AP Statistics 6.3 Justifying a Claim Based on a Confidence Interval for a Population Proportion- MCQs - Exam Style Questions

For a one–sample proportion interval at a fixed confidence level,
\[ \hat{p} \pm z^* \sqrt{\frac{\hat{p}\,(1-\hat{p})}{n}}. \] The width is \(2z^*\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}\). With the same confidence level and the same sample size, \(z^*\) and \(n\) are unchanged.
The only change is \(\hat{p}\): originally \(\hat{p}_{\text{orig}} = \) midpoint of \((0.14,0.20)=0.17\); revised \(\hat{p}_{\text{rev}}=0.27\).
The function \(\hat{p}(1-\hat{p})\) is maximized at \(\hat{p}=0.5\) and increases as \(\hat{p}\) moves closer to \(0.5\).
Since \(0.27\) is closer to \(0.5\) than \(0.17\), we have \[ \hat{p}_{\text{rev}}(1-\hat{p}_{\text{rev}}) > \hat{p}_{\text{orig}}(1-\hat{p}_{\text{orig}}), \] so the standard error and the margin of error are larger, making the revised interval wider than the original.
✅ Answer: (D)