Answer
a)178
b)176
Work Step by Step
a) By using the given formula with $\hat{p}\cdot\hat{q}=0.25.$
$n=\frac{N\cdot z_{\alpha/2}^2\cdot 0.25}{ z_{\alpha/2}^2\cdot 0.25+(N-1)\cdot E^2}=\frac{200\cdot 1.96^2\cdot 0.25}{1.96^2\cdot 0.25+(200-1)\cdot0.025^2}\approx178.$
b)By using the given formula with $\hat{q}=1-\hat{p}.$
$n=\frac{N\cdot z_{\alpha/2}^2\cdot \hat{p}\cdot (1-\hat{p})}{ z_{\alpha/2}^2\cdot \hat{p}\cdot (1-\hat{p})+(N-1)\cdot E^2}=\frac{200\cdot 1.96^2\cdot 0.38\cdot(1-0.38)}{1.96^2\cdot 0.38\cdot(1-0.38)+(200-1)\cdot0.025^2}\approx176.$
