Under the CLT, for a population with mean mu and variance sigma^2, the distribution of the sample mean from large samples is approximately Normal with which mean and variance?

Under the CLT, for a population with mean mu and variance sigma^2, the distribution of the sample mean from large samples is approximately Normal with which mean and variance?

• A. Mean mu; variance sigma^2
• B. Mean mu; variance sigma^2/n
• C. Mean mu; variance sigma^2
• D. Mean mu; variance sigma^2/n^2

Answer: (B)

A key idea is that averaging many independent observations makes the sampling distribution of the mean center around the population mean while the spread shrinks as you collect more data. Specifically, the mean of the sample mean is the population mean mu, and its variance is the population variance divided by the sample size, sigma^2/n. So the sampling distribution of the mean is approximately Normal with mean mu and variance sigma^2/n (standard deviation sigma/√n). This reflects how larger samples reduce variability in the average. The other options would keep the variance at sigma^2 or shrink it to sigma^2/n^2, which isn’t what the CLT predicts.

A key idea is that averaging many independent observations makes the sampling distribution of the mean center around the population mean while the spread shrinks as you collect more data. Specifically, the mean of the sample mean is the population mean mu, and its variance is the population variance divided by the sample size, sigma^2/n. So the sampling distribution of the mean is approximately Normal with mean mu and variance sigma^2/n (standard deviation sigma/√n). This reflects how larger samples reduce variability in the average.

The other options would keep the variance at sigma^2 or shrink it to sigma^2/n^2, which isn’t what the CLT predicts.