Answer to Question #139116 in Statistics and Probability for Susan Gerea

Question

When is the sample mean normally distributed? (5 marks)

Answer 1

answer

When samples are drawn repeatedly from the same population and their sample means obtained, we get a distribution of $n$ sample means that is normally distributed. Based on the Central limit theorem, the sample mean is normally distributed if the sample is large, that is the sample size is atleast 30. This is mostly true regardless of the distribution of the underlying population.

However, when the sample size is smaller than 30, the distribution of sample means will be normally distributed if the population from which the sample is drawn is approximately normally distributed with mean $\mu$ and standard deviation $\sigma$

In such cases, the mean of the sample means estimates the population mean.