Question

1. a 99% confidence interval for a population proportion is calculated using data from a random sample and this interval ends up being from 0.132 to 0.248. which one of the following must be the 90% confidence interval calculated from the same data? a. 0.153 to 0.227 b. 0.146 to 0.234 c. 0.127 to 0.253 d. 0.156 to 0.224 e. 0.168 to 0.232

Explanation:

Step1: Calculate the sample proportion

The sample proportion \( \hat{p}\) is the mid - point of the confidence interval. For a confidence interval \((a,b)\), \(\hat{p}=\frac{a + b}{2}\).
Given the \(99\%\) confidence interval \((0.132,0.248)\), then \(\hat{p}=\frac{0.132+0.248}{2}=\frac{0.38}{2}=0.19\)

Step2: Analyze the width of confidence intervals

The width of a confidence interval for a proportion is \(w = 2\times z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\). The \(z\) - value for a \(99\%\) confidence interval: \(z_{0.005}=2.576\), and for a \(90\%\) confidence interval: \(z_{0.05}=1.645\). Since \(z_{0.05}

Step3: Check the mid - point of each option

• Option A: Mid - point \(\frac{0.153 + 0.227}{2}=\frac{0.38}{2}=0.19\)
• Option B: Mid - point \(\frac{0.146+0.234}{2}=\frac{0.38}{2}=0.19\)
• Option C: Mid - point \(\frac{0.127 + 0.253}{2}=\frac{0.38}{2}=0.19\)
• Option D: Mid - point \(\frac{0.156+0.224}{2}=\frac{0.38}{2}=0.19\)
• Option E: Mid - point \(\frac{0.168+0.232}{2}=\frac{0.4}{2}=0.2\) (rejected as mid - point is not \(0.19\))

Step4: Check the width

The width of the \(99\%\) confidence interval \(w_{99}=0.248 - 0.132=0.116\)
The width of a confidence interval \(w = 2\times z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\). Let \(w_{90}\) be the width of the \(90\%\) confidence interval. Since \(\frac{w_{90}}{w_{99}}=\frac{z_{0.05}}{z_{0.005}}\), \(w_{90}=w_{99}\times\frac{z_{0.05}}{z_{0.005}}\)
\(w_{90}=0.116\times\frac{1.645}{2.576}\approx0.074\)

• Option A: Width \(0.227 - 0.153=0.074\)
• Option B: Width \(0.234 - 0.146 = 0.088\)
• Option C: Width \(0.253-0.127 = 0.126\)
• Option D: Width \(0.224 - 0.156=0.068\)

Answer:

A. \(0.153\) to \(0.227\)