What sample size is needed to give a margin of error within $\pm 1.5 \%$ in estimating a population proportion with $95 \%$ confidence?
Round your answer up to the nearest integer.
Sample size $=$
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Answer
The sample size needed to give a margin of error within \(\pm 1.5 \%\) in estimating a population proportion with 95% confidence is \(\boxed{4269}\).
Step 1 ：Define the constants: the Z-score for a 95% confidence level is 1.96, the estimated proportion is 0.5, and the margin of error is 0.015.
Step 2 ：Calculate the sample size using the formula \(n = \frac{{Z^2 \cdot p \cdot (1-p)}}{{E^2}}\).
Step 3 ：Round up the result to the nearest integer to get the final sample size.
Step 4 ：The sample size needed to give a margin of error within \(\pm 1.5 \%\) in estimating a population proportion with 95% confidence is \(\boxed{4269}\).
