Calculating a confidence interval for the difference of proportions | AP Statistics | Khan Academy

Duncan is investigating if residents of a city support the construction of a new high school. He's curious about the difference of opinion between residents in the north and south parts of the city. He obtained separate random samples of voters from each region. Here are the results:

In the north, 54 out of 120 said they want the school; 66 said they didn't. In the south, 77 said they wanted the school; 63 said they didn't.

Duncan wants to use these results to construct a 90% confidence interval to estimate the difference in the proportion of residents in these regions who support the construction project, ( p_s - p_n ). So these are the true parameters for the difference between these two populations. Assume that all of the conditions for inference have been met.

All right, which of the following is a correct 90% confidence interval based on Duncan's sample? So pause this video and see if you can figure that out. You will need a calculator, and depending on your calculator, you might need a z-table as well.

In a previous video, we introduced the idea of a two-sample z-interval. We talked about the conditions for inference. Lucky for us, they say the conditions for inference have been met, so we can go straight to calculating the confidence interval.

That confidence interval is going to be the difference between the sample proportions, so ( \hat{p}_s ) (the sample proportion in the south) minus the sample proportion in the north. It's going to be that difference plus or minus our critical value ( z^* ) times our estimate of the standard deviation of the sampling distribution of the difference between the sample proportions.

That is going to be our estimate. It is going to be ( \hat{p}_s (1 - \hat{p}_s) ) all of that over the sample size in the south, plus ( \hat{p}_n (1 - \hat{p}_n) ) all of that over the sample size in the north.

Okay, so our sample proportion in the south—I'll later use a calculator to get a decimal value, but this is going to be in the south—we have 77 out of 140 supported. So this is going to be 77 out of 140. In the north, this is going to be 54 out of 120 (54 out of 120).

What is my critical z-value? Well, here I'm going to have to either use a calculator or a z-table. Remember, we have a 90% confidence interval, and so let me see. I'll draw it right over here. If this is a normal distribution and you want to have a 90% confidence interval, that means you're containing 90% of the distribution, which means each of these tails will combine—they would have 10%—but each of them would have five percent, five percent of the distribution.

So I'm going to look at a z-table that figures out how many standard deviations below the mean do I need to be in order to get five percent. Right over here, that's going to tell me, well, if I'm that far below or above, that's going to be my critical z-value. So let me get that z-table out.

So I care about five percent, and I'm using this in a bit of a reverse direction. But let's see, five percent, so this is a little over five percent, getting closer to five percent, even closer to five percent. Now we've gotten right below five percent, so we're going to be in between this and this.

I could just split the difference, and I could just say 1.6. Let's just say 1.645 to go right in between. So this is going to be approximately equal to 1.645. Then let's see, we know what ( \hat{p}_s ) is, we know what ( \hat{p}_n ) is. In the south, our sample size is 140, and in the north, our sample size is 120.

And so now I just have to type all of this into the calculator, which is going to get a little hairy, but we will do it together. For the sake of time, we'll accelerate this typing into the calculator, but I'm going to start with calculating the upper bound. Then we'll calculate the lower bound, and then I think I've closed all my parentheses.

I think we’re ready to get the upper bound, which is going to be equal to 0.218 or approximately 0.202. So we can immediately look at our choices and see where that is the upper bound. This one is looking pretty good—0.202.

But let’s get the lower bound now. So I got my calculator back. Instead of retyping everything, I'm just going to put a minus here. So I go to second… and just so you see what I'm doing, second entry—I see the entry back, and then I can just change the part right before the radical.

So we are going to… all right, so this just needs to be a minus. Click enter, and there you have it! Our lower bound is negative 0.002, and that is indeed this choice right over here. So there we go; we have picked our choice.