The Central Limit Theorem specifically calls out a sample mean to be following a normal distribution (for $n \ge 30$ ). But I am referring to a certain text, and it is calculating $z$ values for sample $skewness$ and sample $kurtosis$ assuming that these follow a normal distribution.
Is the book correct?
In short, if we take unlimited number of samples each of size $n$ (where $n \ge 30$ ) then for each sample $skewness$ and $kurtosis$ will vary. So these are random variables.
Question: Is the sampling distribution of these random variables normal? And if it is, what is the mean $\mu$ and standard deviation $\sigma$ for that?