Answer
a. $801$ homes
b. $1068$ homes
Work Step by Step
Given $\hat p=0.25, \hat q=1-\hat p=0.75 $
a. At a 95% confidence the critical z-value is $z_{\alpha/2}=1.96 $
With $E=0.03,$ the equation $E=z_{\alpha/2}\times\sqrt {\frac{\hat p\hat q}{n}}$
becomes $1.96\times\sqrt {\frac{0.25\times0.75}{n}}=0.03$
Thus $n=(\frac{1.96}{0.03})^2\times0.25\times0.75=800.3\approx801$ (round up to the next integer here)
b. With no estimate of the sample proportion, we use $\hat p=\hat q=0.5 $
With $E=0.03,$ the equation $E=z_{\alpha/2}\times\sqrt {\frac{\hat p\hat q}{n}}$
becomes $1.96\times\sqrt {\frac{0.5\times0.5}{n}}=0.03$
Thus $n=(\frac{1.96}{0.03})^2\times0.5\times0.5=1067.1\approx1068$ (round up to the next integer here)
