66 lines
2.3 KiB
TeX
66 lines
2.3 KiB
TeX
\documentclass[addpoints,10pt]{exam}
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\usepackage{url}
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\usepackage{color}
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\usepackage{hyperref}
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\pagestyle{headandfoot}
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\runningheadrule
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\firstpageheadrule
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\firstpageheader{Scientific Computing}{afternoon assignment day 02}{10/22/2014}
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%\runningheader{Homework 01}{Page \thepage\ of \numpages}{23. October 2014}
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\firstpagefooter{}{}{}
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\runningfooter{}{}{}
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\pointsinmargin
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\bracketedpoints
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%\printanswers
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\shadedsolutions
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\begin{document}
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%%%%%%%%%%%%%%%%%%%%% Submission instructions %%%%%%%%%%%%%%%%%%%%%%%%%
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\sffamily
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%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{questions}
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\question When the p-value is small, we reject the null
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hypothesis. For example, if you want to test whether two means are
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not equal, the null hypothesis is ``means are equal''. If e.g. $p\le
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0.05$ then we take it as sufficient evidence that the null
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hypothesis is not true. Therefore, we assume that the means are not
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equal (which is what you want to show).
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In this exercise we will look at what kind of p-values we expect if
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the null hypothesis is true. In our example, this would be the case
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if the true means of two datasets are actually equal.
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\begin{parts}
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\part Think about how you expect the p-values to behave in that
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situation.
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\part Simulate the situation in which the means are equal by
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repeating the following at least $1000$ times:
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\begin{enumerate}
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\item Generate two arrays {\tt x} and {\tt y} with $10$ normally
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(Gaussian) distributed random numbers using {\tt randn}. By
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construction, the true means behind the random number are zero.
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\item Perform a two sample t-test ({\tt ttest2}) on {\tt x} and
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{\tt y}. Store the p-value.
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\end{enumerate}
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\part Plot a histogram of the $1000$ p-values. What do you think
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is the distribution the p-values (i.e. if you repeated this
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experiment many more times, how would the histogram look like)?
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\part Given what you find, think about whether the following
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strategy is statistically valid: You collect $10$ data points from
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each group and perform a test. If the test is not significant, you
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collect $10$ more and repeat the test. If the test tells you that
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there is a significant difference you stop. Otherwise you repeat
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the procedure until the test is significant.
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\end{parts}
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\end{questions}
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\end{document}
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