Chapter 7, Survey Sampling
Solution 22
Since the confidence interval is $\, \bar X \pm s_{\bar X} \,$, thus by comparing it with $\, \bar X \pm z(\alpha/2)s_{\bar X} \,$, it follows that $\, z(\alpha/2) =1 \,$.
Thus we can compute $\, \alpha \,$ as:
$$
\,
\begin{align*}
&\Rightarrow z(\alpha/2) = 1 \\
&\Rightarrow \Phi^{-1}(1-\alpha/2) = 1 && \text{Using the definition of $z$} \\
&\Rightarrow \alpha = 2(1-\Phi(1)) \\
&\Rightarrow \alpha = 0.3174
\end{align*}
\,
$$
Thus the size of confidence interval is $\, 1-\alpha = 0.6826 \,$.
$$\tag*{$\blacksquare$} $$