done

statistics/assignments/day3.tex

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