Which statement correctly describes the Central Limit Theorem?

Which statement correctly describes the Central Limit Theorem?

• A. It only applies to normally distributed populations.
• B. It states that the sampling distribution of the sample mean approaches normal as n grows, regardless of population distribution, given finite variance.
• C. It asserts that the sample mean equals the population mean exactly with probability 1.
• D. It implies the standard deviation of the sample mean is constant with n.

Answer: (B)

The key idea being tested is how the distribution of the sample mean behaves as you take larger samples. The Central Limit Theorem says that no matter what the population shape is, as long as the population has finite variance, the distribution of the sample mean becomes approximately normal when the sample size is large. This normal distribution is centered at the population mean, and its spread shrinks with more data (the standard deviation of the sample mean is the population standard deviation divided by the square root of the sample size). That’s why the statement describing this behavior is the best choice: it captures the main outcome of the theorem—the normal shape of the sampling distribution of the mean that emerges with large n, regardless of the original population distribution (provided the variance is finite). Why the other ideas don’t fit: the CLT does not require the population to be normally distributed (it actually applies to a wide range of shapes; if the population is normal, the sample mean is normal for any n, but the theorem is about the broader case). The sample mean does not equal the population mean with probability 1 in general; it is an estimator whose distribution centers at the population mean. And the standard deviation of the sample mean decreases with n (it equals sigma/√n), so it is not constant as n grows.

The key idea being tested is how the distribution of the sample mean behaves as you take larger samples. The Central Limit Theorem says that no matter what the population shape is, as long as the population has finite variance, the distribution of the sample mean becomes approximately normal when the sample size is large. This normal distribution is centered at the population mean, and its spread shrinks with more data (the standard deviation of the sample mean is the population standard deviation divided by the square root of the sample size).

That’s why the statement describing this behavior is the best choice: it captures the main outcome of the theorem—the normal shape of the sampling distribution of the mean that emerges with large n, regardless of the original population distribution (provided the variance is finite).

Why the other ideas don’t fit: the CLT does not require the population to be normally distributed (it actually applies to a wide range of shapes; if the population is normal, the sample mean is normal for any n, but the theorem is about the broader case). The sample mean does not equal the population mean with probability 1 in general; it is an estimator whose distribution centers at the population mean. And the standard deviation of the sample mean decreases with n (it equals sigma/√n), so it is not constant as n grows.