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\title{Simple Hypotheses\footnote{\copyright 2019 Joseph C Watkins}}
\author{Worksheet 19}
\date{}
\maketitle

\begin{enumerate}
\item The body temperature in degrees Fahrenheit of $n=40$ randomly chosen healthy adults is measured. The standard deviation $\sigma$ is known to be 0.68 degrees Fahrenheit. The sample mean for the measurements is $\bar x=98.37$.

\begin{enumerate}
\item Find a 99\% confidence interval for the mean body temperature and explain its meaning.
\item Consider the simple hypothesis for the mean body termperature in degrees Fahrenheit
$$H_0: \mu=98.6\quad\mbox{versus}\quad H_1:\mu=98.4.$$
Give the critical value for $\bar x$, the sample mean body temperature when the significance level  $\alpha=0.05,0.02$, and 0.01.
\item Do these critical values increase, decrease, or stay the same in the number of healthy adults chosen increases to $n=50$? Explain your answer.
\item The data have sample mean $\bar x = 98.37$. For $n=40$ and each value of $\alpha$, report whether or not we reject the null hypothesis? Explain your answer.
\item Find the power of the test in part (b) for each significance level.
\begin{center}
\begin{tabular}{c||ccc}
&\multicolumn{3}{c}{significance} \\ 
&0.05&0.02&0.01 \\ \hline
power&&&\\
\end{tabular}
\end{center}
\item Fill in the following table with the number of observations $n^*$ necessary for the given level of significance $\alpha$ and power $1-\beta$.
\begin{center}
\begin{tabular}{c||ccc}
&\multicolumn{3}{c}{significance} \\ 
power&0.05&0.02&0.01 \\ \hline
0.80&&&\\
0.90&&&\\
0.95&&&
\end{tabular}
\end{center}
\item Which value $n^*$ is the highest? lowest? Explain your answer.

\item For $\alpha = 0.01$, indicate the power using the critical value for the test and drawing the density of the sample mean $\bar X$ for the null and alternative hypothesis.
\end{enumerate}
\end{enumerate}
\end{document}