Suppose that I know that a large population consists of equal numbers of men and women. I sample 1000 men and ask whether they have ever eaten an avocado. Some number M say yes. Then I sample 1000 women and ask them the same question. Some number W say yes.
Now, I know how to construct a confidence interval for the proportion of men who have eaten an avocado, or the proportion of women who have done the same:
But given this data, how can I construct a confidence interval for the proportion of people in the total population who have eaten one?
In theory I could pretend that I had asked a random sample of 2000 people in the general population, and that (M + W) had said yes, and I could use the resulting Wilson score interval (for example). But my sample of 1000 men and 1000 men should be a better approximation of the population than a random sample of 2000 would be, so presumably the actual confidence interval should be a bit tighter in this case.