Question

fill in the blank
to decrease the margin of error without decreasing the level of confidence, the sample size needs to be
to decrease the margin of error without decreasing the level of confidence, the sample size needs to be
increased.
kept the same.
decreased.

Explanation:

The formula for the margin of error \(E = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\) (for a confidence interval of the population mean when \(\sigma\) is known). Here, \(z_{\alpha/2}\) is related to the level of confidence. If the level of confidence is not decreased, \(z_{\alpha/2}\) remains the same. Also, assume \(\sigma\) (population standard deviation) is constant. To decrease \(E\), since \(E\) and \(\sqrt{n}\) are inversely - related (\(E\propto\frac{1}{\sqrt{n}}\)), we need to increase \(n\) (sample size).

Answer:

increased.