Distribution of the Sample Mean

The Central Limit Theorem.

Given a population described by any probability distribution having mean μ and finite variance σ², the sampling distribution of the sample meanX ̅computed from samples of size n from this population will be approximately normal with meanμ (the population mean) and variance σ²/n (the population variance divided by n) when the sample size n is large.

[Practice Problems] A population has a non-normal distribution with mean μ and variance σ². The sampling distribution of the sample mean computed from samples of large size from that population will have:

A.    the same distribution as the population distribution.

B.     its mean approximately equal to the population mean.

C.     its variance approximately equal to the population variance.

[Solutions] B

Given a population described by any probability distribution (normal or non-normal) with finite variance, the central limit theorem states that the sampling distribution of the sample mean will be approximately normal, with the mean approximately equal to the population mean, when the sample size is large.