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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?


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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

This is not even so much about variance as it is about point estimates. Charles Manski, an econometrician, has developed the concept of partial identification (google it). It basically starts off saying that we want to assume as little as possible about the breakdown of the population. So suppose we know that in the sample of 100, there are 70 males and 30 females, and there are 40 respondents from subpopulation A and 60 respondents from subpopulation B. The two extreme allocations are:

  1. As many subpopulation A respondents as possible are males: there are 40 males in A, there are 30 males in B, and there are 30 females in B. The proportion of males in A is 100%, and the proportion of males in B is 50%.

  2. As few subpopulation A respondents as possible are males: there are 10 males in A, there are 30 females in A, and there are 60 males in B. The proportion of males in A is 25%, the proportion of males in B is 100%.

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 $(0.25 - 1.96\times\sqrt{0.25\cdot0.75/40}, 1)$.

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