If the sampling size n doubles, what happens to the variance of the sample mean?

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Multiple Choice

If the sampling size n doubles, what happens to the variance of the sample mean?

Explanation:
The main idea is that the variability of the sample mean gets smaller as you collect more data. If each observation has variance sigma^2 and you take n independent observations, the variance of their average is sigma^2/n. So when you double the sample size, you get sigma^2/(2n), which is exactly half of the original variance. In other words, the average becomes more precise with more data. The standard deviation of the sample mean also decreases, by a factor of sqrt(2).

The main idea is that the variability of the sample mean gets smaller as you collect more data. If each observation has variance sigma^2 and you take n independent observations, the variance of their average is sigma^2/n. So when you double the sample size, you get sigma^2/(2n), which is exactly half of the original variance. In other words, the average becomes more precise with more data. The standard deviation of the sample mean also decreases, by a factor of sqrt(2).

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