Which of the following best describes what the Central Limit Theorem states in the context of confidence intervals?

• A. The sample variance $s^2$ is always less than the population variance $\sigma ^2$.
• B. For a sufficiently large sample size, the sampling distribution of the sample mean $\overline{x}$ will be approximately normal regardless of the shape of the population distribution.
• C. The population mean $\mu$ is always equal to the sample mean $\overline{x}$ for any sample size.
• D. Confidence intervals can only be constructed when the population is normally distributed.

Answer: (B)

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. Recognize that the CLT states: For a sufficiently large sample size, the distribution of the sample mean will be approximately normal, regardless of the shape of the population distribution.
3. Note that this property allows us to use normal distribution techniques, such as constructing confidence intervals, even when the original population distribution is not normal.
4. Realize that the CLT does not say anything about the sample variance always being less than the population variance, nor does it claim the population mean equals the sample mean for any sample size.
5. Also, understand that confidence intervals can be constructed without the population being normally distributed, as long as the sample size is large enough for the CLT to apply.