We just said that the sampling distribution of the sample mean is always normal. ... The central limit theorem (CLT) is a theorem that gives us a way to turn a non-normal distribution into a normal distribution.

Are sampling distributions of proportions always normal?

Note: The sampling distribution of a sample proportion p^​p, with, hat, on top is approximately normal as long as the expected number of successes and failures are both at least 10.

Why would a sampling distribution not be normal?

If the population is skewed and sample size small, then the sample mean won't be normal. When doing a simulation, one replicates the process many times. Using 10,000 replications is a good idea. If the population is normal, then the distribution of sample mean looks normal even if .