Question

if the sample size is multiplied by 4, what happens to the standard deviation of the distribution of sample means?

○ a. the standard error is doubled.
○ b. the standard error is increased by a factor of 4.
○ c. the standard error is decreased by a factor of 4.
○ d. the standard error is halved.

Explanation:

Step1: Recall the formula for standard deviation of sample means

The formula for the standard deviation of the distribution of sample means (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 multiplying \(n\) by \(4\)

Let the original sample size be \(n\), so the original standard error is \(\sigma_{\bar{x}_1}=\frac{\sigma}{\sqrt{n}}\). When the sample size is \(n_2 = 4n\), the new standard error is \(\sigma_{\bar{x}_2}=\frac{\sigma}{\sqrt{4n}}\).
Since \(\sqrt{4n}=2\sqrt{n}\), then \(\sigma_{\bar{x}_2}=\frac{\sigma}{2\sqrt{n}}=\frac{1}{2}\times\frac{\sigma}{\sqrt{n}}\).

Answer:

D. The standard error is halved.