Election Poll Margin of Error via the CLT

A pollster surveys $n = 1{,}600$ voters to estimate the proportion $p$ supporting a candidate. Suppose the true proportion is $p = 0.5$. Using the CLT, find the probability that the sample proportion $\hat{p}$ falls within $0.02$ of the true value, i.e., approximate $P(|\hat{p} - 0.5| < 0.02)$.

You may use $\Phi(1.60) \approx 0.9452$.