Which of the following is not a characteristic of the distribution of sample means?

• A. As the sample size increases, the distribution approaches normality according to the Central Limit Theorem.
• B. It has a standard deviation equal to the $\frac{\sigma }{\sqrt{n}}$ (population standard deviation divided by the square root of the sample size).
• C. It has a mean equal to the $\mu$ (population mean).
• D. It is always perfectly symmetric regardless of the population distribution.

Answer: (D)

Verified step by step guidance

1. Understand that the distribution of sample means, also called the sampling distribution of the mean, has specific characteristics derived from the Central Limit Theorem (CLT).
2. Recall that according to the CLT, as the sample size increases, the distribution of sample means approaches a normal distribution, regardless of the shape of the population distribution.
3. Recognize that the mean of the sampling distribution of the sample mean is equal to the population mean, which means \(\mu_{\bar{x}} = \mu\).
4. Know that the standard deviation of the sampling distribution (called the standard error) is the population standard deviation divided by the square root of the sample size, expressed as \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\).
5. Identify that the statement 'It is always perfectly symmetric regardless of the population distribution' is incorrect because the sampling distribution only approaches normality (and thus symmetry) as the sample size becomes large; for small samples from a non-normal population, the distribution of sample means may not be symmetric.