Estimating a P-value from a simulation | AP Statistics | Khan Academy

So we have a question here on p-values. It says Evie read an article that said six percent of teenagers were vegetarians, but she thinks it's higher for students at her school. To test her theory, Evie took a random sample of 25 students at her school, and 20 percent of them were vegetarians.

So just from this first paragraph, some interesting things are being said. It's saying that the true population proportion, if we believe this article of teenagers that are vegetarian, we could say that is six percent. Now for her school, there is a null hypothesis that the proportion of students at her school that are vegetarian. So this is at her school that the true proportion—the null would be it's just the same as the proportion of teenagers as a whole. So that would be the null hypothesis.

You can see that she's generating an alternative hypothesis, but she thinks it's higher for students at her large school. So her alternative hypothesis would be the proportion—the true population parameter for her school is greater than six percent.

To see whether or not you could reject the null hypothesis, you take a sample, and that's exactly what Evie did. She took a random sample of 25 students, and you calculate the sample proportion. Then you figure out what is the probability of getting a sample proportion this high or greater. If it's lower than a threshold, then you will reject your null hypothesis. That probability we call the p-value.

The p-value is equal to the probability that your sample proportion—and she's doing this for students at her school—is going to be greater than or equal to 20 if you assumed that your null hypothesis was true. So if you assumed that the true proportion at your school was six percent vegetarians, but you took a sample of 25 students where you got 20, what is the probability of getting 20 or greater for a sample of 25?

Now, there are many ways to approach it, but it looks like she is using a simulation to see how likely a sample like this was to happen by random chance alone. Evie performed a simulation; she simulated 40 samples of n equals 25 students from a large population where six percent of the students were vegetarian. She recorded the proportion of vegetarians in each sample.

Here are the sample proportions from her 40 samples. So what she's doing here with the simulation, this is an approximation of the sampling distribution of the sample proportions if you were to assume that your null hypothesis is true. It says below that Evie wants to test her null hypothesis, which is that the true proportion at her school is six percent versus the alternative hypothesis that the true proportion at her school is greater than six percent, where p is the true proportion of students who are vegetarian at her school.

Then they ask us, based on these simulated results, what is the approximate p-value of the test? They say the sample result—the sample proportion here was 20. We saw that right over here. If we assume that this is a reasonably good approximation of our sampling distribution of our sample proportions, there's 40 data points here.

How many of these samples do we get a sample proportion that is greater than or equal to twenty percent? Well, you can see this is twenty percent right over here, twenty hundredths, and so you see we have three right over here that meet this constraint. That is 3 out of 40.

So if we think this is a reasonably good approximation, we would say that our p-value is going to be approximately 3 out of 40. That, if the true population proportion for the school were six percent, if the null hypothesis were true, then approximately three out of every 40 times you would expect to get a sample with 20 or larger being vegetarians.

So three fortieths is what? Let's see if I multiply both the numerator and the denominator by two and a half. This is approximately equal to—I say two and a half because to go from forty to a hundred, and then two and a half times three would be 7.5. So I would say this is approximately 7.5 percent.

This is actually a multiple-choice question, and if we scroll down, we do indeed see approximately 7.5 percent right over there.