Which statement best describes the Central Limit Theorem's general claim?

Which statement best describes the Central Limit Theorem's general claim?

• A. It states that the sample mean has the same distribution as the population mean
• B. It states that the distribution of sample means approaches normal as n increases
• C. It states that the variance of sample means equals population variance
• D. It states that the probability of any single observation is zero

Answer: (B)

The main idea being tested is that the sampling distribution of the sample mean becomes normal as the sample size grows, regardless of the population’s shape. As you take many samples of size n and compute their means, those means form a distribution that approaches a bell curve when n gets larger. This normal distribution has a center at the population mean μ and a spread (standard deviation) of σ/√n, where σ^2 is the population variance. In other words, larger samples give more precise estimates of the true mean, and the distribution of those estimates becomes normal. This is why the statement describing the normal approach of the distribution of sample means as n increases is the best representation of the Central Limit Theorem. It captures the behavior of the statistic (the sample mean) rather than the fixed population parameter or individual observations. Other options don’t fit: the CLT doesn’t claim the sample mean shares the exact distribution of the population mean, nor that the variance of sample means equals the population variance, nor that the probability of a single observation is zero.

The main idea being tested is that the sampling distribution of the sample mean becomes normal as the sample size grows, regardless of the population’s shape. As you take many samples of size n and compute their means, those means form a distribution that approaches a bell curve when n gets larger. This normal distribution has a center at the population mean μ and a spread (standard deviation) of σ/√n, where σ^2 is the population variance. In other words, larger samples give more precise estimates of the true mean, and the distribution of those estimates becomes normal.

This is why the statement describing the normal approach of the distribution of sample means as n increases is the best representation of the Central Limit Theorem. It captures the behavior of the statistic (the sample mean) rather than the fixed population parameter or individual observations. Other options don’t fit: the CLT doesn’t claim the sample mean shares the exact distribution of the population mean, nor that the variance of sample means equals the population variance, nor that the probability of a single observation is zero.