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

• A. The sample mean is always equal to the population mean, regardless of 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 standard deviation of the sampling distribution of the sample mean is equal to $\frac{\sigma }{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size.
• D. The mean of the sampling distribution of the sample mean equals the population mean $\mu$.

Answer: (A)

Verified step by step guidance

1. Step 1: Understand the Central Limit Theorem (CLT). The CLT states that the sampling distribution of the sample mean will approach a normal distribution as the sample size \(n\) becomes large, regardless of the shape of the population distribution.
2. Step 2: Identify the key conclusions of the CLT: (a) The sampling distribution of the sample mean approaches normality as \(n\) increases, (b) The mean of the sampling distribution of the sample mean equals the population mean \(\mu\), and (c) The standard deviation of the sampling distribution (also called the standard error) is \(\sigma / \sqrt{n}\), where \(\sigma\) is the population standard deviation.
3. Step 3: Analyze the statement 'The sample mean is always equal to the population mean, regardless of sample size.' This implies that every sample mean exactly equals the population mean, which is not true because sample means vary from sample to sample.
4. Step 4: Recognize that the CLT does not claim the sample mean equals the population mean for every sample, but rather that the average of all possible sample means equals the population mean.
5. Step 5: Conclude that the incorrect statement is the one claiming the sample mean is always equal to the population mean, as this contradicts the variability inherent in sampling.