Which of the following statements about sampling distributions for the sample mean is false?

• A. The sampling distribution of the sample mean has less variability than the population distribution.
• B. The standard deviation of the sampling distribution of the sample mean is equal to the population standard deviation $\sigma$.
• C. As the sample size increases, the sampling distribution of the sample mean becomes approximately normal, regardless of the population distribution.
• D. The mean of the sampling distribution of the sample mean $\mu$ is equal to the population mean $\mu$.

Answer: (B)

Verified step by step guidance

1. Step 1: Understand the concept of the sampling distribution of the sample mean. It is the probability distribution of all possible sample means of a given size drawn from the population.
2. Step 2: Recall that the mean of the sampling distribution of the sample mean is equal to the population mean, which can be expressed as \(\mu_{\bar{x}} = \mu\).
3. Step 3: Recognize that the standard deviation of the sampling distribution of the sample mean, called the standard error, is \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the population standard deviation and \(n\) is the sample size. This means the standard deviation of the sampling distribution is less than the population standard deviation unless \(n=1\).
4. Step 4: Understand the Central Limit Theorem, which states that as the sample size \(n\) increases, the sampling distribution of the sample mean approaches a normal distribution regardless of the shape of the population distribution.
5. Step 5: Identify the false statement by comparing each option to these principles. The false statement is the one claiming that the standard deviation of the sampling distribution of the sample mean is equal to the population standard deviation, which contradicts the formula for the standard error.