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

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

Answer: (D)

Verified step by step guidance

1. Step 1: Understand the Central Limit Theorem (CLT) and its key conclusions. The CLT states that as the sample size increases, the sampling distribution of the sample mean approaches a normal distribution, regardless of the population's original distribution.
2. Step 2: Recognize that the mean of the sampling distribution of the sample mean is equal to the population mean. This means the expected value of the sample mean is the population mean.
3. Step 3: Recall 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 \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\).
4. Step 4: Identify that the statement "The sample mean will always be exactly equal to the population mean for any sample size" is not a conclusion of the CLT. The sample mean is a random variable and will vary from sample to sample; it only approaches the population mean on average as sample size increases.
5. Step 5: Conclude that the incorrect statement is the one claiming the sample mean is always exactly equal to the population mean, which contradicts the variability inherent in sampling.