In the context of confidence intervals for a population mean, what happens to the expected value of the sample mean $m$ as the sample size increases?

• A. The expected value of $m$ remains equal to the population mean.
• B. The expected value of $m$ becomes more variable.
• C. The expected value of $m$ increases.
• D. The expected value of $m$ decreases.

Answer: (A)

Verified step by step guidance

1. Recall that the sample mean \(\overline{X}\) is an unbiased estimator of the population mean \(\mu\). This means that the expected value of the sample mean is equal to the population mean, i.e., \(E(\overline{X}) = \mu\).
2. Understand that the expected value (or mean) of the sample mean does not depend on the sample size \(n\). It remains constant regardless of how large or small the sample is.
3. Recognize that while the expected value stays the same, the variability (or standard error) of the sample mean decreases as the sample size increases. This is because the standard error is given by \(\sigma / \sqrt{n}\), where \(\sigma\) is the population standard deviation.
4. Therefore, increasing the sample size makes the sample mean more precise (less variable), but it does not change its expected value.
5. Summarize that the expected value of the sample mean remains equal to the population mean as the sample size increases.