Question

12 multiple choice 4 points select the most appropriate response. what will happen (other things being equal) if you increase the sample size used to construct a given confidence interval? it will be narrower it will be wider it will not change 13 multiple choice 4 points

Explanation:

Step1: Recall the formula for confidence interval

The formula for a confidence interval (for example, for the population mean when population standard deviation \(\sigma\) is known) is \(\bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\), where \(n\) is the sample size. The margin of error \(E = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\).

Step2: Analyze the effect of \(n\) on the margin of error

As \(n\) increases, \(\frac{\sigma}{\sqrt{n}}\) decreases. Since \(E\) is directly proportional to \(\frac{\sigma}{\sqrt{n}}\) (because \(z_{\alpha/2}\) and \(\sigma\) are constant in this "other things being equal" scenario), the margin of error \(E\) decreases.

Step3: Relate margin of error to the width of the confidence interval

The width of the confidence interval is \(2E\). If \(E\) decreases, then \(2E\) (the width of the confidence interval) decreases.

Answer:

It will be narrower