Answer
The power of test is 0.2090.
Work Step by Step
If the true population mean is 48.9, then the average of the sampling distribution is
\[{{\mu }_{{\bar{x}}}}=0.49\]
and standard deviation is
\[\begin{align}
& {{\sigma }_{{\hat{p}}}}=\frac{S}{\sqrt{n}} \\
& =\frac{6}{\sqrt{24}} \\
& =1.22474
\end{align}\]
The Type II error can be calculated as follows:
\[\begin{align}
& \beta =P\left( \text{Type II error} \right) \\
& =P\left( \text{not rejecting }{{H}_{0}}\text{ when }{{H}_{0}}\text{is true} \right) \\
& =P\left( \bar{x}>\frac{47.9-48.9}{{6}/{\sqrt{24}}\;} \right) \\
& =P\left( z>\frac{-1}{1.22474} \right)
\end{align}\]
\[\begin{align}
& =P\left( z>-0.81 \right) \\
& =0.7910 \\
\end{align}\]
Probability of \[{{H}_{0}}\] rejection when \[{{H}_{1}}\] is true is\[1-\beta \]. The value \[1-\beta \] is referred to as the power of the test. The power of test is the probability that null hypothesis is correctly rejected. It is calculated as follows:
\[\begin{align}
& 1-\beta =1-0.7910 \\
& =0.2090
\end{align}\]
Hence, the power of test is 0.2090.
