Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p
-Margin of error: $0.011$; confidence level: $92 \% ; \hat { \mathrm { p } }$ and $\hat { \mathrm { q } }$ unknown
A) 40
B) 1
C) 6328
D) 6327

Question

Quizplus · Accepted Answer

To construct a confidence interval for the population proportion when the margin of error (E) and confidence level are given but $\hat{p}$ and $\hat{q}$ are unknown, we use the formula for the margin of error in a proportion confidence interval: $E = Z \sqrt{\frac{\hat{p}\hat{q}}{n}}$. However, without $\hat{p}$ and $\hat{q}$, we maximize the product $\hat{p}\hat{q}$ by assuming $\hat{p} = \hat{q} = 0.5$ (since $\hat{p}\hat{q}$ is maximized when $\hat{p} = \hat{q} = 0.5$). Rearranging the formula to solve for $n$ gives $n = \frac{Z^2 \hat{p}\hat{q}}{E^2}$. Given $E = 0.011$ and a confidence level of 92%, which corresponds to a $Z$ value approximately equal to 1.75 (since Z values for common confidence levels like 90%, 95%, and 99% are approximately 1.645, 1.96, and 2.576, respectively, and 92% is between 90% and 95%, so the Z value is estimated between those Z values for 90% and 95%). Substituting the values gives $n = \frac{(1.75)^2 \times 0.5 \times 0.5}{(0.011)^2}$, which calculates to approximately 6327. However, since sample size must be a whole number and we cannot have a fraction of a sample, rounding to the nearest whole gives 6328.

Use the Given Degree of Confidence and Sample Data to Construct