L Let $Y$ Be a Bernoulli Random Variable with Success Probability $\operatorname { Pr } ( Y = 1 ) = p$

Essay

L Let $Y$ be a Bernoulli random variable with success probability $\operatorname { Pr } ( Y = 1 ) = p$ , and let $Y _ { 1 } , \ldots , Y _ { n }$ be i.i.d. draws from this distribution. Let $\hat { p }$ be the fraction of successes (1s) in this sample. Given the following statement
$$\operatorname { Pr } ( - 1.96 < z < 1.96 ) = 0.95$$
and assuming that $\hat { p }$ being approximately distributed $N \left( p , \frac { p ( 1 - p ) } { n } \right)$ , derive the $95 \%$ confidence interval for $p$ by solving the above inequalities.