each day and measure the volume of liquid in each cup. Past measurements have shown that the standard deviation of soda dispensed by this machine is about 0.24 ounces.
(a) Define the parameter and speclfy the hypotheses.
(b) Describe the sampling distribution of the sample mean filling volume assuming the null hypothesis is true and that the true standard deviation is 0.24 ounces. Skatch the sampling distribution.
(c) Using a significance level of \( 1 \% \), what values of \( x \) would provide sufficient evidence to reject the null hypothesis? Identify this rejection region on your sampling dilstribution from part (b).
(d) Recognizing that the sample mean changes from sample to-sample and that the null hypothesis may or may not be true, describe the four outcomes for the conclusion of this inference procedure. Then describe a consequence for each of these outcomes.
\begin{tabular}{|c|c|c|c|}
\hline \multirow{2}{*}{\multicolumn{2}{|c|}{ Table of error types }} & \multicolumn{2}{|c|}{ Null hypothesis \( \left(H_{0}\right) \) is } \\
\hline & & True & False \\
\hline \multirow{2}{*}{\begin{tabular}{c}
Decision \\
about null \\
hypothesis \( \left(H_{0}\right) \)
\end{tabular}} & Reject & \begin{tabular}{c}
Outcome 1 \\
Type l error \\
(talse positive) \\
(probability = \\
\end{tabular} & \begin{tabular}{c}
Outcome 3 \\
Correct inference \\
(true positive) \\
(probability = \( 1-\beta \) )
\end{tabular} \\
\hline & \begin{tabular}{l}
Fail to \\
reject
\end{tabular} & \begin{tabular}{c}
Outcome 2 \\
Correct inference \\
(true negative) \\
(probabillity \( =1-\alpha \) )
\end{tabular} & \begin{tabular}{c}
Outcome 4 \\
Type Il error \\
(false negative) \\
(probability \( =\beta \) )
\end{tabular} \\
\hline
\end{tabular}
(e) Identify the four outcomes on your two sampling distributions in part (e).
(g) Find the probabilties of the four outcomes. What do we call these probabilities?
(h) List two reasons why the power of this test is so low.
(i) List two ways to increase the power of this test.