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\firstpageheader{Scientific Computing}{afternoon assignment day 02}{10/22/2014}
%\runningheader{Homework 01}{Page \thepage\ of \numpages}{23. October 2014}
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\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}





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