## Elementary Statistics (12th Edition)

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:$n=\frac{z^2_{\frac{\alpha}{2}}\cdot0.25}{E^2}.$ Here, $\hat{p}$ is unknown, hence: $1-\alpha=0.8$, hence $\frac{\alpha}{2}=0.1.$ By using the table the z-score belonging to 0.1:$z_{\frac{\alpha}{2}}=1.28.$ Hence $n=\frac{1.28^2\cdot0.25}{0.04^2}=256.$