
\begin{question}
A survey with 51 persons was conducted to analyze the
design of an advertising campaign. Each respondent was asked to
evaluate the overall impression of the advertisement on an
eleven-point scale from 0 (bad) to 10 (good). The evaluations are
summarized separately with respect to type of occupation of the
respondents in the following figure.

\setkeys{Gin}{width=0.8\textwidth}
\includegraphics{anova-002}

To analyze the influence of occupation on the evaluation of the
advertisement an analysis of variance was performed:

\begin{Schunk}
\begin{Soutput}
  Res.Df    RSS Df Sum of Sq     F     Pr(>F)
1     50 53.549                              
2     47 34.018  3    19.531 8.995 8.1265e-05
\end{Soutput}
\end{Schunk}

Which of the following statements are correct?

\begin{answerlist}
  \item It can be shown that the evaluation of the respondents depends on their occupation. (Significance level $5\%$)
  \item The fraction of explained variance is larger than $60$\%.
  \item A one-sided alternative was tested for the mean values.
  \item The fraction of explained variance is smaller than $45$\%.
  \item The test statistic is larger than $7.5$.
\end{answerlist}
\end{question}

\begin{solution}
In order to be able to answer the questions the fraction of
explained variance has to be determined. The residual sum of squares
when using only a single overall mean value ($\mathit{RSS}_0$) as
well as the residual sum of squares when allowing different mean
values given occupation ($\mathit{RSS}_1$) are required. Both are
given in the \texttt{RSS}~column of the ANOVA~table.  The
fraction of explained variance is given by $1 -
\mathit{RSS}_1/\mathit{RSS}_0 = 1 - 34.018/53.549 =
0.365$.  

The statements above can now be evaluated as right or wrong.

\begin{answerlist}
  \item True. The $p$~value is $ 8.13e-05 $ and hence  significant. It can   be shown that the evaluations differ with respect to the occupation of the respondents.
  \item False. The fraction of explained variance is $0.365$ and hence \textit{not} larger than 0.6.
  \item False. An ANOVA always tests the null hypothesis, that all mean values are equal against the alternative hypothesis that they are different.
  \item True. The fraction of explained variance is $0.365$ and hence  smaller than 0.45.
  \item True. The test statistic is $F = 8.995$ and hence  larger than $7.5$.
\end{answerlist}
\end{solution}

%% META-INFORMATION
%% \extype{mchoice}
%% \exsolution{10011}
%% \exname{Analysis of variance}

