State the central limit theorem in terms of the sampling distribution of the sample mean.

• A. The distribution of the sample mean becomes more skewed as n increases.
• B. As n increases, the distribution of the sample mean approaches Normal(μ, σ^2/n), regardless of the population distribution, provided finite variance.
• C. The population variance decreases as sample size grows.
• D. The sample mean always equals the population mean for any sample.

Answer: (B)

The central idea is that the average of many independent observations from a population with finite variance tends to have a normal shape. As the sample size n grows, the sampling distribution of the sample mean X̄ becomes approximately Normal(μ, σ^2/n): it centers at the population mean μ and its spread is σ/√n. So larger samples produce averages that cluster more tightly around μ, and the distribution looks more bell-shaped regardless of the original population shape.

This is why the statement that the distribution of the sample mean approaches Normal(μ, σ^2/n) as n increases is the best fit. It also reflects that the variability of the sample mean decreases like σ^2/n, not that the population variance changes.

The other ideas don’t fit: the distribution becoming more skewed with larger n is incorrect—it becomes more normal; the population variance doesn’t shrink with sample size; and the sample mean, while an unbiased estimator of μ, does not equal μ for every finite sample (only on average over many samples).