Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p
-Margin of error: $0.07$; confidence level: $90 \%$; from a prior study, $\hat{\mathrm { p }}$ is estimated by $0.19$.
A) 255
B) 6
C) 85
D) 75

Question

Quizplus · Accepted Answer

To construct a confidence interval for the population proportion $p$, we use the formula for the sample size $n$ given by $n = \left( \frac{Z_{\alpha/2} \cdot \sqrt{\hat{p}(1-\hat{p})}}{E} \right)^2$, where $Z_{\alpha/2}$ is the Z-score corresponding to the confidence level, $\hat{p}$ is the estimated proportion, and $E$ is the margin of error. For a $90\%$ confidence level, $Z_{\alpha/2} \approx 1.645$, $\hat{p} = 0.19$, and $E = 0.07$. Plugging these values into the formula gives $n \approx 85$.

Use the Given Degree of Confidence and Sample Data to Construct