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

\begin{center}   
{\Large \textbf{STA 431 Quiz 10}}\\
\vspace{1 mm}
\end{center}


%\vspace{3mm}

\begin{enumerate} 

% 

\item (7 points) On the left is a centered original model, and on the right is a familiar surrogate model. Let

\begin{center}
\begin{tabular}{cc}
\parbox{1.5 in}{
        \begin{eqnarray*}
        d_1 & = & \lambda_1 F + e_1 \\
        d_2 & = & \lambda_2 F + e_2 \\
        d_3 & = & \lambda_3 F + e_3 
        \end{eqnarray*} 
} % End parbox               
&
\parbox{1.5 in}{
        \begin{eqnarray*}
        d_1 & = & F^\prime + e_1^\prime \\
        d_2 & = & \lambda_2^\prime F^\prime + e_2^\prime \\
        d_3 & = & \lambda_3^\prime F^\prime + e_3^\prime 
        \end{eqnarray*} 
} % End parbox
\end{tabular}
\end{center}

where $F \sim N(0,\phi)$, $e_1$, $e_2$ and $e_3$ are normal and independent of $F$ and each other with expected value zero, $Var(e_1)=\omega_1,Var(e_2)=\omega_2$, and $Var(e_3)=\omega_3$. The $\lambda_j$ (factor loadings) are non-zero constants.

        \begin{enumerate}
            \item In terms of the original model, what is $F^\prime$? \vspace{10mm}
            \item Fill in the table. For each symbol from the surrogate model, write what it equals in terms of the original model. Show any necessary work below the table.

        \begin{displaymath}
            \renewcommand{\arraystretch}{2.0}
                \begin{array}{|c||c|c|c|c|c|c|} \hline
            \mbox{Surrogate} & \phi^\prime & \lambda_2^\prime &\lambda_3^\prime  
                             & \omega_1^\prime & \omega_2^\prime & \omega_3^\prime \\ \hline
            \mbox{Original} & \hspace{12mm} & \hspace{12mm} & \hspace{12mm} & 
                              \hspace{12mm} & \hspace{12mm} & \hspace{12mm} \\ \hline
            \end{array} 
            \renewcommand{\arraystretch}{1.0}
            \end{displaymath} 

        \end{enumerate} % End of surrogate model question

\vspace{50mm}




\item (3 points) \label{R} For the R part of the assignment (last question), you fit a 2-factor model to the \texttt{poverty} data. Give an estimate and a 95\% confidence interval for the correlation between factors. The answer is three numbers from your printout. Write the numbers in the space below and circle them on your printout.

\end{enumerate} % End of quiz questions

 \vspace{20mm}
 
\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}