Statistics - Subpopulation Random Sampling

Hi guys,

I have a statistics question that I've been wrapping my brain around but I can't seem to come up with a good answer.

Let's say we have a population N. In N, there are 2 sub populations A and B. We know the percentages of the sub populations. So for example, A is 40% of N and B is 60% of N.

Now we're given a statistic of N. So for example, let's say we're asking the entire population N if they are male or female.

We get the following results:

Male - 70% Female - 30%

However, we don't know which of these people comes from which of these sub populations. All we know, is that globally 70% are male.

Now, it would seem to me that we would also expect that 70% of A and 70% of B are male. However, the question I'm struggling with is the variance. How do we calculate the variance of the percentage of subpopulation A responses that are male?

Thanks

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This is not correct. We do not know that 70% of population A is male and 70% of population B is male. It may be that the populations are correlated. For example, if A = population has beards, B = population without beards. Then 100% of population A is male.

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Clearly we need some assumption of independence: assume membership in population A and maleness are independent. Now given that there are $m$ males in the population and $n$ members of population A, the number of males in population A has a hypergeometric distribution with parameters $N$, $m$ and $n$. This has expected value $\frac{nm}{N}$ and variance $\frac{nm(N-n)(N-m)}{N^2(N-1)}$. –  Robert Israel Jun 21 '11 at 18:13
Now, for a more sophisticated analysis, you can start putting confidence limits on this using the extreme probabilities we've calculated above. So the confidence interval for the proportion of males in A would (probably -- I am not totally sure) be $(0.25 - 1.96\times\sqrt{0.25\cdot0.75/40}, 1)$.