Question

The distribution of sample means is not always a normal distribution under what circumstances

Answer 1

The distribution of sample means will not be normal when it is grounded on small samples (n < 30) carefully chosen from a population that is not normal because if n < 30, you would the use the t distribution. Normal distribution would only be use if the sample size is large. Large in this case relies on the requirements for accuracy and the shape of the underlying population.