<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[12pt,a4paper,pdftex]{exam}

\newcommand{\exercisetopic}{Resampling}
\newcommand{\exercisenum}{X2}
\newcommand{\exercisedate}{December 14th, 2020}

\input{../../exercisesheader}

\firstpagefooter{Prof. Dr. Jan Benda}{}{jan.benda@uni-tuebingen.de}

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\begin{document}

\input{../../exercisestitle}

\begin{questions}

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\question \qt{Read chapter 7 of the script on ``resampling methods''!}\vspace{-3ex}

\question \qt{Bootstrap the standard error of the mean}
We want to compute the standard error of the mean of a data set by
means of the bootstrap method and compare the result with the formula
``standard deviation divided by the square-root of $n$''.
\begin{parts}
  \part Download the file \code{thymusglandweights.dat} from Ilias.
  This is a data set of the weights of the thymus glands of 14-day old chicken embryos
  measured in milligram.
  \part Load the data into Matlab (\code{load} function).
  \part Compute histogram, mean, and standard error of the mean of the first 80 data points.
  \part Compute the standard error of the mean of the first 80 data
  points by bootstrapping the data 500 times. Write a function that
  bootstraps the standard error of the mean of a given data set. The
  function should also return a vector with the bootstrapped means.
  \part Compute the 95\,\% confidence interval for the mean from the
  bootstrap distribution (\code{quantile()} function) --- the
  interval that contains the true mean with 95\,\% probability.
  \part Use the whole data set and the bootstrap method for computing
  the dependence of the standard error of the mean from the sample
  size $n$.
  \part Compare your result with the formula for the standard error
  $\sigma/\sqrt{n}$.
\end{parts}
\begin{solution}
  \lstinputlisting{bootstrapmean.m}
  \lstinputlisting{bootstraptymus.m}
  \includegraphics[width=0.5\textwidth]{bootstraptymus-datahist}
  \includegraphics[width=0.5\textwidth]{bootstraptymus-meanhist}
  \includegraphics[width=0.5\textwidth]{bootstraptymus-samples}
\end{solution}


\question \qt{Student t-distribution}
The distribution of Student's t, $t=\bar x/(\sigma_x/\sqrt{n})$, the
estimated mean $\bar x$ of a data set of size $n$ divided by the
estimated standard error of the mean $\sigma_x/\sqrt{n}$, where
$\sigma_x$ is the estimated standard deviation, is not a normal
distribution but a Student-t distribution.  We want to compute the
Student-t distribution and compare it with the normal distribution.
\begin{parts}
\part Generate 100000 normally distributed random numbers.
\part Draw from these data 1000 samples of size $n=3$, 5, 10, and
50. For each sample size $n$ ...
\part ... compute the mean $\bar x$ of the samples and plot the
probability density of these means.
\part ... compare the resulting probability densities with corresponding
normal distributions.
\part ... compute Student's $t=\bar x/(\sigma_x/\sqrt{n})$ and compare its
distribution with the normal distribution with standard deviation of
one. Is $t$ normally distributed? Under which conditions is $t$
normally distributed?
\end{parts}
\newsolutionpage
\begin{solution}
  \lstinputlisting{tdistribution.m}
  \includegraphics[width=1\textwidth]{tdistribution-n03}\\
  \includegraphics[width=1\textwidth]{tdistribution-n05}\\
  \includegraphics[width=1\textwidth]{tdistribution-n10}\\
  \includegraphics[width=1\textwidth]{tdistribution-n50}
\end{solution}

\end{questions}

\end{document}</pre></body></html>