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\title{Stat 185 - Problems off of past exams}
\author{}
\date{}

\begin{document}
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\noindent Our final exam will be comprehensive and based off of past exams. Treat the following list of problems list of problems like a review sheet!

\begin{itemize}
\item Exam I

\begin{figure}[htbp]
\begin{center}
\includegraphics*[width=4in]{yards_hist.pdf}
\caption{A histogram of total yards gained by each team for the 2016 season}
\label{fig:hist}
\end{center}
\end{figure}


        \begin{enumerate}
        	\item[5.] On average, the 127 college football teams in Division I gained 5322 yards throughout the 2016 season 
		with a standard deviation of 887 yards.
            \begin{enumerate}
              \item Use a normal model to estimate the number of teams that rushed more than 6000 yards.
              \item A histogram for this data is shown in figure \ref{fig:hist}. Based on the histogram, do you see any issues with using a 
              normal model for this estimate?
            \end{enumerate}
        \end{enumerate}

\item Exam 2
    \begin{table}
            \centering
            \begin{tabular}{l||c|c|c|c}
            Region & Western NC & Piedmont & Eastern NC & Total  \\
            \hline
            Enrollment & 1508       & 1541     & 232        & 3281   \\
            \end{tabular}
            \caption{In state UNCA Enrollment by NC Region}
            \label{table-unca_enrollment}
    \end{table}
    
	\begin{enumerate}
         \item[3.] Referring to our random sample of 50 of 3281 total students from table \ref{table-unca_enrollment}, use a normal 
        distribution to estimate the probability that \text{more than} 28 of those 50 students are 
        \textit{not} from Western NC.

      	\item[4.] Suppose now that we are interested in the proportion of in-state UNCA students who are from outside Western NC 
        	\textit{this} year. Since data on the entire student body is not yet available for this year, we draw a simple random sample of
        	64 in-state UNCA students and find that 34 of them are from outside Western NC. Use this data to write down a 95\% confidence interval 
        	for the proportion of in-state UNCA students who from outside Western NC.

        	\item[5.] UNCA claims that 90\% of it's students are from North Carolina but we suspect that it might be less than that. 
		Suppose we draw a random sample
        	of 50 UNCA students and find that 41 of them are in state. Let's use this to explore the claim that 90\% of UNCA students are in-state vs the possibility
        	that fewer than 90\% of UNCA students are in-state.
        	\begin{enumerate}
        		\item Construct appropriate null and alternative hypotheses to explore our question.
        		\item Compute the associated $p$-value.
        		\item Based on a 95\% level of confidence, what is the conclusion of our hypothesis test?
        	\end{enumerate}
        \end{enumerate}

\item Exam 3
\begin{enumerate}
	\item In planning for an early afternoon, outside wedding ceremony on November 18, I looked up the temperature at 12:54PM for the last nineteen November $18^{\text{th}}$s.
	I found an average temperature of 52.3 degrees with a standard deviation of 11.8 degrees. Use this data to write down a 95\% confidence interval for the temperature.

	\item An organic farmer sells bags of Brussels sprouts labeled as one pound each. 
	Let's use a $t$-test to investigate the farmer's claim that bags each contain one pound.
	I purchased one bag each of 4 weeks during the Summer farmer's market, 
	weighed them at home and recorded the following weights:
        \begin{center}\begin{tabular}{c|c|c|c}
          \hline
          1.05 & 1.15 & 1.2 & 1.1
        \end{tabular}\end{center}
        \begin{enumerate}
            	\item Write down the hypothesis statement for the problem.
            	\item Write down a formula showing that the mean is $\mu=1.125$
		\item Write down a formula showing that the standard deviation is $\sigma = 0.06454972$.
		\item Assuming the mean $\mu_0 = 1$ is correct, compute the $t$-score for the observed mean.
		\item Use a $t$-table to determine the outcome of the hypothesis test.
        \end{enumerate}


	\item A recent study surveyed people's attitudes on the relative importance of economic issues. In particular, the study examined the question - what is more important:
	\begin{itemize}
		\item Reducing the federal deficit or
		\item Cutting taxes for corporations
	\end{itemize}
	The results are summarized in table 1.
        \begin{table}[h]
            \centering
            \label{tbl:deficit_vs_taxes}
            \begin{tabular}{l || c c}
                 & Democrats & Republicans  \\
                 \hline
                Deficit over taxes & 97 & 66  \\
                Taxes over deficit & 43 & 48  \\
            \end{tabular}
            \caption{Views on deficit reduction over tax reduction}
        \end{table}
	Let $p_D$ denote the proportion of democrats who think that reducing the deficit is the more important of the two issues and let $p_R$
	denote the proportion of republicans who think that reducing the deficit is the more important of the two issues. We suspect that
	$p_D<p_R$ so let's use a hypothesis test to examine this question.
	\begin{enumerate}
		\item Compute the observed proportions $\hat{p}_R$ and $\hat{p}_D$, as well as the difference $$\hat{p} = \hat{p}_D-\hat{p}_R.$$
		\item Compute the standard error and the test statistic.
		\item Use a normal table to compute the $p$-value.
		\item State the conclusion of the test.
		\item Why was it OK to use a normal table?
	\end{enumerate}
	
	\item A statistics professor teaching two sections of the same introductory statistics course is concerned that exam scores in one section (let's call it section A) are noticeably 
	lower than exam scores in another section (section B). To test this hypothesis, he computed the mean and standard deviation for the combined scores on the first two exams for
	both classes and found the following:
	\begin{description}
		\item[Section A:] Mean=118.4, std dev = 40, number of students = 30
		\item[Section B:] Mean=133.5, std dev = 32.7, number of students = 30
	\end{description}
	Viewing these data as samples from a random process, find
	\begin{enumerate}
		\item The difference between the two means.
		\item The associated standard error and test statistic.
		\item Compute the $p$-value using a normal distribution.
		\item From the computations, can we conclude with a 95\% level of confidence that section A scores lower than class B?
	\end{enumerate}

	\item A professor using an 
        open source introductory statistics book predicts that 60\% of the students will 
        purchase a hard copy of the book, 25\% will print it out from the web, and 15\% will 
        read it online. At the end of the semester he asks his students to complete a survey 
        where they indicate what format of the book they used. Of the 126 students, 71 said 
        they bought a hard copy of the book, 30 said they printed it out from the web, and 25 
        said they read it online.
        \begin{enumerate}
            \item State the hypotheses for testing if the professor's predictions were inaccurate.
            \item How many students did the professor expect to buy the book, print the book, and 
            read the book exclusively online?
            \item Calculate the chi-squared statistic, the degrees of freedom associated with it, 
            and the $t^*$-value for a 95\% level of confidence
            \item Based on the $t^*$-value calculated in part (d), what is the conclusion of the 
            hypothesis test? Interpret your conclusion in this context.
	\end{enumerate}	
\end{enumerate}

\end{itemize}

\end{document}  
