Question 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.
LPSSXJ The Asker · Probability and Statistics
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.
Transcribed Image Text: 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.
More Transcribed Image Text: 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.
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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) . ...