Show that the confidence interval (4.70) [ left(bar{X}+t_{n-1,0.025} sqrt{S^{2} / n}, quad bar{X}-t_{n-1,0.025} sqrt{S^{2} / n} ight)

Show that the confidence interval (4.70)

\[ \left(\bar{X}+t_{n-1,0.025} \sqrt{S^{2} / n}, \quad \bar{X}-t_{n-1,0.025} \sqrt{S^{2} / n}\right) \]

is formed from the pivotal quantity \(\left(\bar{X}, \sqrt{S^{2} / n}\right)\) as in equation (4.73).

\(\left(\bar{X}+t_{n-1,0.025} \sqrt{S^2 / n}, \quad \bar{X}-t_{n-1,0.025} \sqrt{S^2 / n}\right) . \tag{4.70}\)

\(\operatorname{Pr}\left(t_{n-1,0.025} \leq(\bar{X}-\mu) / \sqrt{S^2 / n} \leq-t_{n-1,0.025}\right)=95 \% . \tag{4.73}\)