Chapter 7 - Estimates and Sample Sizes - 7-2 Estimating a Population Proportion - Basic Skills and Concepts - Page 341: 33

a)1537 b)1449

If $\hat{p}$ is known:$n=\frac{z^2_{\frac{\alpha}{2}}\cdot \hat{p}\cdot (1-\hat{p})}{E^2}.$ If $\hat{p}$ is unknown:$n=\frac{z^2_{\frac{\alpha}{2}}\cdot0.25}{E^2}.$ a) Here, $\hat{p}$ is unknown, hence $n=\frac{1.96^2\cdot0.25}{0.025^2}=1537.$ b)Here, $\hat{p}$ is known, it is 38%=0.38, hence $n=\frac{1.96^2\cdot(0.38)\cdot(1-0.38)}{0.025^2}=1449.$