[bootstrap] fixed english text
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\chapter{\tr{Bootstrap methods}{Bootstrap Methoden}}
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\label{bootstrapchapter}
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\selectlanguage{english}
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Bootstrapping methods are applied to create distributions of
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statistical measures via resampling of a sample. Bootstrapping offers several
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advantages:
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@ -12,9 +10,9 @@ advantages:
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\item Fewer assumptions (e.g. a measured sample does not need to be
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normally distributed).
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\item Increased precision as compared to classical methods. %such as?
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\item General applicability: The bootstrapping methods are very
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\item General applicability: the bootstrapping methods are very
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similar for different statistics and there is no need to specialize
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the method depending on the investigated statistic measure.
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the method to specific statistic measures.
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\end{itemize}
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\begin{figure}[tp]
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@ -22,27 +20,26 @@ advantages:
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\includegraphics[width=0.8\textwidth]{2012-10-29_16-41-39_523}\\[2ex]
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\includegraphics[width=0.8\textwidth]{2012-10-29_16-29-35_312}
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\titlecaption{\label{statisticalpopulationfig} Why can't we measure
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the statistical population but only draw samples?}{}
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properties of the full population but only draw samples?}{}
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\end{figure}
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Reminder: in statistics we are interested in properties of the
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``statistical population'' (in German: \determ{Grundgesamtheit}), e.g. the
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Reminder: in statistics we are interested in properties of a
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\enterm{statistical population} (\determ{Grundgesamtheit}), e.g. the
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average length of all pickles (\figref{statisticalpopulationfig}). But
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we cannot measure the lengths of all pickles in the statistical
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population. Rather, we draw samples (simple random sample
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\enterm[SRS|see{simple random sample}]{SRS}, in German:
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\determ{Stichprobe}). We then estimate a statistical measure of interest
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(e.g. the average length of the pickles) within this sample and
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hope that it is a good approximation of the unknown and immeasurable
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real average length of the statistical population (in German aka
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\determ{Populationsparameter}). We apply statistical methods to find
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out how precise this approximation is.
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If we could draw a large number of \textit{simple random samples} we could
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calculate the statistical measure of interest for each sample and
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estimate the probability distribution using a histogram. This
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distribution is called the \enterm{sampling distribution} (German:
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\determ{Stichprobenverteilung},
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we cannot measure the lengths of all pickles in the
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population. Rather, we draw samples (\enterm{simple random sample}
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\enterm[SRS|see{simple random sample}]{SRS}, \determ{Stichprobe}). We
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then estimate a statistical measure of interest (e.g. the average
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length of the pickles) within this sample and hope that it is a good
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approximation of the unknown and immeasurable true average length of
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the population (\determ{Populationsparameter}). We apply statistical
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methods to find out how precise this approximation is.
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If we could draw a large number of \enterm{simple random samples} we
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could calculate the statistical measure of interest for each sample
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and estimate its probability distribution using a histogram. This
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distribution is called the \enterm{sampling distribution}
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(\determ{Stichprobenverteilung},
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\subfigref{bootstrapsamplingdistributionfig}{a}).
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\begin{figure}[tp]
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@ -67,16 +64,14 @@ Commonly, there will be only a single SRS. In such cases we make use
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of certain assumptions (e.g. we assume a normal distribution) that
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allow us to infer the precision of our estimation based on the
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SRS. For example the formula $\sigma/\sqrt{n}$ gives the standard
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error of the mean which is the standard deviation of the distribution
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of average values around the mean of the statistical population
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estimated in many SRS
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error of the mean which is the standard deviation of the sampling
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distribution of average values around the true mean of the population
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(\subfigref{bootstrapsamplingdistributionfig}{b}).
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%explicitely state that this is based on the assumption of a normal distribution?
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Alternatively, we can use ``bootstrapping'' to generate new samples
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from the one set of measurements (resampling). From these bootstrapped
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samples we calculate the desired statistical measure and estimate
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their distribution (\enterm{bootstrap distribution},
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from one set of measurements (resampling). From these bootstrapped
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samples we compute the desired statistical measure and estimate their
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distribution (\enterm{bootstrap distribution},
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\subfigref{bootstrapsamplingdistributionfig}{c}). Interestingly, this
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distribution is very similar to the sampling distribution regarding
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its width. The only difference is that the bootstrapped values are
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@ -89,7 +84,7 @@ Bootstrapping methods create bootstrapped samples from a SRS by
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resampling. The bootstrapped samples are used to estimate the sampling
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distribution of a statistical measure. The bootstrapped samples have
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the same size as the original sample and are created by randomly drawing with
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replacement, that is, each value of the original sample can occur
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replacement. That is, each value of the original sample can occur
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once, multiple time, or not at all in a bootstrapped sample.
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@ -107,10 +102,10 @@ of the statistical population.
