Let X1,X2,…,X25 be a random sample from a N(\mu,1) distribution where \mu \in \mathbb{R} is unknown. Consider testing of the hypothesis H_0 : \mu=5.2 against H_1 : \mu=5.6 . Then null hypothesis is reject if and only if \frac{1}{25} \sum_{i=1}^{25} X_i >k , for some constant k. IF the size of the test is 0.05, then the probability of type-II error equals __________________________ (round off to 2 decimal places)
Since, X_1,X_2,\dots,X_n \overset{iid}{\sim} N(\mu,1) , \bar{X} = \frac{1}{25}\sum_{i=1}^{25}X_i \sim N(\mu,1/25) \\
\begin{align*}
So, & \\
& P(\bar{X} > k / H_0) = 0.05 \\
\implies & P(Z > 5(k-5.2) / Z \sim N(0,1) ) =0.05 \\
\implies& \Phi(5(k-5.2)) = 0.95 \\
\implies& 5(k-5.2) = 1.645 \\
\implies& k = 0.329+5.2=5.529
\end{align*} \\
\textit{Thus, the probability of type 2 error is } \\
\begin{align*}
P(\bar{X}<5.829 / H_1) &= P(Z< 5(5.529-5.6)) \\
&=P(Z< -0.355) \\ &= \Phi(-0.355) = 1 - \Phi(0.355) \\
&=1-0.6387 = 0.3613
\end{align*}