Question

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

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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 relationship between \(\sigma_{\bar{x}}\) and \(n\)

From the formula \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\), we can see that \(\sigma_{\bar{x}}\) and \(n\) are inversely related. As \(n\) (the sample size) increases, the value of \(\sqrt{n}\) increases. Since \(\sigma_{\bar{x}}\) is equal to \(\sigma\) (a constant, assuming the population standard deviation \(\sigma\) is fixed) divided by \(\sqrt{n}\), when \(\sqrt{n}\) increases, \(\sigma_{\bar{x}}\) decreases.

Answer:

B. \(n\) increases