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Name \underline{\hspace{60mm}} \\ 
$\,$ \\
Student Number \underline{\hspace{60mm}}
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{\Large \textbf{STA 442/2101 f2012 Quiz 1}}\\
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\begin{enumerate} 

    \item (4 points) A random sample of size $n=80$ was drawn from a distribution with density $f(y) = \theta e^{-\theta y}$ for $y>0$, where the parameter $\theta>0$. 
    \begin{enumerate}
        \item Find the maximum likelihood estimate of $\theta$. Show your work. Don't bother wth a second derivative test. Your answer is a symbolic expression. \textbf{Circle your final answer}.
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        \item Results included $\sum_{i=1}^n y_i = 81.36$, $\sum_{i=1}^n y_i^2 = 157.78$, and $\sum_{i=1}^n \log(y_i) = -45.22$. Give the maximum likelihood estimate in numeric form. Your answer is a number. \textbf{Circle it}.
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    \end{enumerate}
    
    \item (3 points) For a standard multiple regression with normal error terms, suppose a 95\% confidence interval for $\beta_6$ is $(-1.2,0.75)$. Assume that the model is \emph{completely correct}. Does the confidence interval mean that $Pr\{-1.2 < \beta_6 < 0.75 \} = 0.95$? Answer Yes or No and briefly explain.
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    \item (3 points) Let $\mathbf{X}$ be a real $n \times p$ matrix, and let $\mathbf{a}$ be a real $p \times 1$ column vector. Show that $\mathbf{a}^\prime (\mathbf{X}^\prime\mathbf{X})\mathbf{a} \geq 0$ (that is, $\mathbf{X}^\prime\mathbf{X}$ is non-negative definite). 

\end{enumerate}


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