Question

If we are not given the information that the sample means are normally distributed, how can we verify that this condition is met?

Why do we require the distribution of sample means to be approximately normally distributed?

Answer 1

If the population is normal, the sampling distribution of will be normal for any sample size.

The less normal the population, the larger
n needs to be for the sampling distribution of to be nearly normal. However, a good rule of thumb is that for almost all populations, the sampling distribution of
will be approximately normal if .

The central limit theorem states that the sampling distribution of the mean of any independent, random variable will be normal or nearly normal, if the sample size is large enough.