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\title{Stat 471: Homework 03}
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\author{Moo K. Chung\\
mchung@stat.wisc.edu}
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{\bf Due Date:} October 27, 1:20pm. No late assignments will be
accepted or graded. There will be $10\%$ bonus points for
exceptional solutions.

\begin{enumerate}
\item  Estimate $p=P(X_1^2 + X_2^2 \geq 5)$ where $X_i$ are iid
$N(0,1)$ using simple Monte-Carlo integration technique.

\item Estimate the above probability using importance sampling
method. Show your importance sampling method gives a better
performance compared to the simple Monte-Carlo integration
technique.

\item Using $\tt{strength.data}$, test if the population variance
of arm strength is $\sigma^2=20^2$ at $\alpha = 0.1$ based on
Monte-Carlo simulation. Do not assume any parametric model.
Estimate the $90\%$ confidence interval for $\sigma^2$ based on
the simulation.

\item Based on the test used in Problem 3, estimate the type II
error and plot the statistical power based on Monte-Carlo
simulation. Comment on your plot.

\item Solve problem 3 using bootstrap. Use 200 bootstrap
replicates. Do not assume any parametric model. Compare your
result to that of problem 3.

\end{enumerate}

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