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

Explanation:

Step1: Recall the formula for standard deviation of sample means

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 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 multiplied by 4 (new sample size \(n_2 = 4n\)), 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:

C. The standard error is halved.