<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[a4paper,12pt,pdftex]{exam}

\newcommand{\ptitle}{EOD waveform}
\input{../header.tex}
\firstpagefooter{Supervisor: Peter Pilz}{phone: 29 74835}%
{email: peter.pilz@uni-tuebingen.de}

\begin{document}
\input{../instructions.tex}

In a scientific study three different substances are tested on three groups of
subjects, a forth group is not treated and serves as a control
group.

Simulate test results for the different groups using normally
distributed random numbers. The standard deviation is the same in all
groups. The effect size $d=\frac{\mu_1 - \mu_2}{\sigma}$ relative to
the control group is $0.0, 0.5,$ and $1.0$ for the three experimental
groups.

\begin{questions}
  \question{} How large is the \emph{power} of the study if each group
  consists of 10 subjects?
  \begin{parts}
    \part{} Execute a large number of experiments (10.000) and estimate how often statistically significant results are yielded with a t-test against control and with alpha=0.05. Give \emph{type II} error and power (1 - \emph{type II} error).
  \end{parts}
  \question{} What is the effect of different goup sizes?
  \begin{parts}
    \part{} Consider the following group sizes ($control:group1:group2:group3$):
    \begin{itemize}
    \item $4:12:12:12$
    \item $7:11:11:11$
    \item $13:9:9:9$
    \item $16:8:8:8$
    \item $19:7:7:7$
    \item $22:6:6:6$
    \item $25:5:5:5$
    \item $28:4:4:4$
    \end{itemize}
    \part{} Plot the \emph{power} as a function of the size of the control group for $d={0.5, 1.0}$. Give the group size relative to the size of the experimental groups.
    \part{} Plot the false positive results for d = 0.0 in relation to relative control group size.
  \end{parts}
  \question{} Can you recommend a group size for the control group?
\end{questions}
\end{document}
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