84 lines
3.5 KiB
TeX
84 lines
3.5 KiB
TeX
\documentclass[12pt,a4paper,pdftex]{exam}
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\newcommand{\exercisetopic}{Resampling}
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\newcommand{\exercisenum}{X2}
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\newcommand{\exercisedate}{December 14th, 2020}
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\input{../../exercisesheader}
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\firstpagefooter{Prof. Dr. Jan Benda}{}{jan.benda@uni-tuebingen.de}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{document}
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\input{../../exercisestitle}
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\begin{questions}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\question \qt{Read chapter 7 of the script on ``resampling methods''!}\vspace{-3ex}
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\question \qt{Bootstrap the standard error of the mean}
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We want to compute the standard error of the mean of a data set by
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means of the bootstrap method and compare the result with the formula
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``standard deviation divided by the square-root of $n$''.
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\begin{parts}
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\part Download the file \code{thymusglandweights.dat} from Ilias.
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This is a data set of the weights of the thymus glands of 14-day old chicken embryos
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measured in milligram.
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\part Load the data into Matlab (\code{load} function).
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\part Compute histogram, mean, and standard error of the mean of the first 80 data points.
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\part Compute the standard error of the mean of the first 80 data
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points by bootstrapping the data 500 times. Write a function that
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bootstraps the standard error of the mean of a given data set. The
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function should also return a vector with the bootstrapped means.
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\part Compute the 95\,\% confidence interval for the mean from the
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bootstrap distribution (\code{quantile()} function) --- the
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interval that contains the true mean with 95\,\% probability.
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\part Use the whole data set and the bootstrap method for computing
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the dependence of the standard error of the mean from the sample
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size $n$.
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\part Compare your result with the formula for the standard error
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$\sigma/\sqrt{n}$.
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\end{parts}
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\begin{solution}
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\lstinputlisting{bootstrapmean.m}
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\lstinputlisting{bootstraptymus.m}
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\includegraphics[width=0.5\textwidth]{bootstraptymus-datahist}
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\includegraphics[width=0.5\textwidth]{bootstraptymus-meanhist}
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\includegraphics[width=0.5\textwidth]{bootstraptymus-samples}
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\end{solution}
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\question \qt{Student t-distribution}
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The distribution of Student's t, $t=\bar x/(\sigma_x/\sqrt{n})$, the
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estimated mean $\bar x$ of a data set of size $n$ divided by the
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estimated standard error of the mean $\sigma_x/\sqrt{n}$, where
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$\sigma_x$ is the estimated standard deviation, is not a normal
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distribution but a Student-t distribution. We want to compute the
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Student-t distribution and compare it with the normal distribution.
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\begin{parts}
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\part Generate 100000 normally distributed random numbers.
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\part Draw from these data 1000 samples of size $n=3$, 5, 10, and
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50. For each sample size $n$ ...
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\part ... compute the mean $\bar x$ of the samples and plot the
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probability density of these means.
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\part ... compare the resulting probability densities with corresponding
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normal distributions.
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\part ... compute Student's $t=\bar x/(\sigma_x/\sqrt{n})$ and compare its
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distribution with the normal distribution with standard deviation of
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one. Is $t$ normally distributed? Under which conditions is $t$
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normally distributed?
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\end{parts}
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\newsolutionpage
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\begin{solution}
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\lstinputlisting{tdistribution.m}
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\includegraphics[width=1\textwidth]{tdistribution-n03}\\
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\includegraphics[width=1\textwidth]{tdistribution-n05}\\
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\includegraphics[width=1\textwidth]{tdistribution-n10}\\
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\includegraphics[width=1\textwidth]{tdistribution-n50}
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\end{solution}
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\end{questions}
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\end{document} |