As I understand it, one result of the central limit theorem is that the sampling distribution of means drawn from any population will be approximately normal. Assume the population consist of Bernoulli trials with a given probability p and we want to estimate p. Then our population consist of zeros and ones and our sample means will be in the interval [0,1]. So I'm a bit confused how such a distribution of sample means can be approximated by a normal distribution which is never 0 in the interval (-infinity, infinity). Is it because the probability is very small outside of [0,1] and thereby still a good approximation. Or am I not understanding the central limit theorem. I have googled a bit and most of what I have found is applying the central limit theorem to the binomial distribution. But this seems to be the easier case and my guess is having to estimate a proportion is very common. Any insight is appreciated.