> In your case, the sum of heights will still converge to the normal distribution of you take the height of n random people (man or woman)
Yes and no. The sample must be independent and identically distributed. In your case the "identical" part is not correct, as men and women have different distributions (both are normal but with different mean and std). However, if both distributions are normal, then their sum is normal (even with different mean and std).
The fact that the sum is normal in this case has nothing to do with the CLT - it's just a quirk of the normal distribution that the sum is normal. Had men/women had non-normal distributions with different means/stds, then the sum would not be normal.
If you sample from the global distribution (all men and women of earth) then samples are identically distributed. It's just a new distribution that is not gaussian, but the sum of samples will converge to a gaussian.
Yes and no. The sample must be independent and identically distributed. In your case the "identical" part is not correct, as men and women have different distributions (both are normal but with different mean and std). However, if both distributions are normal, then their sum is normal (even with different mean and std).
The fact that the sum is normal in this case has nothing to do with the CLT - it's just a quirk of the normal distribution that the sum is normal. Had men/women had non-normal distributions with different means/stds, then the sum would not be normal.