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\title{Stat 185 - Exam 2}
\author{}
\date{}

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\begin{flushright}
	Name:\underline{\hspace{2.2in}}
\end{flushright}

\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.

\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 I've got a bunch of 12 sided dice with five sides labeled 1, two sided labeled 2, and five sides labeled 3
	\begin{enumerate}
		\item Suppose I roll one such die. What  is the probability that I roll a 2 or a 3?
		\item Suppose I roll two such dice. What  is the probability that I roll a 2 and a 3?
		\item Suppose I roll 1 such die. Write down a specific formula to show that the expected value of my roll is 3.
		\item Suppose I roll 100 such dice and add the results. What is the expected value of my sum?
%		\item Suppose I roll 1 such die. Write down a specific formula to show that the standard deviation of my roll is $\sqrt{5/6}$.
%		\item Suppose I roll 100 such dice and add the results. What is the standard deviation of my sum?
	\end{enumerate}


%	\item Consider a random variable $X$ with the following distribution:
%	\begin{center}
%        \begin{tabular}{c||c|c|c}
%            $X$ & 0 & 1 \\
%    		\hline
%            $P(X)$ & $1-p$ & $p$
%        \end{tabular}
%        \end{center}
%        \begin{enumerate}
%        		\item Write down the specific formula to show that $E(X) = p$.
%		\item Write down the specific formula to evaluate $\sigma(X)$. \\
%		Note that this expression for $\sigma(X)$ simplifies to $\sqrt{p(1-p)}$ but you need not perform this simplification.
%        \end{enumerate}
        

        \item Table \ref{table-unca_enrollment} displays in-state UNCA enrollments for last Fall by region.        
        Suppose we pick a random sample of 50 of those 3281 total students. What is the probability that exactly 28 of those 50 students are 
        \textit{not} from Western NC?
        
        \item Referring again 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 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.
	\newpage
	\item 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}




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