Probability of sample proportions example | Sampling distributions | AP Statistics | Khan Academy

We're told suppose that 15% of the 1750 students at a school have experienced extreme levels of stress during the past month. A high school newspaper doesn't know this figure, but they are curious what it is. So they decide to ask us 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 proportion is 15%, which they tell us up here, they say 15% of the population of the seventeen hundred and fifty 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 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. There are four choices here. I'll scroll down a little bit and see if you can answer this on your own.

The way that we're going to tackle this is we're going to think about the sampling distribution of our sample proportions. First, we're going to say, "Well, is this sampling distribution approximately normal?" Is it approximately normal? If it is, then we can use its mean and 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.

For the first part, how do we decide this? 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 1 minus our population proportion is greater than or equal to 10. Then if both of these are true, then our sampling distribution of our sample proportions is going to be approximately normal.

In this case, the newspaper is asking 160 students, that's the sample size. So 160. The true population proportion is 0.15, and that needs to be greater than or equal to 10. Let's see, this is going to be 16 plus 8, which is 24, and 24 is indeed greater than or equal to 10, so that checks out.

Then if I take our sample size times 1 minus P, well, 1 minus 0.15 is going to be 0.85, 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.

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. Our standard deviation of our sampling distribution of our sample proportions is going to be equal to the square root of P times 1 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.

Now let's get our calculator out. So I'm going to take the square root of 0.15 times 0.85 divided by 160. Let me close those parentheses. So what is this going to give me? 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.

You could draw your classic bell curve for a normal distribution. Something like this and our normal distribution is going to have a mean. It's going to have a mean right over here. So this is the mean of our sampling distribution, so this is going to be equal to the same thing as our population proportion, 0.15.

We also know that our standard deviation here is going to be approximately equal to 0.028. What we want to know is 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 we could say that 10% would be right over here. I'll say 0.10, and so the probability that in a sample of 160 you get a proportion for that sample, a sample proportion that is larger than 10%, would be this area right over here. So this right over here would be the probability that your sample proportion is greater than they say is more than 10%, is more than 0.1.

I could write 1.0 just like that. Then to calculate it, I can get out our calculator again. So here I'm going to go to my distribution menu right over there, and then I'm going to do a normal cumulative distribution function. So let me click enter there.

What is my lower bound? Well, my lower bound is 10%, 0.1. What is my upper bound? Well, we'll just make this 1 because that is the highest proportion you could have for a sampling distribution of sample proportions. Now what is our mean? Well, we already know that's 0.15. What is the standard deviation of our sampling distribution? Well, it's approximately 0.028.

Then I can click enter. If you're taking an AP exam, you actually should write this. You should tell the graders what you're actually typing in your normal CDF function. But if we click enter right over here, and then enter there, we have it. It's approximately 96%. So this is approximately 0.96.

And out of our choices, it would be this one right over here. If you're taking this on the AP exam, you would say that you called normal CDF, where you have your lower bound, and you would put in your 0.10. You would say that you used an upper bound of 1. You would say that you gave a mean of 0.15, and then you gave a standard deviation of 0.028, just so people know that you knew what you were doing. But hopefully, this is helpful.