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\title{$t$ Procedures}
\author{Worksheet 22\footnote{\copyright 2019 Joseph C Watkins}}
\date{}
\maketitle

\begin{enumerate}
\item Genetic evolution is based on mutation. Consequently, one fundamental question in evolutionary biology is the rate of {\em de novo} mutations, i.\,e., those mutations that neither parent possessed. For example, these mutations can occur in the germs cells, the egg or the sperm of the parent. To investigate this question in humans, Kong et al, sequenced the entire genomes of 78 Icelandic trios (mother, father, offspring) and recorded  the number of {\em de novo} mutations in the offspring.
The mean $\bar x = 63.24$ and the standard deviation $s = 14.18$.
\begin{enumerate}
\item Give a 95\% confidence interval for the mean number of mutations.
\item A mutation rate of one mutation in one-hundred million base pairs would lead to a mean number of {\em de novo} mutations of 60. State a hypothesis test for this situation.
\item What does the confidence interval in (a) say about a test with a 5\% significance level?
\item Give the $t$ statistic for this situation and state whether or not the test can be rejected at the 1\% level. Explain your answer.
\item Using the standard deviation above, fill in the following table for the power for the given alternatives.
\begin{center}
\begin{tabular}{cccc}
alternative&significance&&number of \\
hypothesis&level&power&observations \\ \hline
63&0.05&0.80\\
63&0.02&0.80 \\
63&0.05&&80 \\
63&0.05&&240 \\
62&0.05&0.80 \\
62&0.02&0.80 \\
62&0.05&&80 \\
62&0.05&&240 \\ \hline
\end{tabular}
\end{center}
\end{enumerate}
\item July and August are the strongest months for monsoon rains in Tucson. Your friend suspects that more rain takes place in July.
\begin{enumerate}
\item State a hypothesis appropriate to you friend's suspicion.
\item For the years 2007 to 2016, here are the data for rainfall total in inches. 
\begin{small}
\begin{center}
\begin{tabular}{c|ccccccccccc}
&2007&2008&2009&2010&2011&2012&2013&2104&2015&2016 \\ \hline
July&4.38&4.03&2.15&1.44&1.69&0.71&1.44&4.58&2.06&3.32 \\ 
August&4.27&1.69&0.86&2.89&1.03&0.00&1.85&3.32&1.80&1.09
\end{tabular}
\end{center}
\end{small}
\item What test procedure will you use in this case? Explain your answer.
\item Compute the appropriate $t$ statistics for the hypothesis in part (a).
\item What is the conclusion concerning your hypothesis? Use you answer in part (c) to explain your conclusion.
\item If you incorrectly used the two-sample $t$ procedure, how  does the $p$-value change?
\end{enumerate}
\end{enumerate}
\end{document}