12. Suppose that a random sample \( X_{1}, \ldots, X_{n} \) is to be taken from the normal distribution with unknown mean \( \mu \) and unknown variance \( \sigma^{2} \), and the following hypotheses are to be tested:

\[

\begin{array}{ll}

H_{0}: & \mu \leq 3, \\

H_{1}: & \mu>3 .

\end{array}

\]

Suppose also that the sample size \( n \) is 17 , and it is found from the observed values in the sample that \( \bar{X}_{n}=3.2 \) and \( (1 / n) \sum_{i=1}^{n}\left(X_{i}-\bar{X}_{n}\right)^{2}=0.09 \). Calculate the value of the statistic \( U \), and find the corresponding \( p \)-value.

\[

\begin{array}{ll}

H_{0}: & \mu \leq 3, \\

H_{1}: & \mu>3 .

\end{array}

\]

Suppose also that the sample size \( n \) is 17 , and it is found from the observed values in the sample that \( \bar{X}_{n}=3.2 \) and \( (1 / n) \sum_{i=1}^{n}\left(X_{i}-\bar{X}_{n}\right)^{2}=0.09 \). Calculate the value of the statistic \( U \), and find the corresponding \( p \)-value.

Community Answer

Honor CodeSolved 1 Answer

See More Answers for FREE

Enhance your learning with StudyX

Receive support from our dedicated community users and experts

See up to 20 answers per week for free

Experience reliable customer service

Get Started

U_{1} has the distribution of X / Y where X has a normal distribution with mean \psi and variance 1 , and Y is independent of X such that m Y^{2} has the \chi^{2} distribution with m degrees of freedom. Notice that -X has a normal distribution with mean -\psi and variance 1 and is independent of Y. So U_{2} has the distribution of -X / Y=-U_{1}. So\operatorname{Pr}\left(U_{2} \leq-c\right)=\operatorname{Pr}\left(-U_{1} \leq-c\right)=\operatorname{Pr}\left(U_{1} \geq c\right) . ...