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\begin{flushright}
Name \underline{\hspace{60mm}} \\ 
$\,$ \\
Student Number \underline{\hspace{60mm}}
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\begin{center}   
{\Large \textbf{STA 312 f2023 Quiz 8}}\\
\vspace{1 mm}
\end{center}   

\begin{enumerate}


\item (4 points) Let $T =  e^{\mathbf{x}^\top \boldsymbol{\beta}} \times \epsilon$, where $\mathbf{x}^\top \boldsymbol{\beta} = \beta_0+\beta_1x_1 + \ldots + \beta_{p-1}x_{p-1}$, and $\epsilon \sim \exp(1)$. The parameters $\beta_0, \ldots, \beta_{p-1}$ are unobserved constants, while the explanatory variable values $x_1, \ldots, x_{p-1}$ are observed constants.

Derive the density of $T$. Show your work. Be sure to indicate where the density is non-zero, and where it is zero.

\newpage


\item (3 points) Consider a Weibull regression model with \emph{exactly two} explanatory variables. If $x_1$ is increased by one unit, the hazard function at time $t$ is multiplied by \underline{\hspace{15mm}}. That's the \emph{hazard} function. Show your work, starting with a convenient expression on the formula sheet. You have more room than you need. \textbf{Circle your answer.} 
\vspace{120mm}

\item (3 points) \label{R} In your analysis of the \texttt{cancer} data, you fit a Weibull regression model with just sex and physician's ECOG rating. You produced an estimate of median survival time for female patients with an \texttt{ecog} rating of 1, together with a 95\% confidence interval. Write the following in the spaces provided.
    \begin{enumerate}
        \item Estimated median survival time. \vspace{10mm}
        \item Lower 95\% confidence limit. \vspace{10mm}
        \item Upper 95\% confidence limit. \vspace{10mm}
    \end{enumerate}
On your printout, circle the three numbers and write ``Question~\ref{R}" beside them.
\end{enumerate}

\begin{center}\textbf{Please attach your printout.} Make sure your name is on it. \end{center}


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