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\begin{flushright}
Name \underline{\hspace{60mm}} \\ 
$\,$ \\
Student Number \underline{\hspace{60mm}}
\end{flushright}   
\vspace{5mm}


\begin{center}   
{\Large \textbf{STA 442/2101 F 2012 Quiz 6}}\\
\vspace{1 mm}
\end{center}



\begin{enumerate} 

    \item (5 points) This question is based on the normal linear model $\mathbf{Y} = \mathbf{X} \boldsymbol{\beta} + \boldsymbol{\epsilon}$. You may use facts like these without proof.

{\small
\vspace{3mm}
\begin{tabular}{ll} \hline
&\\
$\widehat{\boldsymbol{\beta}} = (\mathbf{X}^\prime \mathbf{X})^{-1} 
                   \mathbf{X}^\prime \mathbf{Y} \sim N_p\left(\boldsymbol{\beta}, 
                   \sigma^2 (\mathbf{X}^\prime \mathbf{X})^{-1}\right)$
&
$SSE/\sigma^2 \sim \chi^2(n-p)$ \\
&\\
$SSE$ and $\widehat{\boldsymbol{\beta}}$ are independent.
&
If $Z\sim N(0,1)$ and $W \sim \chi^2(\nu)$ are independent, \\
& then $T = \frac{Z}{\sqrt{W/\nu}} \sim t(\nu)$. \\ 
& \\ \hline
\end{tabular}
} % End size

We seek a statistic that can be used to construct tests or a confidence interval for the linear combination $\mathbf{a}^\prime \boldsymbol{\beta}$, where $\mathbf{a}$ is a $p \times 1$ vector of constants.

        \begin{enumerate}
            \item What is the distribution of $\mathbf{a}^\prime \widehat{\boldsymbol{\beta}}$? Just write down the answer if you can. Show work only if you have to.
\vspace{40mm}            
            \item Standardize your answer by subtracting off the expected value and dividing by the standard deviation. Call this random variable $Z$. What is the distribution of $Z$?

\newpage

            \item Divide $Z$ by the square root of a chi-squared random variable, divided by its degrees of freedom, and simplify. Call this statistic $T$. Show a little work.
\vspace{80mm}
            \item How do you know that the numerator and denominator are independent?
\vspace{30mm}
        \end{enumerate}




    \item (5 points) This question is based on your printout from the SAT data --- the model with two explanatory variables, Math test score and Verbal test score. \textbf{\emph{Write your answers in the blanks below, and also circle them on your printout.} On the printout, label the answers 2a, 2b etc.}
    \begin{enumerate}
        \item \underline{\hspace{10mm}} What proportion of the variation (\emph{not} remaining variation) in first-year Grade point Average is explained by Math test score and Verbal test score? The answer is a single number from your printout.
        \item \underline{\hspace{10mm}} Give the test statistic for testing $H_0: \beta_1=\beta_2=0$. Your answer is a single number, a value of $t$ or $F$, from your printout.
        \item \underline{\hspace{10mm}} Controlling for Verbal score, is Math score related to first-year grade point average? Give the value of the test statistic. Your answer is a single number, a value of $t$ or $F$, from your printout.
        \item \underline{\hspace{10mm}} Controlling for Math score, is Verbal score related to first-year grade point average? Give the value of the test statistic. Your answer is a single number, a value of $t$ or $F$, from your printout.
        \item \underline{\hspace{10mm}}~~~\underline{\hspace{10mm}} Give  a 95\% prediction interval for a student who got 650 on the Verbal and 700 on the Math SAT. Your answer is a set of two numbers from your printout, the lower prediction limit and the upper prediction limit.
    \end{enumerate}
\end{enumerate} 
\begin{center}\textbf{Attach your printout for Question 2 (Homework Question 8). Make sure your name is written on the printout.}\end{center}
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