Question

fill in the blanks. the margin of error____with an increased sample size and____with an increase in confidence level. a. increases; increases b. increases; decreases c. decreases; increases d. decreases; decreases

Explanation:

The formula for the margin of error \(E = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\). When the sample size \(n\) increases, since \(n\) is in the denominator, the value of \(\frac{\sigma}{\sqrt{n}}\) decreases, so the margin of error \(E\) decreases. When the confidence level increases, the value of \(z_{\alpha/2}\) (the critical value) increases. For example, for a \(90\%\) confidence level \(z_{\alpha/2}=1.645\), for a \(95\%\) confidence level \(z_{\alpha/2} = 1.96\), and for a \(99\%\) confidence level \(z_{\alpha/2}=2.576\). As \(z_{\alpha/2}\) increases, the margin of error \(E\) increases.

Answer:

C. decreases; increases