\documentclass[12pt]{article}\usepackage{ammaths}\en

\begin{document}

\module{A statistics exercise}{01}{05}{1 period}

\prereq{Statistics. This is the last episode before the test.}

\object{\begin{itemize}
        \item Check the understanding of the main statistical notions and methods.
	\item Peer-to-peer teaching to ascertain and consolidate the understanding.
	\end{itemize}}

\mater{\begin{itemize}
	\item Test papers.
	\item Slideshow.
	\end{itemize}}

\modpart{Cooperative work}{45 mins}

\begin{description}
\item[Step 1 (10 minutes) :] Each student is handed out a test paper and starts answering to the questions on the paper.
\item[Step 2 (15 minutes) :] Each student gives his paper to the student behind. This second student has to check the answers of the first one, make any corrections that may be appropriate, than continue answering the questions.
\item[Step 2 (20 minutes) :] Each student gives his paper to the student behind. This third student has to check the answers of the first and second, make any corrections that may be appropriate, than continue answering the questions.
\end{description}


\modpart{Correction}{10 mins}

Papers are finally given to a fourth student who will mark it and grade it according to the answers given by teacher with a slideshow. Papers are collected at the end of the hour and every students is credited of the average marks of the three papers he or she contributed to.
 



\pagebreak

\moddocdis{A statistics exercise}{01}{05}{Test paper}

\begin{multicols}{2}

In a casino game, people can win between $0$ and $20$ pounds. The fee to play this game is $10$ pounds.
A study has been carried over 200 people, each playing 1 game.
Their winnings after the games are shown in the table below. The
first line gives the possible winnings, gathered in intervals. The
second line shows the number of people whose winnings at the end of the game was in each
interval.


\begin{flushright}
\begin{tabular}{rp{5cm}}
Student 1 & \rule{0pt}{15pt}\\[2pt]
\cline{2-2}
\multicolumn{2}{c}{} \\[2pt]
Student 2 & \rule{0pt}{15pt}\\[2pt]
\cline{2-2}
\multicolumn{2}{c}{} \\[5pt]
Student 3 & \rule{0pt}{15pt}\\[2pt]
\cline{2-2}
\end{tabular}

\vspace{0.7cm}\ 

\end{flushright}

\end{multicols}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline Winnings & $[0;4[$ & $[4;6[$ & $[6;8[$ & $[8;10[$ & $[10;14[$
& $[14;20[$ \\
\hline Absolute frequency & $38$ & $50$ & $54$ & $46$ & $10$ & $2$ \\
\hline
\end{tabular}
\end{center}

For all computations, whenever necessary, the middle of each
interval will be used.

\begin{enumerate}
\item Represent this data as a histogram.
\begin{center}
\psset{unit=1cm}
\begin{pspicture}(0,-0.5)(10,6)
\psgrid[gridlabels=0,gridcolor=gray](0,0)(10,6)
\end{pspicture}
\end{center}
\item Compute the average of these data and represent it as a blue
vertical line on the histogram.
\begin{center}
\begin{tabular}{|p{14cm}|}
\hline
  \rule{0pt}{15pt}\\[5pt]
\hline
\end{tabular}
\end{center}
\item Compute the median and the quartiles of these data and represent it as a red
vertical line on the histogram.
\begin{center}
\begin{tabular}{|p{14cm}|}
\hline
  \rule{0pt}{15pt}\\[5pt]
\hline
\end{tabular}
\end{center}
\end{enumerate}

Another study has been carried over 500 other people. The results are shown in the table below.

\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Winning & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline
Frequency & 27 & 34 & 37 & 34 & 42 & 41 & 39 & 41 & 35 & 29 \\
\hline
\hline
Winning & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 \\
\hline
Frequency & 25 & 23 & 24 & 17 & 12 & 9 & 11 & 8 & 5 & 7 \\
\hline
\end{tabular}
\end{center}

\begin{enumerate}\setcounter{enumi}{3}
\item Find out the minimum, maximum and range (or spreading) of these $500$ values.
\begin{center}
\begin{tabular}{|p{14cm}|}
\hline
  \rule{0pt}{15pt}\\[5pt]
\hline
\end{tabular}
\end{center}
\item Compute the average winnings of these $500$ values.
\begin{center}
\begin{tabular}{|p{14cm}|}
\hline
  \rule{0pt}{15pt}\\[5pt]
\hline
\end{tabular}
\end{center}
\item Find out the median of these values and show all the steps in your method.
\begin{center}
\begin{tabular}{|p{14cm}|}
\hline
  \rule{0pt}{60pt}\\[5pt]
\hline
\end{tabular}
\end{center}

