Example constructing and interpreting a confidence interval for p | AP Statistics | Khan Academy

We're told Della has over 500 songs on her mobile phone, and she wants to estimate what proportion of the songs are by a female artist. She takes a simple random sample—that's what SRS stands for—of 50 songs on her phone and finds that 20 of the songs sampled are by a female artist. Based on this sample, which of the following is a 99% confidence interval for the proportion of songs on her phone that are by a female artist?

So, like always, pause this video and see if you can figure it out on your own. Della has a library of 500 songs right over here, and she's trying to figure out the proportion that are sung by a female artist. She doesn't have the time to go through all 500 songs to figure out the true population proportion ( p ), so instead, she takes a sample of 50 songs, and ( n ) is equal to 50.

From that, she calculates a sample proportion, which you could denote with ( \hat{p} ), and she finds that 20 out of the 50 are sung by a female—20 out of 50—which is the same thing as 0.4. Then, she wants to construct a 99% confidence interval.

Before we even go about constructing the confidence interval, we want to check to make sure that we're making some valid assumptions; we're using a valid technique. So, before we actually calculate the confidence interval, let's just make sure that our sampling distribution is not distorted in some way, and so that we can, with confidence, make a confidence interval.

The first condition is to make sure that your sample is truly random, and they tell us that it's a simple random sample, so we'll take their word for it. The next condition is to assume that your sampling distribution of the sample proportions is approximately normal. There, you want to be confident, or you want to see that in your sample you have at least 10 successes and at least 10 failures. Well, here we have 20 successes, which means, well, ( 50 - 20 ), we have 30 failures. So both of those are more than 10 and thus meet that condition.

The last condition is sometimes called the independence test or the independence rule, or the 10 percent rule if you're doing this sample with replacement. If she were to look at one song, test whether it's a female or not, and then put it back in her pile and then look at another song, then each of those observations would truly be independent. But we don't know that. In fact, we'll assume that she didn't do it with replacement, and so if you don't do it with replacement, you can assume rough independence for each observation of a song if this is no more than 10% of the population.

It looks like it is exactly 10% of the population, so Della just squeezes through on our independence test right over there. So, with that out of the way, let's just think about what the confidence interval is going to be. Well, it's going to be her sample proportion plus or minus—there's going to be some critical value, and this critical value is going to be dictated by our confidence level we want to have, and then that critical value times the standard deviation of the sampling distribution of the sample proportions, which we don't know.

Instead of having that, we use the standard error of the sample proportion, and in this case, it would be ( \hat{p} \times (1 - \hat{p}) ) all of that over ( n ), our sample size—all of that over 50. So what’s this going to be? We're going to get ( \hat{p} ), our sample proportion here is 0.4 plus or minus—I'll save the ( z^* ) here, our critical value, for a little bit. We're going to use a z table for that.

So we're going to have 0.4 right over there, and ( 1 - 0.4 ) is times 0.6, all of that over 50. We can already look at some choices that look interesting here—this choice and this choice both look interesting, and the main thing we have to reason through is which one has the correct critical value. Do we want to go 1.96 standard errors above and below our sample proportion, or do we want to go 2.576 standard errors above and below our sample proportion?

The key is the 99% confidence level. Now, if we have a 99% confidence level, one way to think about it is—let me just do my best shot at drawing a normal distribution here. If you want a 99% confidence level, that means you want to contain the 99—the middle 99 under the curve right over here in that area. If this is 99, then this right over here is going to be 0.5%, and this right over here is 0.5%. We want the z value that's going to leave 0.5% above it, and so that's actually going to be 99.5% is what we want to look up on the table.

That's because many z tables, including the one that you might see on something like an AP stats exam, they will have the area up to and including a certain value, and so they're not going to leave this free right over here. So let's just look up 99.5 on our z table. All right, so let me move this down so you can see it. All right, that's our z table. Let's see, we're at 99. Okay, it's going to be right in this area right over here, and so that is 2.5—looks like 2.57 or 2.58, around that.

This right over here is about 2.57; it's between 2.57 and 2.58, which gives us enough information to answer this question. It’s definitely not going to be this one right over here—we have 2.576, which is indeed between 2.57 and 2.58.

So, let's remind ourselves we've been able to construct our confidence interval right over here. But what does that actually mean? That means that if we were to repeatedly take samples of size 50 and repeatedly use this technique to construct confidence intervals, that roughly 99% of those intervals constructed this way are going to contain our true population parameter.