Which of the following is true of the Central Limit Theorem?

• A. It states that, for a sufficiently large sample size, the sampling distribution of the sample mean $ṅ$ will be approximately normal regardless of the shape of the population distribution.
• B. It states that the sample variance $s^2$ is always less than the population variance ${\sigma }^2$.
• C. It states that the sampling distribution of the sample mean $ṅ$ is always uniform.
• D. It states that the population mean $\mu$ is always equal to the sample mean $ṅ$.

Answer: (A)

Verified step by step guidance

1. Understand that the Central Limit Theorem (CLT) is a fundamental concept in statistics that describes the behavior of the sampling distribution of the sample mean.
2. Recall that the CLT states that when you take sufficiently large random samples from any population (regardless of the population's distribution shape), the distribution of the sample means will tend to be approximately normal (bell-shaped).
3. Note that the CLT does not say anything about the sample variance always being less than the population variance; this is not a property guaranteed by the theorem.
4. Recognize that the CLT does not claim the sampling distribution of the sample mean is uniform; instead, it approaches a normal distribution as sample size increases.
5. Understand that the CLT does not state the population mean is always equal to the sample mean, but rather that the mean of the sampling distribution of the sample mean equals the population mean.