\item According to these results, would you play this game ?
\begin{center}
\begin{tabular}{|p{14cm}|}
\hline
  \rule{0pt}{15pt}\\[5pt]
\hline
\end{tabular}
\end{center}
\end{enumerate}
We now have $700$ values : $200$ in the first table, gathered in intervals, and $500$ in the second table.

\begin{enumerate}\setcounter{enumi}{7}
\item Use the two averages to compute the average of the $700$ values. Show your computation.
\begin{center}
\begin{tabular}{|p{14cm}|}
\hline
  \rule{0pt}{30pt}\\[5pt]
\hline
\end{tabular}
\end{center}
\item Can you deduce in the same way the median of the $700$ values ? If so, do it. If not, explain why.
\begin{center}
\begin{tabular}{|p{14cm}|}
\hline
  \rule{0pt}{30pt}\\[5pt]
\hline
\end{tabular}
\end{center}


\end{enumerate}


\pagebreak

\moddocdis{A statistics exercise}{01}{05}{Correction}


In a casino game, people can win between $0$ and $20$ pounds. The fee to play this game is $10$ pounds.
A study has been carried over 200 people, each playing 1 game.
Their winnings after the games are shown in the table below. The
first line gives the possible winnings, gathered in intervals. The
second line shows the number of people whose winnings at the end of the game was in each
interval.

\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline Winnings & $[0;4[$ & $[4;6[$ & $[6;8[$ & $[8;10[$ & $[10;14[$
& $[14;20[$ \\
\hline Absolute frequency & $38$ & $50$ & $54$ & $46$ & $10$ & $2$ \\
\hline
\end{tabular}
\end{center}

For all computations, whenever necessary, the middle of each
interval will be used.

\begin{enumerate}
\item Represent this data as a histogram.
\begin{center}
\newrgbcolor{zzttqq}{0.6 0.2 0}
\psset{unit=1cm}
\scalebox{0.5}{
\psset{xunit=1.0cm,yunit=1cm}
\begin{pspicture*}(-1,-1)(12,6)
\psgrid[subgriddiv=0,gridlabels=0,gridcolor=gray,subgriddiv=5,subgridcolor=lightgray](0,0)(10,6)
\psset{xunit=0.5cm,yunit=0.1cm,algebraic=true,dotstyle=*,dotsize=3pt 0,linewidth=0.8pt,arrowsize=3pt 2,arrowinset=0.25}
\psaxes[labelFontSize=\scriptstyle,xAxis=true,yAxis=true,Dx=2,Dy=10,ticksize=-2pt 0,subticks=2]{->}(0,0)(-1,-6)(21,57)
\psframe[linecolor=zzttqq,fillcolor=zzttqq,fillstyle=solid,opacity=0.1](0,0)(4,19)
\psframe[linecolor=zzttqq,fillcolor=zzttqq,fillstyle=solid,opacity=0.1](4,0)(6,50)
\psframe[linecolor=zzttqq,fillcolor=zzttqq,fillstyle=solid,opacity=0.1](6,0)(8,54)
\psframe[linecolor=zzttqq,fillcolor=zzttqq,fillstyle=solid,opacity=0.1](8,0)(10,46)
\psframe[linecolor=zzttqq,fillcolor=zzttqq,fillstyle=solid,opacity=0.1](10,0)(14,10)
\psframe[linecolor=zzttqq,fillcolor=zzttqq,fillstyle=solid,opacity=0.1](14,0)(20,0.6)
\psframe[linecolor=zzttqq,fillcolor=zzttqq,fillstyle=solid,opacity=0.1](14,40)(16,50)
\rput(18,45){10 people}
\psline[linecolor=blue](6.36,0)(6.36,57)
\psline[linecolor=red](7,0)(7,57)
\end{pspicture*}}



