Which statement best describes the Central Limit Theorem?

Which statement best describes the Central Limit Theorem?

• A. As sample size grows, the distribution of sample means approaches normality, regardless of the population shape.
• B. The population distribution becomes normal as sample size grows.
• C. It only applies to normally distributed populations.
• D. The standard deviation of the sample mean equals the population standard deviation.

Answer: (A)

The main idea being tested is how the distribution of the sample mean behaves as you take larger samples. The Central Limit Theorem says that the distribution of the sample mean becomes approximately normal as the sample size grows, even if the original population distribution is not normal, provided the population has finite variance and the observations are independent. This is why the statement focusing on normality of the sampling distribution of the mean with increasing n is correct; it’s about how the means from many samples spread around the true mean, not about the population itself changing shape. As the sample size increases, the spread of that sampling distribution—the standard error, which is the population standard deviation divided by the square root of n—shrinks, making the sample mean cluster more tightly around the true mean. The other options aren’t accurate for this idea: the population distribution does not become normal merely by taking larger samples, the theorem applies regardless of the population’s shape (not only for normally distributed populations), and the standard deviation of the sample mean is not the population standard deviation but the population standard deviation divided by the square root of n.

The main idea being tested is how the distribution of the sample mean behaves as you take larger samples. The Central Limit Theorem says that the distribution of the sample mean becomes approximately normal as the sample size grows, even if the original population distribution is not normal, provided the population has finite variance and the observations are independent. This is why the statement focusing on normality of the sampling distribution of the mean with increasing n is correct; it’s about how the means from many samples spread around the true mean, not about the population itself changing shape. As the sample size increases, the spread of that sampling distribution—the standard error, which is the population standard deviation divided by the square root of n—shrinks, making the sample mean cluster more tightly around the true mean.

The other options aren’t accurate for this idea: the population distribution does not become normal merely by taking larger samples, the theorem applies regardless of the population’s shape (not only for normally distributed populations), and the standard deviation of the sample mean is not the population standard deviation but the population standard deviation divided by the square root of n.