Question

what happens to the mean and standard deviation of the distribution of sample means as the size of the sample decreases?

a. the mean of the sample means increases and the standard error stays constant.
b. the mean of the sample means stays constant and the standard error increases.
c. the mean of the sample means decreases and the standard error increases.
d. the mean of the sample means stays constant and the standard error decreases.

Explanation:

The mean of the sampling distribution of the sample means, \(\mu_{\bar{x}}\), is equal to the population mean \(\mu\), i.e., \(\mu_{\bar{x}}=\mu\). So, it does not depend on the sample size \(n\).

The standard deviation of the sampling distribution of the sample means (standard error), \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the population standard deviation. As \(n\) (sample size) decreases, \(\frac{\sigma}{\sqrt{n}}\) increases (since the denominator \(\sqrt{n}\) gets smaller).

Answer:

B. The mean of the sample means stays constant and the standard error increases.