Which statement about the Central Limit Theorem enables inference as sample size grows?

• A. It guarantees the population distribution is normal.
• B. It shows that sample means are always exactly equal to the population mean.
• C. It enables normal-based inference for large samples by the distribution of the sample mean approaching normal.
• D. It states that all moments converge to fixed values.

Answer: (C)

The central idea is that as you collect more data, the distribution of the sample mean becomes approximately normal, even if the underlying population isn’t. This lets you use normal-based methods to make inferences about the population mean from the sample mean in large samples.

Specifically, with enough observations, the sampling distribution of the sample mean has mean μ and standard deviation σ/√n, so it approximates a normal distribution N(μ, σ^2/n). Because of that, you can construct confidence intervals and perform hypothesis tests using the normal (or z) framework, which is the essence of normal-based inference for large samples.

The other statements miss the point: the Central Limit Theorem doesn’t require the population to be normal, it doesn’t claim the sample mean is always exactly equal to the population mean (there’s always some sampling error, though it vanishes as n grows), and it isn’t about all moments converging to fixed values.