According to the law of large numbers, as the sample size increases, which of the following statements is true about the $\mathrm{sample mean}$?

• A. The $\mathrm{sample mean}$ always equals the $\mathrm{population mean}$.
• B. The $\mathrm{sample mean}$ tends to get closer to the $\mathrm{population mean}$.
• C. The $\mathrm{sample mean}$ becomes more variable and less predictable.
• D. The $\mathrm{sample mean}$ moves further away from the $\mathrm{population mean}$.

Answer: (B)

Verified step by step guidance

1. Understand the Law of Large Numbers (LLN), which states that as the sample size \(n\) increases, the sample mean \(\overline{X}\) tends to get closer to the population mean \(\mu\).
2. Recognize that the LLN does not guarantee the sample mean will always equal the population mean, but rather that the difference between them becomes smaller on average as \(n\) grows.
3. Recall that the variability of the sample mean decreases with larger sample sizes, because the standard error of the mean is given by \(\frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the population standard deviation.
4. Note that because the standard error decreases, the sample mean becomes more stable and predictable, not more variable or less predictable.
5. Conclude that the correct interpretation is: the sample mean tends to get closer to the population mean as the sample size increases.