\documentclass[10pt]{article} 
%\usepackage{amsbsy} % for \boldsymbol and \pmb 
\usepackage{graphicx} % To include pdf files!
\usepackage{amsmath}
\usepackage{amsbsy}
\usepackage{amsfonts}
%\usepackage[colorlinks=true, pdfstartview=FitV, linkcolor=blue, citecolor=blue, urlcolor=blue]{hyperref} % For links
\usepackage{fullpage}           % Good for US Letter paper

\topmargin=-0.75in
\textheight=9.5in
\usepackage{fancyhdr} 
\renewcommand{\headrulewidth}{0pt}  % Otherwise there's a rule under the header
\setlength{\headheight}{15.2pt}
\fancyhf{}
\pagestyle{fancy} 

\cfoot{Page \thepage {} of 2}
% \pagestyle{empty} % No page numbers


\begin{document}
\enlargethispage*{1000 pt} 

\begin{flushright}
Name \underline{\hspace{60mm}} \\ 
$\,$ \\
Student Number \underline{\hspace{60mm}}
\end{flushright}   
\vspace{2mm}

\begin{center}   
{\Large \textbf{STA 431 Quiz 5}}\\
\vspace{1 mm}
\end{center}

\noindent
% \emph{Calculators are allowed.} 

%\vspace{3mm}

\begin{enumerate} 

\item \label{R} \emph{Do not answer this question if you do not have a printout for Question 7 of this weeks's assignment.} In Question 7, you tested $H_0: \alpha_1=0$.  

    \begin{enumerate}
        \item (2 points) Write the value of the test statistic and the $p$-value in the space below. These are numbers from your printout. On your printout, circle the numbers and write ``Question~\ref{R}" beside them.  \vspace{40mm}
        \item (2 points) In terms of the influence of smoking on cancer (which is the point of all this), what do you conclude from this test? If a conclusion is justified, draw a directional conclusion. \emph{Be guided by the $\alpha=0.05$ significance level, but do not mention it.} You have a lot mote room than you need.
    \end{enumerate} 

\newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item In one version of the instrumental variables model, the instrumental variable $z$ has a direct influence on the explanatory variable $x$. The model equations and path diagram are

% Put model equations and path diagram side by side.
\begin{tabular}{cc}
\raisebox{.5in}{\parbox{2in} {
\begin{eqnarray*}
    X_i & = & \alpha_1 + \beta_1Z_i +\epsilon_{i1} \\ 
    Y_i & = & \alpha_2 + \beta_2X_i +\epsilon_{i2} \\
\end{eqnarray*}    
}} % End parbox and then raisebox
& \includegraphics[width=3in]{InstruVarB}
\end{tabular}

where $E(Z_i)=\mu_z$, $Var(Z_i) = \sigma^2_z$, $E(\epsilon_{i1}) = E(\epsilon_{i2}) = 0$, $Var(\epsilon_{i1})=\psi_1$, $Var(\epsilon_{i2})=\psi_2$ and $Cov(\epsilon_{i1},\epsilon_{i2}) = \psi_{12}$.

    \begin{enumerate}
        \item (2 points) Calculate $\sigma_{12} = Cov(Z_i,X_i)$. Show a little work and \textbf{circle your answer}. 
        \vspace{65mm}
        \item (2 points) Calculate $\sigma_{13} = Cov(Z_i,Y_i)$. Show a little work and \textbf{circle your answer}. 
        \vspace{75mm}
        \item (2 points) Give the formula for a Method of Moments estimate of the parameter $\beta_2$ in terms of $\widehat{\sigma}_{ij}$ values. % \textbf{Circle your answer}.  
\vspace{20mm}
    \end{enumerate}

\end{enumerate} % End of quiz questions

\noindent
\textbf{Please attach your printout to the quiz paper. The printout should show your \emph{complete} R input and output.} Make sure your name and student number appear on the printout.

\end{document}

% From a lecture slide (Omitted variables)
{\LARGE $\boldsymbol{\Sigma} =$}                      
\renewcommand{\arraystretch}{2.0}
\begin{tabular}{|c||c|c|c|}  \hline
    &   $X$             &             $Y$                  &       $Z$     \\ \hline
$X$ & $\beta_1^2\sigma^2_z+\sigma^2_1$ & $\beta_2(\beta_1^2\sigma^2_z+\sigma^2_1)+c$ & $\beta_1\sigma^2_z$ \\ \hline
$Y$ & $\cdot$ & $\beta_1^2\beta_2^2\sigma^2_z + \beta_2^2\sigma^2_1 + 2\beta_2c + \sigma^2_2$ & $\beta_1\beta_2\sigma^2_z$ \\  \hline
$Z$ & $\cdot$ & $\cdot$ & $\sigma^2_z$ \\  \hline
\end{tabular} 
\renewcommand{\arraystretch}{1.0}