Which statement about the law of large numbers and the sample mean is true?

• A. As sample size grows, the sample mean becomes more variable.
• B. The sample mean equals the population mean for any sample.
• C. The distribution of the sample mean becomes normal as sample size increases.
• D. As sample size grows, the sample mean converges to the population mean.

Answer: (D)

The law of large numbers shows that as you collect more observations, the average of those observations gets closer to the true population average. In other words, the sample mean converges to the population mean as the sample size grows.

This means the statement that as sample size grows, the sample mean converges to the population mean is the accurate reflection of the law of large numbers.

The other ideas don’t fit. The sample mean does not become more variable with larger samples; its variability actually decreases as n increases (the standard error shrinks like σ/√n). It’s also not true that the sample mean equals the population mean for any single sample—the sample mean is an unbiased estimator, but it only equals the true mean in the long run or in expectation, not for every sample. The claim about the distribution becoming normal with larger sample size speaks to the central limit theorem, not the law of large numbers; the CLT describes the shape of the distribution, while the LLN describes convergence of the mean to the true value.