Answer
Because the margin of error is inversely proportional to the square root of the sample size.
Work Step by Step
$E_1=z_{\frac{α}{2}}.\sqrt {\frac{p ̂(1-p ̂)}{n_1}}$
If $n_2=4n_1$
$E_2=z_{\frac{α}{2}}.\sqrt {\frac{p ̂(1-p ̂)}{n_2}}$
$E_2=z_{\frac{α}{2}}.\sqrt {\frac{p ̂(1-p ̂)}{4n_1}}$
$E_2=z_{\frac{α}{2}}.\sqrt {\frac{1}{4}}.\sqrt {\frac{p ̂(1-p ̂)}{n_1}}$
$E_2=\frac{1}{2}.z_{\frac{α}{2}}.\sqrt {\frac{p ̂(1-p ̂)}{n_1}}$
$E_2=\frac{1}{2}E_1$
