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Name \underline{\hspace{60mm}} \\ 
$\,$ \\
Student Number \underline{\hspace{60mm}}
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{\Large \textbf{STA 312 f2023 Quiz 4}}\\
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\begin{enumerate}

\item (3 points) The expected value of a Weibull random variable $T$ with parameters $\alpha$ and $\lambda$ is $E(T) = \frac{1}{\lambda} \, \Gamma\left(\frac{1}{\alpha}+1\right)$. You don't have to show this. To obtain the standard error of \emph{estimated} $E(T)$ for your R work, you needed to calculate \.{g}$(\alpha,\lambda)$. Show the calculation of \.{g}$(\alpha,\lambda)$ in the space below. \textbf{Circle your final answer}.
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 \item (2 points) \label{comp} For Question 1 of Assignment 4, you analyzed numerical data from a Weibull distribution, and you produced a 95\%  confidence interval for $E(T)$. Write the confidence interval in the space below: Just two numbers. On your printout, circle the numbers and write ``Question~\ref{comp}" beside them. \textbf{The code that produced the confidence interval for $E(T)$ must be shown.}

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\item (5 points) Let $T$ be a continuous random variable with $P(T>0)=1$, density $f(t)$ and cumulatve distribution function $F(t) = P(T \leq t)$.  Prove $h(t) = \frac{f(t)}{S(t)}$. You may use anything on the formula sheet except the fact you are proving. 


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\noindent
Please attach the printout with your answer to Question~\ref{comp} of this quiz (Question 1d of the assignment). \textbf{The code that produced the confidence interval for $E(T)$ must be shown.} Make sure your name and student number are written on the printout.

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