If the population is normally distributed, the sampling distribution of the sample mean is normal for any sample size.

If the population is normally distributed, the sampling distribution of the sample mean is normal for any sample size.

• A. True
• B. False
• C. Only if n ≥ 30
• D. Only if population variance is finite

Answer: (A)

When the population is normally distributed, the sample mean is a linear combination of independent normal observations, and the sum (or average) of independent normal variables is itself normal. So the sampling distribution of the sample mean is exactly normal for any sample size n. Its mean is the population mean μ, and its variance is the population variance σ^2 divided by n. This stays true no matter how small or large n is, which is why the statement is true. The central limit theorem is about non-normal populations needing larger n for approximate normality, not about normal populations requiring a minimum n.

When the population is normally distributed, the sample mean is a linear combination of independent normal observations, and the sum (or average) of independent normal variables is itself normal. So the sampling distribution of the sample mean is exactly normal for any sample size n. Its mean is the population mean μ, and its variance is the population variance σ^2 divided by n. This stays true no matter how small or large n is, which is why the statement is true. The central limit theorem is about non-normal populations needing larger n for approximate normality, not about normal populations requiring a minimum n.