What happens to the mean and variance of the sampling distribution of the sample mean as the sample size grows?

• A. The mean remains μ and the variance decreases with larger n
• B. The mean increases without bound
• C. The mean becomes zero
• D. The variance increases with n

Answer: (A)

As the sample size grows, the estimate of the population mean becomes more precise while remaining centered at that mean. The sampling distribution of the sample mean has expected value equal to the population mean μ, and its variance is σ²/n. So increasing n keeps the mean at μ but makes the variance shrink (the standard error is σ/√n and gets smaller as n grows). This reflects the idea that larger samples yield more reliable estimates of μ.

The other possibilities don’t fit: the mean does not increase without bound, nor does it become zero, and the variance does not rise with n.