If all possible samples of size n are drawn from a normal population, the probability distribution of the sample mean is an exact normal distribution.

Question

If all possible samples of size n are drawn from a normal population, the probability distribution of the sample mean  is an exact normal distribution.

Quizplus · Accepted Answer

This statement is true and is known as the Central Limit Theorem. It states that if independent random samples of size n are drawn from a population (not necessarily normal), then the distribution of the sample means will be approximately normal if n is large enough. As n increases, the sample means become more normally distributed, with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of n. When n is sufficiently large (generally considered to be greater than or equal to 30), the distribution of the sample mean is essentially normal, regardless of the shape of the original population distribution.

If All Possible Samples of Size N Are Drawn from a Normal