Definition 10.4.1. Confidence Intervals for p.
Given a point estimate \(\tilde{p}\) for p, a confidence interval for p is a range of values which contains the actual value of p with high probability. In notation, a two-sided confidence interval for p is of the form
\begin{equation*}
\tilde{p} - E_1 \lt p \lt \tilde{p} + E_2
\end{equation*}
with
\begin{equation*}
P(\tilde{p} - E_1 \lt p \lt \tilde{p} + E_2) = 1 - \alpha
\end{equation*}
where \(\alpha\) is near 0 and \(E_k \gt 0\text{.}\) One-sided confidence intervals for p can be similarly described
\begin{equation*}
P( p \lt \tilde{p} + E_2) = 1 - \alpha
\end{equation*}
or
\begin{equation*}
P(\tilde{p} - E_1 \lt p) = 1 - \alpha.
\end{equation*}