Answer
22457
Work Step by Step
Given $p\in(0.224,0.235)$, we can find $\hat p=\frac{0.224+0.235}{2}=0.2295, \hat q=1-\hat p=0.7705, E=\frac{0.235-0.224}{2}=0.0055 $
At a 95% confidence the critical z-value is $z_{\alpha/2}=1.96 $
With $E=0.0055,$ the equation $E=z_{\alpha/2}\times\sqrt {\frac{\hat p\hat q}{n}}$
becomes $1.96\times\sqrt {\frac{0.2295\times0.7705}{n}}=0.0055$
Thus $n=(\frac{1.96}{0.0055})^2\times0.2295\times0.7705=22456.5\approx22457$ (round up to the next integer here)
