Question: Let \({{X}_{1}},{{X}_{2}},....,{{X}_{n}}\) be \(N\left( \mu ,{{\sigma }^{2}} \right)\) independent random variables. Let \(\bar{X}=\frac{1}{n}\sum\limits_{i=1}^{n}{{{X}_{i}}}\) and \(\alpha \in \left( 0,1 \right)\)

(a) Assume that n = 25 and \(\Pr \left( |\bar{X}-\mu |>2{{z}_{\alpha /2}} \right)=\alpha \). Find \({{\sigma }^{2}}\)

(b) Assume that n = 25 and that \({{\sigma }^{2}}\) is unknown. Let \({{S}^{2}}=\frac{1}{24}\sum\limits_{i=1}^{25}{{{\left( {{X}_{i}}-\bar{X} \right)}^{2}}}\). Find c such that \(\Pr \left( \frac{5|\bar{X}-\mu |}{S}>c \right)=\alpha \)