We want to investigate the proportion of antibody possession for a specific disease among Korean adults. Let $p$ be the proportion of antibody possession in the population, and let $\hat { p }$ be the proportion of antibody possession in a sample of $n$ people randomly selected from the population. Find the minimum value of $n$ such that the probability that $| \hat { p } - p | \leq 0.16 \sqrt { \hat { p } ( 1 - \hat { p } ) }$ is at least 0.9544. (where $Z$ is a random variable following the standard normal distribution and $\mathrm { P } ( 0 \leq Z \leq 2 ) = 0.4772$.) [4 points]
We want to investigate the proportion of antibody possession for a specific disease among Korean adults. Let $p$ be the proportion of antibody possession in the population, and let $\hat { p }$ be the proportion of antibody possession in a sample of $n$ people randomly selected from the population. Find the minimum value of $n$ such that the probability that $| \hat { p } - p | \leq 0.16 \sqrt { \hat { p } ( 1 - \hat { p } ) }$ is at least 0.9544. (where $Z$ is a random variable following the standard normal distribution and $\mathrm { P } ( 0 \leq Z \leq 2 ) = 0.4772$.) [4 points]