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\title{Stat 185 - Final Exam}
\author{}
\date{}

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\begin{flushright}
	Name:\underline{\hspace{2.2in}}
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\noindent Please write all your solutions (answers with organized supporting work) on separate paper, put your name on this sheet, and staple it all together when you turn it in. \\
I don't plan to grade anything written on the exam sheet itself.

\section*{The basics}

If you score 92\% or higher on this section, then you \textit{will} pass the class.

\begin{enumerate}
	\item Suppose we randomly select 200 college students and measure their heights in feet. We find that our data has 
	an average of $5.8$ with a standard deviation of $0.37$. We wish to write down a 95\% confidence interval for this data.
        \begin{enumerate}
 		\item Find the standard error associated with this sample.
		\item Write down a 95\% confidence interval for the average height of college students based on this data.
        \end{enumerate}
        
	\item Suppose we randomly select 4 college students, measure their heights in feet and find them to be
	   \begin{center}\begin{tabular}{c|c|c|c}
          \hline
          5.7 & 5.2 & 6.1 & 5.9
        \end{tabular}\end{center}
        \begin{enumerate}
        	\item Write down a formula showing that the mean of these heights is $5.725$.
		\item Write down a formula showing that the standard deviation of these heights is approximately $0.386$
		\item Find the standard error associated with this sample.
		\item Write down a 90\% confidence interval for the average height of college students based on this data.
        \end{enumerate}

	\item Supposedly, approximately 11\% of the population is left handed, but we think it might be higher than that. 
	Suppose that in a random sample of 85 people, we find 18 left handers. Let's use this data to explore the question 
	of whether the 10\% estimate is truly correct vs whether there might be more than 11\%.
	\begin{enumerate}
		\item Write down the Null and Alternative Hypotheses for this problem.
		\item Compute the standard error, test statistic, and $p$-value.
		\item State the conclusion of the hypothesis test and your reasons why.
	\end{enumerate}
\end{enumerate}

\section*{A bit more}

\begin{enumerate}
	\setcounter{enumi}{3}    
         \item The table below lists student enrollment by state region for all 3281 NC students. Suppose we draw a random sample of 30 of those students. Use a normal 
        distribution to estimate the probability that \text{more than} 17 of those 30 students are 
        \textit{not} from Eastern NC.
        
        \begin{center}
            \begin{tabular}{l||c|c|c|c}
            Region & Western NC & Piedmont & Eastern NC & Total  \\
            \hline
            Enrollment & 1508       & 1541     & 232        & 3281   \\
            \end{tabular}
	\end{center}


	
	\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 three exams for
	both classes and found the following:
	\begin{description}
		\item[Section A:] Mean=169, std dev = 59, number of students = 30
		\item[Section B:] Mean=204, std dev = 52, 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 Suppose you ask two friends to play rock-paper-scissors and count the 
	times each option is played. The table belowsummarizes the data.
	Use a one-way $\chi^2$-test to evaluate whether these data support the hypothesis that players choose between these three options randomly, 
	or if certain options are favored above others.
	
\begin{center}
\begin{tabular}{c c c}
Rock    & Paper & Scissors   \\
\hline
43      & 21    & 35    
\end{tabular}
\end{center}

%	\item Data relating favorite outdoor activities is shown in table 1 below. Suppose I run a \texttt{chisq.test} command on this table and generate the following output:
%	\begin{verbatim}
%		Pearson's Chi-squared test
%		X-squared = 10.424, df = 6, p-value = 0.1079
%	\end{verbatim}
%	\begin{enumerate}
%		\item Draw a picture illustrating how \texttt{X-squared} and \texttt{p-value} fit together.
%		\item What hypothesis statement is associated with this test?
%		\item How should you interpret the $p$-value?
%	\end{enumerate}
\end{enumerate}

\section*{A regression problem}

\begin{enumerate}     
	\setcounter{enumi}{6}
	\item I ran a linear regression on a random sample of 100 racers in the Peachtree Road Race to determine if there is a relationship between race time and age.
	The results are shown in the following table:
	
	\begin{center}\begin{tabular}{rrrrr}
  \hline
 & Estimate & Std. Error & t value & Pr($>$$|$t$|$) \\ 
  \hline
(Intercept) & 46.31345   & 1.14506  & 40.446  &  <2e-16 \\ 
  age & 0.03906 &   0.03062  & 1.276   & 0.205 \\ 
   \hline
\end{tabular}\end{center}
	\vspace{0cm}
        \begin{enumerate}
        	\item Write down the hypothesis statement that we would check with this linear model.
             \item Write out the formula for linear model.
             \item What does the model predict for the race time of a runner who is 32 years old?
             \item State the conclusion of the hypothesis test from part (a).
        \end{enumerate}
     
\end{enumerate}

%\begin{table}[h]
%\centering
%\begin{tabular}{l|cccc}
%& NE & SE & NW & SW \\
%\hline             
%Mountain Bike & 24 & 52 & 36 & 37 \\
%Road Bike     & 32 & 42 & 38 & 45 \\
%Rock Climb    & 19 & 37 & 28 & 55
%\end{tabular}
%\caption{Favorite outdoor activities by region}
%\label{my-label}
%\end{table}

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