Question

22 multiple choice 4 points
select the most appropriate response.
which of the following is a benefit of increasing the sample size when trying to estimate the mean from a sample average?
a reduction in the bias of the estimate
a reduction in the variability of the estimate
an increase in the width of the resulting confidence interval
there will be no benefit because the population has not changed
23 multiple choice 4 points

Explanation:

• Bias is a property of the sampling method, not the sample size. So increasing sample size doesn't reduce bias.
• The formula for the standard error of the mean is \(SE=\frac{\sigma}{\sqrt{n}}\) (where \(\sigma\) is the population standard deviation and \(n\) is the sample size). As \(n\) increases, \(SE\) decreases. A smaller standard error means less variability in the estimate of the mean.
• The formula for a confidence interval for the mean (when population standard deviation \(\sigma\) is known) is \(\bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\). As \(n\) increases, the margin of error \(z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\) decreases, so the width of the confidence interval decreases.
• Increasing sample size does have a benefit as it affects the precision (variability) of the estimate.

Answer:

B. A reduction in the variability of the estimate