Question

the standard deviation of the sampling distribution of the sample mean decreases when a) x increases b) n increases c) n decreases

Explanation:

Step1: Recall the formula for the standard deviation of the sampling distribution of the sample mean

The formula for the standard deviation of the sampling distribution of the sample mean (also known as the standard error) is \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the population standard deviation and \(n\) is the sample size.

Step2: Analyze the effect of changing \(n\)

If we consider the relationship \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\), when \(n\) (the sample size) increases:
Let \(n_1 < n_2\). Then \(\sqrt{n_1}<\sqrt{n_2}\), and \(\frac{\sigma}{\sqrt{n_1}}>\frac{\sigma}{\sqrt{n_2}}\). So as \(n\) increases, \(\sigma_{\bar{x}}\) (the standard deviation of the sampling distribution of the sample mean) decreases.
When \(n\) decreases: Let \(n_3>n_4\). Then \(\sqrt{n_3}>\sqrt{n_4}\), and \(\frac{\sigma}{\sqrt{n_3}}<\frac{\sigma}{\sqrt{n_4}}\). So as \(n\) decreases, \(\sigma_{\bar{x}}\) increases.
The value of \(x\) (an individual data point) is not directly related to the formula for the standard deviation of the sampling distribution of the sample mean \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\)

Answer:

C. \(n\) decreases