The example of misuse of the CLT illustrates that a mixture of normals is not a normal distribution. For example the height of people is a mixture of two normals, one for men and the other for women, this mixture has two modes and so it is not bell shaped and is not a normal distribution.
2) The central limit theorem applies to the distribution of the sample average. It applies whenever the samples are iid and the second moment is finite. The fact that the samples are coming from a mixture of normals doesn't change that.
> Human height isn't actually bimodal even if some histograms display two modes.
"Bimodal" doesn't really have a precise definition or test, if you don't assume normal distrubtions.
That paper argues that only if means are separated by 2σ should the distribution be considered bimodal.
But there are many measures of bimodality. [1] [2] [3] [4] [5] [6]
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In any case, I would be very much surprised if the population couldn't be selected enough (age, race, country, diet, family) to have human height be bimodal by any measure.
CLT is about the sum.
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)
> 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.
