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We are given that the $\hat{p}$denote the proportion in the sample who say that they support the increase.

We need to find that sample would be needed to guarantee that the standard deviation of $\hat{p}$is no more than $0.01$.

First of all , we will use standard deviation for the estimation because it is proportion to population,

$SD$$\left(\hat{p}\right)=\sqrt{\frac{p脳\left(1-p\right)}{n}}$, here $\hat{p}$ denote the proportion in the sample who say that they support the increase, we have to find out $n$.

Now standard deviation should not be more than$0.01$

$\sqrt{\frac{0.4脳0.6}{n}}鈮�0.01鈬�\frac{0.4脳0.6}{n}={0.01}^2$

We will multiply $n$ on other side, by simplifying we will find value of $n$;

$n鈮�\frac{0.4脳0.6}{{0.01}^2}鈬�n鈮�2400$

If the sample size is at least $2400$ then condition would be satisfied which is standard deviation would be $0.01$ or less . So answer is $2400$.