If a population distribution is known to be normally distributed, the sample mean must be equal to the population mean.

If a population distribution is known to be normally distributed, the sample mean must be equal to the population mean.

• A. True
• B. False
• C. Only if the sample size is infinite
• D. Always equal to the population mean

Answer: (B)

The sample mean is an unbiased estimator of the population mean, but it is not guaranteed to equal it in every sample. When the population is normally distributed, the sampling distribution of the sample mean X̄ is normal with mean μ and variance σ²/n. That means X̄ is a random value centered at μ, and only in expectation does it equal μ. As n grows, the spread σ/√n shrinks, so X̄ tends to be very close to μ, but it still isn’t guaranteed to be exactly μ for a finite sample. Only in the theoretical infinite-sample limit would X̄ converge to μ with certainty. So the statement is false.

The sample mean is an unbiased estimator of the population mean, but it is not guaranteed to equal it in every sample. When the population is normally distributed, the sampling distribution of the sample mean X̄ is normal with mean μ and variance σ²/n. That means X̄ is a random value centered at μ, and only in expectation does it equal μ. As n grows, the spread σ/√n shrinks, so X̄ tends to be very close to μ, but it still isn’t guaranteed to be exactly μ for a finite sample. Only in the theoretical infinite-sample limit would X̄ converge to μ with certainty. So the statement is false.