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

• A. The mean of the sampling distribution of the sample mean is equal to the population mean. $\mu \_x=\mu$
• 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 sample mean will always be equal to the population mean for any sample size. $x=\mu$
• 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. ${\sigma }_x=\frac{\sigma }{\sqrt{n}}$

Answer: (C)

Verified step by step guidance

1. Step 1: Understand the Central Limit Theorem (CLT) - it states that the sampling distribution of the sample mean will approach a normal distribution as the sample size increases, regardless of the population's distribution shape.
2. Step 2: Identify the key conclusions of the CLT: (a) The mean of the sampling distribution of the sample mean equals the population mean, (b) The sampling distribution approaches normality as sample size grows, and (c) The standard deviation of the sampling distribution (standard error) is the population standard deviation divided by the square root of the sample size, expressed as \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\).
3. Step 3: Analyze the statement 'The sample mean will always be equal to the population mean for any sample size' - this implies exact equality for every sample, which is not true because sample means vary due to sampling variability.
4. Step 4: Recognize that the CLT describes the behavior of the distribution of sample means, not the equality of any single sample mean to the population mean.
5. Step 5: Conclude that the statement about the sample mean always equaling the population mean is NOT a conclusion of the CLT, while the other statements are valid conclusions.