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

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

• A. True
• B. False
• C. Only for large samples
• D. Only if population variance is finite

Answer: (A)

When the population is normally distributed, the sampling distribution of the sample mean is exactly normal for any sample size. This happens because if each observation Xi is independent and follows a normal distribution with mean mu and variance sigma^2, then the sum of those normals is normal with mean n mu and variance n sigma^2. The sample mean X̄ = (1/n) Σ Xi scales that sum, so X̄ follows a normal distribution with mean mu and variance sigma^2 / n. In other words, X̄ ~ N(mu, sigma^2 / n), with standard error sigma / sqrt(n). This holds for any n, not just large samples, which is why the statement is true. The finite variance note is automatic for a normal population, so it isn’t a separate condition.

When the population is normally distributed, the sampling distribution of the sample mean is exactly normal for any sample size. This happens because if each observation Xi is independent and follows a normal distribution with mean mu and variance sigma^2, then the sum of those normals is normal with mean n mu and variance n sigma^2. The sample mean X̄ = (1/n) Σ Xi scales that sum, so X̄ follows a normal distribution with mean mu and variance sigma^2 / n. In other words, X̄ ~ N(mu, sigma^2 / n), with standard error sigma / sqrt(n). This holds for any n, not just large samples, which is why the statement is true. The finite variance note is automatic for a normal population, so it isn’t a separate condition.