\end{center}
\item Compute the average of these data and represent it as a blue
vertical line on the histogram.
\begin{center}
\begin{tabular}{|p{14cm}|}
\hline
$$\bar{x}=\frac{2\times 38+5\times 50+7\times 54+9\times 46+12\times 10+17\times 2}{200}=6.36$$\\
\hline
\end{tabular}
\end{center}
\item Compute the median and the quartiles of these data and represent it as a red
vertical line on the histogram.
\begin{center}
\begin{tabular}{|p{14cm}|}
\hline
\corrige There are 200 values.\\
\corrige It's an even number so the median is between the 100th and the 101st value.\\
\corrige The first quartile is the $\dfrac{200}{4}=50$th value.\\
\corrige The third quartile is the $\dfrac{3\times200}{4}=150$th value.\\
\corrige To find the actual values, we need the cumulative absolute frequencies.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline W. & $[0;4[$ & $[4;6[$ & $[6;8[$ & $[8;10[$ & $[10;14[$
& $[14;20[$ \\
\hline A.F. & $38$ & $50$ & $54$ & $46$ & $10$ & $2$ \\
\hline C.A.F. & $38$ & $88$ & $142$ & $188$ & $198$ & $200$ \\
\hline
\end{tabular}
\end{center}
\corrige So, finally,
$$Med=[6;8[\textrm{ ; }Q_1=[4;6[\textrm{ and }Q_3=[8;10[.$$
or 
$$Med=7\textrm{ ; }Q_1=5\textrm{ and }Q_3=9.$$
\\\hline
\end{tabular}
\end{center}
\end{enumerate}

Another study has been carried over 500 other people. The results are shown in the table below.

\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Winning & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline
Frequency & 27 & 34 & 37 & 34 & 42 & 41 & 39 & 41 & 35 & 29 \\
\hline
\hline
Winning & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 \\
\hline
Frequency & 25 & 23 & 24 & 17 & 12 & 9 & 11 & 8 & 5 & 7 \\
\hline
\end{tabular}
\end{center}

\begin{enumerate}\setcounter{enumi}{3}
\item Find out the minimum, maximum and range (or spreading) of these $500$ values.
\begin{center}
\begin{tabular}{|p{14cm}|}
\hline
{\corrige The minimum is $0$. The maximum is $19$. The range is $19-0=19$.}\\[5pt]
\hline
\end{tabular}
\end{center}
\item Compute the average winnings of these $500$ values.
\begin{center}
\begin{tabular}{|p{14cm}|}
\hline
 $$\dfrac{0\times27+1\times34+\cdots+19\times7}{500}=6.99$$
\corrige The average is $6.99$.\\
\hline
\end{tabular}
\end{center}
\item Find out the median of these values and show all the steps in your method.
\begin{center}
\begin{tabular}{|p{14cm}|}
\hline
\corrige There are 500 values. It's an even number so the median is between the 250th and the 251st value.
Using the C.A.F. (cumulative absolute frequencies) we see that the median is $6$.\\
\hline
\end{tabular}
\end{center}

\item According to these results, would you play this game ?
\begin{center}
\begin{tabular}{|p{14cm}|}
\hline
\corrige Why not, but it's more likely to lose money than to win, as the fee is 10 pounds and the average winnings after
500
games only amounts to $6,99$ pounds.\\
\hline
\end{tabular}
\end{center}
\end{enumerate}
We now have $700$ values : $200$ in the first table, gathered in intervals, and $500$ in the second table.

\begin{enumerate}\setcounter{enumi}{7}
\item Use the two averages to compute the average of the $700$ values. Show your computation.
\begin{center}
\begin{tabular}{|p{14cm}|}
\hline
 \corrige In this computation, we have to consider the fact that there are $200$ values in the first set of data and
$500$ in
the second. So the global mean is $$\frac{200\times 6.36+500\times 6.99}{700}=6.81.$$ \\
\hline
\end{tabular}
\end{center}
\item Can you deduce in the same way the median of the $700$ values ? If so, do it. If not, explain why.
\begin{center}
\begin{tabular}{|p{14cm}|}
\hline
\corrige  We can't find the value of the median, as in one case the data is gathered in intervals, and in the second
it's
just listed.\\
\corrige We could gather the 700 values in classes, but it wouldn't be very accurate.\\
\hline
\end{tabular}
\end{center}


\end{enumerate}


\end{document}