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error of the mean.}{The --- usually unknown --- sampling
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distribution of the mean is distributed around the true mean of
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the statistical population ($\mu=0$, red). The bootstrap
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distribution of the means calculated for many bootstrapped samples
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distribution of the means computed from many bootstrapped samples
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has the same shape as the sampling distribution but is centered
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around the mean of the SRS used for resampling. The standard
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deviation of the bootstrap distribution (blue) is thus an estimator for
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deviation of the bootstrap distribution (blue) is an estimator for
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the standard error of the mean.}
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\end{figure}
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@ -137,8 +132,8 @@ distribution is the standard error of the mean.
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\section{Permutation tests}
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Statistical tests ask for the probability that a measured value
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originates from the null hypothesis. Is this probability smaller than
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Statistical tests ask for the probability of a measured value
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to originate from a null hypothesis. Is this probability smaller than
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the desired significance level, the null hypothesis may be rejected.
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Traditionally, such probabilities are taken from theoretical
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@ -148,36 +143,37 @@ data. An alternative approach is to calculate the probability density
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of the null hypothesis directly from the data itself. To do this, we
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need to resample the data according to the null hypothesis from the
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SRS. By such permutation operations we destroy the feature of interest
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while we conserve all other features of the data.
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while we conserve all other statistical properties of the data.
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\begin{figure}[tp]
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\includegraphics[width=1\textwidth]{permutecorrelation}
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\titlecaption{\label{permutecorrelationfig}Permutation test for
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correlations.}{Let the correlation coefficient of a dataset with
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200 samples be $\rho=0.21$. The distribution of the null
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hypothesis, yielded from the correlation coefficients of
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permuted and uncorrelated datasets is centered around zero
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(yellow). The measured correlation coefficient is larger than the
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hypothesis (yellow), optained from the correlation coefficients of
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permuted and therefore uncorrelated datasets is centered around
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zero. The measured correlation coefficient is larger than the
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95\,\% percentile of the null hypothesis. The null hypothesis may
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thus be rejected and the measured correlation is statistically
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significant.}
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thus be rejected and the measured correlation is considered
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statistically significant.}
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\end{figure}
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A good example for the application of a permutaion test is the
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statistical assessment of correlations. Given are measured pairs of
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data points $(x_i, y_i)$. By calculating the correlation coefficient
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we can quantify how strongly $y$ depends on $x$. The correlation
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coefficient alone, however, does not tell whether it is statistically
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coefficient alone, however, does not tell whether the correlation is
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significantly different from a random correlation. The null hypothesis
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for such a situation would be that $y$ does not depend on $x$. In
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order to perform a permutation test, we now destroy the correlation by
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permuting the $(x_i, y_i)$ pairs, i.e. we rearrange the $x_i$ and
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$y_i$ values in a random fashion. By creating many sets of random
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pairs and calculating the resulting correlation coefficients, we yield
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a distribution of correlation coefficients that are a result of
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randomness. From this distribution we can directly measure the
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statistical significance (figure\,\ref{permutecorrelationfig}).
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order to perform a permutation test, we need to destroy the
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correlation by permuting the $(x_i, y_i)$ pairs, i.e. we rearrange the
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$x_i$ and $y_i$ values in a random fashion. Generating many sets of
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random pairs and computing the resulting correlation coefficients,
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yields a distribution of correlation coefficients that result
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randomnly from uncorrelated data. By comparing the actually measured
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correlation coefficient with this distribution we can directly assess
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the significance of the correlation
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(figure\,\ref{permutecorrelationfig}).
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\begin{exercise}{correlationsignificance.m}{correlationsignificance.out}
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Estimate the statistical significance of a correlation coefficient.
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@ -190,10 +186,8 @@ Estimate the statistical significance of a correlation coefficient.
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generating uncorrelated pairs. For this permute $x$- and $y$-values
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\matlabfun{randperm()} 1000 times and calculate for each
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permutation the correlation coefficient.
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\item Read out the 95\,\% percentile from the resulting null
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hypothesis distribution and compare it with the correlation
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coefficient calculated for the original data.
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\item Read out the 95\,\% percentile from the resulting distribution
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of the null hypothesis and compare it with the correlation
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coefficient computed from the original data.
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\end{enumerate}
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\end{exercise}
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\selectlanguage{english}
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@ -15,7 +15,7 @@
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\else
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\newcommand{\stitle}{}
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\fi
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\header{{\bfseries\large Exercise 7\stitle}}{{\bfseries\large Statistics}}{{\bfseries\large December 2nd, 2019}}
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\header{{\bfseries\large Exercise 8\stitle}}{{\bfseries\large Statistics}}{{\bfseries\large December 2nd, 2019}}
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\firstpagefooter{Prof. Dr. Jan Benda}{Phone: 29 74573}{Email:
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jan.benda@uni-tuebingen.de}
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\runningfooter{}{\thepage}{}
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