Which statement best describes the practical importance of the Central Limit Theorem for estimation?

Which statement best describes the practical importance of the Central Limit Theorem for estimation?

• A. It guarantees that the sample proportion is unbiased for any sample size.
• B. It ensures the sampling distribution of the sample mean is exactly normal for all n.
• C. It enables confidence intervals and hypothesis tests for the mean when n is large, regardless of the population shape.
• D. It implies that the variance of the sample mean is zero for large n.

Answer: (C)

The practical takeaway is that the sampling distribution of the mean becomes approximately normal as the sample size grows, even if the population distribution isn’t normal, provided the population variance is finite. This normal-approximation lets us perform real-world inferences about the mean when n is large, using standard confidence intervals and hypothesis tests. That’s why the best statement emphasizes using the normal approximation for the mean with large n to enable confidence intervals and tests, regardless of the population’s shape. The other ideas don’t fit: the CLT doesn’t guarantee unbiasedness for the sample proportion at any size, it doesn’t make the sampling distribution exactly normal for all n (only approximately so for large n), and it doesn’t imply the variance of the sample mean is zero (it decreases with n as sigma^2/n, but is not zero).

The practical takeaway is that the sampling distribution of the mean becomes approximately normal as the sample size grows, even if the population distribution isn’t normal, provided the population variance is finite. This normal-approximation lets us perform real-world inferences about the mean when n is large, using standard confidence intervals and hypothesis tests.

That’s why the best statement emphasizes using the normal approximation for the mean with large n to enable confidence intervals and tests, regardless of the population’s shape. The other ideas don’t fit: the CLT doesn’t guarantee unbiasedness for the sample proportion at any size, it doesn’t make the sampling distribution exactly normal for all n (only approximately so for large n), and it doesn’t imply the variance of the sample mean is zero (it decreases with n as sigma^2/n, but is not zero).