You plan to conduct a survey on your college campus to learn about the political awareness of students. You want to estimate the true proportion of college students on your campus who voted in the 2012 presidential election with $95 \%$ confidence and a margin of error no greater than five percent. How many students must you interview?
Answer
Solving this equation, we find that we need to interview at least \(\boxed{385}\) students.
Step 1 ：We want to estimate the true proportion of college students on our campus who voted in the 2012 presidential election with 95% confidence and a margin of error no greater than five percent.
Step 2 ：We use the formula for sample size in a proportion estimation: \(n = \frac{{Z^2 \cdot p \cdot (1-p)}}{{E^2}}\), where \(Z\) is the Z-score for the desired confidence level, \(p\) is the estimated proportion, and \(E\) is the desired margin of error.
Step 3 ：For a 95% confidence level, the Z-score is 1.96. We don't have an estimate for the proportion of students who voted, so we'll use 0.5 as a conservative estimate. The desired margin of error is 0.05.
Step 4 ：Substituting these values into the formula, we get \(n = \frac{{(1.96)^2 \cdot 0.5 \cdot (1-0.5)}}{{(0.05)^2}}\).
Step 5 ：Solving this equation, we find that we need to interview at least \(\boxed{385}\) students.
