Which statement about the Central Limit Theorem is true?

• A. With a large enough sample size, the sampling distribution of the mean is approximately normal regardless of population shape.
• B. It requires the population to be perfectly normal.
• C. It ensures the population mean equals the sample mean.
• D. It guarantees that any sample mean equals the population mean.

Answer: (A)

The central idea being tested is how the distribution of the sample mean behaves as you take larger samples. The Central Limit Theorem says that, as the sample size grows, the distribution of the mean of those samples becomes approximately normal, even if the underlying population distribution is not normal—provided the observations are independent (or weakly dependent) and have finite variance. This is why the statement is true: with a large enough sample, the sampling distribution of the mean takes on a bell-shaped form, which is the normal distribution.

This also implies that the mean of that sampling distribution is the population mean (the average across many samples centers around the true mean), and the spread shrinks as the sample size increases (the standard deviation of the sampling distribution is sigma divided by the square root of n). It does not require the population to be perfectly normal, and it does not guarantee that any single sample mean will exactly equal the population mean or that every sample mean equals the population mean.