In this situation, there are different formula for the mean and standard error, but the same logic and procedure for finding confidence intervals remains.
Demonstration of these ideas is shown in an example.
and SE() =
The shape will be approximately normal for sufficiently large samples. This is reasonable to assume if each sample individually meets the rule of thumb for the approximation.
The sampling distribution for the difference in sample proportions will be approximately normal since each sample is sufficiently large. (There are at least 5 children of each type in each sample.)
We may estimate the standard error by plugging in our sample proportions instead of the unknown population proportions, finding = .0660.
Using our general formula for confidence intervals,
(.38 - .40) +/- (1.96)(.0660)or
-.02 +/- .13There is very little evidence that the true two population proportions differ by a substantial amount.
Bret Larget, email@example.com