Which statement about the Law of Large Numbers is true?

Which statement about the Law of Large Numbers is true?

• A. The sample mean converges to the true mean as sample size grows.
• B. The population mean changes as more data are collected.
• C. The sampling distribution becomes uniform with larger samples.
• D. The variance of the sample mean becomes negative as n increases.

Answer: (A)

The tested idea is that the average of a growing sample settles at the true population mean. This is the Law of Large Numbers: as you collect more data, the sample mean gets closer to the true mean, and the variability of that sample mean decreases (its standard error is roughly sigma divided by the square root of n). The population mean itself is fixed and does not change with more data, so saying it changes is not correct. The sampling distribution does not become uniform just from larger samples; with more data it centers around the true mean and, by the central limit theorem, tends toward a normal shape with decreasing spread. The variance of the sample mean cannot be negative; it equals sigma^2/n and shrinks toward zero as n grows. So the statement that the sample mean converges to the true mean as sample size grows is the correct one.

The tested idea is that the average of a growing sample settles at the true population mean. This is the Law of Large Numbers: as you collect more data, the sample mean gets closer to the true mean, and the variability of that sample mean decreases (its standard error is roughly sigma divided by the square root of n). The population mean itself is fixed and does not change with more data, so saying it changes is not correct. The sampling distribution does not become uniform just from larger samples; with more data it centers around the true mean and, by the central limit theorem, tends toward a normal shape with decreasing spread. The variance of the sample mean cannot be negative; it equals sigma^2/n and shrinks toward zero as n grows. So the statement that the sample mean converges to the true mean as sample size grows is the correct one.