Question

1. a 90% confidence interval is constructed in order to estimate the proportion of college students who are involved in at least one extracurricular activity. the interval is centered at 0.64, with a margin of error of 0.027. which one of the following intervals must be a 95% confidence interval constructed using the same sample of data? a. 0.620 to 0.662 b. 0.608 to 0.672 c. 0.611 to 0.669 d. 0.604 to 0.676 e. 0.615 to 0.665

Explanation:

Step1: Recall the relationship between confidence level and margin of error

As the confidence level increases, the margin of error also increases. For a 90% confidence interval, the margin of error \(E_{90}=0.027\). For a 95% confidence interval, \(E_{95}>E_{90}\) since \(z - \)score for 95% confidence (\(z = 1.96\)) is larger than \(z - \)score for 90% confidence (\(z=1.645\)) and \(E=z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\) (where \(\hat{p}\) is the sample proportion and \(n\) is the sample size, and here \(\hat{p}=0.64\) and \(n\) is fixed).

Step2: Calculate the lower and upper bounds of the 90% confidence interval

The 90% confidence interval is \(\hat{p}-E_{90}\) to \(\hat{p}+E_{90}\), which is \(0.64 - 0.027=0.613\) to \(0.64 + 0.027 = 0.667\).

Step3: Analyze each option

• Option A: The width of the interval \(0.662-0.620 = 0.042\), and the margin of error \(E=\frac{0.662 - 0.620}{2}=0.021<0.027\).
• Option B: The width of the interval \(0.672 - 0.608=0.064\), and the margin of error \(E=\frac{0.672 - 0.608}{2}=0.032\).
• Option C: The width of the interval \(0.669 - 0.611 = 0.058\), and the margin of error \(E=\frac{0.669 - 0.611}{2}=0.029\).
• Option D: The width of the interval \(0.676 - 0.604=0.072\), and the margin of error \(E=\frac{0.676 - 0.604}{2}=0.036\).
• Option E: The width of the interval \(0.665 - 0.615=0.05\), and the margin of error \(E=\frac{0.665 - 0.615}{2}=0.025<0.027\).

Since the margin of error for 95% confidence interval must be larger than \(0.027\), we check the margin of error for each option. The margin of error for a confidence interval is calculated as \(E=\frac{\text{Upper bound}-\text{Lower bound}}{2}\).

Answer:

D. \(0.604\) to \(0.676\)