Confidence intervals for proportions
Example constructing and interpreting a confidence interval for p
- [Instructor] We're told that 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, n is equal to 50, and from that she calculates a sample proportion, which we could denote with p hat. And she finds that 20 out of the 50 are sung by a female, 20 out of the 50 which is the same thing as 0.4. And then she wants to construct a 99% confidence interval. So before we even go about constructing the confidence interval, you wanna check to make sure that we're making some valid assumptions or 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. So 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. And there you wanna be confident or you wanna 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 minus 20, we have 30 failures. So both of those are more than 10, and so meets that condition. And then the last condition is, sometimes called the independence test or the independence rule or the 10% rule. If you were doing this sample with replacement, so 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. And so it looks like it is exactly 10% of the population, so Della just squeezes through on our independence test right over there. So that out of the way let's just think about what the confidence interval's 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 wanna have, and then that critical value times the standard deviation of the sampling distribution of the sample proportions which we don't know. And so instead of having that, we use the standard error of the sample proportion. And in this case it would be p hat times one minus p hat all of that over n our sample size, all of that over 50. So what's this going to be? We're gonna get p hat, our sample proportion here, is 0.4 plus or minus, I'll save the z star here, our critical value for a little bit. We're gonna use a z-table for that. And so we're gonna have 0.4 right over there, one minus 0.4 is times 0.6 all of that over 50. So 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 a correct critical value. Do we wanna go 1.96 standard errors above and below our sample proportion? Or do we wanna go 2.576 standard errors above and below our sample proportion? And the key is the 99% confidence level. Now if we have a 99% confidence level, one way to think about it is, so let me just do my best shot at drawing a normal distribution here. And so if you want a 99% confidence level, that means you wanna contain the 99%, the middle 99%, under the curve right over here, that area. And so 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 wanna look up on the table. And 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, 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 gonna be right in this area right over here. And so that is 2.5, looks like 2.57, or 2.5, 2.58 around that. And so 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.
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