2. Suppose that nine observations are selected at random from the normal distribution with unknown mean \( \mu \) and unknown variance \( \sigma^{2} \), and for these nine observations it is found that \( \bar{X}_{n}=22 \) and \( \sum_{i=1}^{n}\left(X_{i}-\bar{X}_{n}\right)^{2}=72 \).

a. Carry out a test of the following hypotheses at the level of significance \( 0.05 \) :

\[

\begin{array}{ll}

H_{0}: & \mu \leq 20 . \\

H_{1}: & \mu \leq 20 .

\end{array}

\]

b. Carry out a test of the following hypotheses at the level of significance \( 0.05 \) by using the two-sided \( t \) test:

\[

\begin{array}{ll}

H_{0}: & \mu=20, \\

H_{1}: & \mu \neq 20 .

\end{array}

\]

c. From the data, construct the observed confidence interval for \( \mu \) with confidence coefficient \( 0.95 . \)

a. Carry out a test of the following hypotheses at the level of significance \( 0.05 \) :

\[

\begin{array}{ll}

H_{0}: & \mu \leq 20 . \\

H_{1}: & \mu \leq 20 .

\end{array}

\]

b. Carry out a test of the following hypotheses at the level of significance \( 0.05 \) by using the two-sided \( t \) test:

\[

\begin{array}{ll}

H_{0}: & \mu=20, \\

H_{1}: & \mu \neq 20 .

\end{array}

\]

c. From the data, construct the observed confidence interval for \( \mu \) with confidence coefficient \( 0.95 . \)

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When \mu_{0}=20, the statistic U given by Eq. (9.5.2) has a t distribution with 8 degrees of freedom. The value of U in this exercise is 2 .(a) We would reject H_{0} if U \geq 1.860. Therefore, we reject H_{0}.(b) We would reject H_{0} if U \leq-2.306 or U \geq 2.306. Therefore, we don't reject H_{0}.(c) We should include in the confidence interval, all values of \mu_{0} for which the value of U given by Eq. (9.5.2) will lie between -2.306 and 2.306 . These values form the interval 19.694<\mu_{0}<24.306. ...