91Ó°ÊÓ

Here\(n = 12{\rm{ and }}{X_1}, \cdots ,{X_n}\~N\left( {\mu ,{\sigma ^2}} \right)\)where both\(\mu \)and\({\sigma ^2}\)are unknown.

We wish to test

\(\begin{array}{l}{H_0}:\mu \ge {\mu _0}\;\;\;\\{H_1}:\mu < {\mu _0}\end{array}\)

Here\({\mu _0} = 3\). The test statistic is

\(T = \frac{{\sqrt n \left( {\bar X - {\mu _0}} \right)}}{{\sqrt {\frac{1}{{n - 1}}} _{i = 1}^n{{\left( {{X_i} - \bar X} \right)}^2}}}\)

Which follows a\(t\)distribution with\(n - 1 = 11\)degrees of freedom. We reject the null for small values of\(T\), where the cutoff for being small is the test statistic being smaller than\(T_{n - 1}^{ - 1}\left( {{\alpha _0}} \right) = - 3.106\)

Thus, we reject the null if

\(\begin{array}{c}\bar X \le 3 - \frac{{3.106}}{{12}}\sqrt {\frac{1}{{11}}_{i = 1}^n{{\left( {{X_i} - \bar X} \right)}^2}} \\ = 3 - 0.078\sqrt {\underbrace {12}_{i = 1}{{\left( {{X_i} - \bar X} \right)}^2}} \end{array}\)
Reject the null if\(\bar X \le 3 - 0.078\sqrt {\prod\limits_{i = 1}^{12} {{{\left( {{X_i} - \bar X} \right)}^2}} } \)