Which of the following is not a conclusion of the central limit theorem?

• A. The sample mean will always be equal to the population mean for any sample size.
• B. The sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population's distribution.
• C. The mean of the sampling distribution of the sample mean equals the population mean.
• D. The standard deviation of the sampling distribution of the sample mean is equal to the population standard deviation divided by the square root of the sample size. $\frac{\sigma }{\sqrt{n}}$

Answer: (A)

Verified step by step guidance

1. Step 1: Understand the Central Limit Theorem (CLT) and its key conclusions. The CLT states that the sampling distribution of the sample mean will approach a normal distribution as the sample size increases, regardless of the population's original distribution.
2. Step 2: Recognize that one conclusion of the CLT is that the mean of the sampling distribution of the sample mean equals the population mean. This means \(\mu_{\bar{x}} = \mu\), where \(\mu_{\bar{x}}\) is the mean of the sample means and \(\mu\) is the population mean.
3. Step 3: Identify that the standard deviation of the sampling distribution of the sample mean (also called the standard error) is the population standard deviation divided by the square root of the sample size, expressed as \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the population standard deviation and \(n\) is the sample size.
4. Step 4: Note that the CLT does not claim the sample mean will always be equal to the population mean for any sample size. Instead, it states that the distribution of the sample mean will center around the population mean, but individual sample means can vary, especially for small sample sizes.
5. Step 5: Conclude that the statement "The sample mean will always be equal to the population mean for any sample size" is NOT a conclusion of the Central Limit Theorem, making it the correct answer to the question.