. So these are going to be independent events and it's going to be with replacement. And so let's say we could pick a one and then a one. We could ... | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
And so let's say we could pick a one and then a one. We could pick a one, then a two, a one, and a three. We could pick a two and then a one. We could pick a two and a two, a two and a three. We could pick a three and a one, a three and a two, or a three and a three. There's three possible balls for the first pick and ... | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
We could pick a two and a two, a two and a three. We could pick a three and a one, a three and a two, or a three and a three. There's three possible balls for the first pick and three possible balls for the second because we're doing it with replacement. And so what is the sample mean in each of these for all of these ... | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
And so what is the sample mean in each of these for all of these combinations? So for this one, the sample mean is one. Here it is 1.5. Here it is two. Here it is 1.5. Here it is two. Here it is 2.5. | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
Here it is two. Here it is 1.5. Here it is two. Here it is 2.5. Here it is two. Here it is 2.5. And then here it is three. | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
Here it is 2.5. Here it is two. Here it is 2.5. And then here it is three. And so we can now plot the frequencies of these possible sample means that we can get. And that plot will be a sampling distribution of the sample means. So let's do that. | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
And then here it is three. And so we can now plot the frequencies of these possible sample means that we can get. And that plot will be a sampling distribution of the sample means. So let's do that. So let me make a little chart right over here, a little graph right over here. So these are the possible, possible sample... | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
So let's do that. So let me make a little chart right over here, a little graph right over here. So these are the possible, possible sample means. We can get a one, we could get a 1.5. We could get a two, we could get a 2.5, or we can get a three. And now let's see the frequency of it. And I will put that over here. | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
We can get a one, we could get a 1.5. We could get a two, we could get a 2.5, or we can get a three. And now let's see the frequency of it. And I will put that over here. And so let's see, how many ones out of our nine possibilities we have, how many were one? Well, only one of the sample means was one. And so the rela... | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
And I will put that over here. And so let's see, how many ones out of our nine possibilities we have, how many were one? Well, only one of the sample means was one. And so the relative frequency, if we just said the number, we could make this line go up one, or we could just say, hey, this is going to be one out of the... | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
And so the relative frequency, if we just said the number, we could make this line go up one, or we could just say, hey, this is going to be one out of the nine possibilities. And so let me just make that, I'll call this right over here. This is 1 9th. Now what about 1.5? Well, let's see, there's one, two of these poss... | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
Now what about 1.5? Well, let's see, there's one, two of these possibilities out of nine. So 1.5, it would look like this. This right over here is two over nine. And now what about two? Well, we can see there's one, two, three. So three out of the nine possibilities, we got a two. | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
This right over here is two over nine. And now what about two? Well, we can see there's one, two, three. So three out of the nine possibilities, we got a two. So we could say this is two, or we could say this is 3 9th, which is the same thing, of course, as 1 3rd. So this right over here is three over nine. And then wh... | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
So three out of the nine possibilities, we got a two. So we could say this is two, or we could say this is 3 9th, which is the same thing, of course, as 1 3rd. So this right over here is three over nine. And then what about 2.5? Well, there's two 2.5s, so two out of the nine times. Another way you could interpret this ... | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
A high school newspaper doesn't know this figure, but they are curious what it is. So they decide to ask a simple random sample of 160 students if they have experienced extreme levels of stress during the past month. Subsequently, they find that 10% of the sample replied yes to the question. Assuming the true proportio... | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
Assuming the true proportion is 15%, which they tell us up here, they say 15% of the population of the 1,750 students actually have experienced extreme levels of stress during the past month. So that is the true proportion. So let me just write that. The true proportion for our population is 0.15. What is the approxima... | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
The true proportion for our population is 0.15. What is the approximate probability that more than 10% of the sample would report that they experienced extreme levels of stress during the past month? So pause this video and see if you can answer it on your own, and there are four choices here. I'll scroll down a little... | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
I'll scroll down a little bit and see if you can answer this on your own. So the way that we're going to tackle this is we're gonna think about the sampling distribution of our sample proportions. And first, we're gonna say, well, is this sampling distribution approximately normal? Is it approximately normal? And if it... | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
Is it approximately normal? And if it is, then we can use its mean and its standard deviation and create a normal distribution that has that same mean and standard deviation in order to approximate the probability that they're asking for. So first, this first part, how do we decide this? Well, the rule of thumb we have... | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
Well, the rule of thumb we have here, and it is a rule of thumb, is that if we take our sample size times our population proportion, and that is greater than or equal to 10, and our sample size times one minus our population proportion is greater than or equal to 10, then if both of these are true, then our sampling di... | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
So 160, the true population proportion is 0.15, and that needs to be greater than or equal to 10. And so let's see, this is going to be 16 plus eight, which is 24, and 24 is indeed greater than or equal to 10, so that checks out. And then if I take our sample size times one minus P, well, one minus 15 hundredths is goi... | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
And this is definitely going to be greater than or equal to 10. Let's see, this is going to be 24 less than 160, so this is going to be 136, which is way larger than 10, so that checks out. And so the sampling distribution of our sample proportions is approximately going to be normal. And so what is the mean and standa... | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
And so what is the mean and standard deviation of our sampling distribution? So the mean of our sampling distribution is just going to be our population proportion. We've seen that in other videos, which is equal to 0.15. And our standard deviation of our sampling distribution, of our sample proportions, is going to be... | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
And our standard deviation of our sampling distribution, of our sample proportions, is going to be equal to the square root of P times one minus P over N, which is equal to the square root of 0.15 times 0.85, all of that over our sample size, 160. So now let's get our calculator out. So I'm gonna take the square root o... | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
And so what is this going to give me? So it's going to give me approximately 0.028, and I'll go to the thousands place here. So this is approximately 0.028. This is going to be approximately a normal distribution. So you could draw your classic bell curve for a normal distribution, so something like this. And our norma... | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
This is going to be approximately a normal distribution. So you could draw your classic bell curve for a normal distribution, so something like this. And our normal distribution is going to have a mean. It's going to have a mean right over here of, so this is the mean of our sampling distribution. So this is going to b... | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
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