Limits for a large sample exact coefficient \(\gamma \) confidence interval \(\left( {A,B} \right)\) for the distribution with parameters \(\mu \) and known \({\sigma ^2}\) is given by
\(\begin{align}A &= {{\bar X}_n} - {\Phi ^{ - 1}}\left( {\frac{{1 + \gamma }}{2}} \right)\frac{\sigma }{{\sqrt n }},\\B &= {{\bar X}_n} + {\Phi ^{ - 1}}\left( {\frac{{1 + \gamma }}{2}} \right)\frac{\sigma }{{\sqrt n }},\end{align}\)
Where \(\gamma \in \left( {0,1} \right)\)
In this case,
\(\begin{align}A &= \arcsin \left( {\sqrt {{{\bar X}_n}} } \right) - {\Phi ^{ - 1}}\left( {\frac{{1 + \gamma }}{2}} \right)\frac{1}{{\sqrt n }}\\B &= \arcsin \left( {\sqrt {{{\bar X}_n}} } \right) + {\Phi ^{ - 1}}\left( {\frac{{1 + \gamma }}{2}} \right)\frac{1}{{\sqrt n }}\end{align}\)
Because \(P\left( {A < \arcsin \left( {\sqrt p } \right) < B} \right) \approx \gamma \)
This yields an approximate confidence interval for p with confidence \(\gamma \)
\(\begin{align}{A_1} &= {\sin ^2}\left( {\arcsin \left( {\sqrt {{{\bar X}_n}} } \right) - {\Phi ^{ - 1}}\left( {\frac{{1 + \gamma }}{2}} \right)\frac{1}{{\sqrt n }}} \right)\\{B_1} &= {\sin ^2}\left( {\arcsin \left( {\sqrt {{{\bar X}_n}} } \right) + {\Phi ^{ - 1}}\left( {\frac{{1 + \gamma }}{2}} \right)\frac{1}{{\sqrt n }}} \right)\end{align}\)
Now, the interval \(\left( {{A_1},{B_1}} \right)\) is the \(\gamma \) coefficient for the interval p.
