\documentclass[11pt]{article}
\usepackage[Mickael]{ammaths}\en
\usealgorithms
\begin{document}\nocolor
\livreteuro{7}{Échantillonage et simulations}{Sampling and simulating}
\vfill
\begin{center}
\includegraphics[height=14cm]{Images/simulating.eps}\\ \smallskip
\textsf{\small \emph{Yucca Muffin}, by Milo Beckman}
\end{center}
\vfill
\begin{center}
\fbox{\parbox{12cm}{\sf\small At the end of this chapter, you should be able to :
\begin{itemize}
\item compute the margin of error and fluctuation interval at $95\%$ confidence for a known probability ;
\item use a fluctuation interval to accept or reject an assumption ;
% \item compute the confidence interval at $95\%$ for a sample ;
% \item use a confidence interval to estimate a probability ;
\item use the calculator to simulate a random experiment.
\end{itemize}}}
\end{center}
\begin{flushright}
\sf\small Aymar de Saint-Seine et Mickaël Védrine\\ Année scolaire 2011/2012
\end{flushright}
\newpage\null\thispagestyle{empty}\newpage%insertion page blange
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Margin of error
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\exocpart{Margin of error and fluctuation interval}
\exoc{{\bf\textsf{Fluctuations when throwing a die}}
The frequency polygons below were obtained by forcing the pupils in a class to throw a a fair die 10 times, 100
times, and 50000 times.
\begin{center}
\psset{xunit=1.5cm,yunit=15cm}
\begin{pspicture}(0,-0.025)(7,0.32)
\psaxes[Dy=0.1](0,0)(7,0.32)
\psline[linecolor=black,linestyle=dotted](0,0)(1,0.1)(2,0.2)(3,0.2)(4,0.3)(5,0.1)(6,0.1)(7,0)
\psline[linecolor=black](0,0)(1,0.18)(2,0.18)(3,0.15)(4,0.18)(5,0.14)(6,0.17)(7,0)
\psline[linecolor=black,linestyle=dashed](0,0)(1,0.1692)(2,0.1644)(3,0.1643)(4,0.1654)(5,0.1684)(6,0.1683)(7,0)
\psline[linecolor=black,linestyle=dashed](5,0.3)(5.5,0.3)\uput[r](5.6,0.3){50\ 000 throws}
\psline[linecolor=black](5,0.275)(5.5,0.275)\uput[r](5.6,0.275){100 throws}
\psline[linecolor=black,linestyle=dotted](5,0.25)(5.5,0.25)\uput[r](5.6,0.25){10 throws}
\end{pspicture}
\end{center}
\begin{enumerate}
\item Read on each graph the approximate frequency of the side 4.
\item Draw on the graph a red horizontal line representing the theoretical probability of getting one specific side.
\item What do you notice about the distance between each graph and the red line ?
\item Write a sentence about the phenomenon showcased by this set of frequency polygons.
\end{enumerate}
}
\exoc{{\bf\textsf{The US 2008 election}}
% Sampling, estimation, margin of error, confidence interval
% 2 hours
In 2008, the American electors had to choose between the Republican John McCain and the Democrat Barack Obama. Surveys
were organised by both parties to \textbf{estimate} the proportion of electors who wanted to vote for
each candidate. As it's impossible to gather the opinions of all the electors, surveys are carried over small parts of
the population, called \textbf{samples}. We will consider that samples are built randomly.
\probpart{-- Sampling fluctuation}
\begin{enumerate}
\item Over a sample of 900 electors, 497 declared that they wanted to vote for Obama. Compute the percentage of
potential Obama electors in this sample.
\item Ten other surveys were organized over the same period. The size of each sample and the number of potential Obama
electors are given in the table below.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Survey & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
Size & 895 & 873 & 900 & 885 & 899 & 842 & 878 & 900 & 897 & 892 \\
\hline
Obama electors & 462 & 493 & 501 & 437 & 467 & 447 & 468 & 495 & 488 & 478 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}
\item Compute the percentage of potential Obama electors in each sample. Round the answers to 2DP.
\item If McCain had only known about the 4th survey, what could he have deduced ?
\item Can you deduce from these surveys the actual percentage of Obama voters ?
\end{enumerate}
\end{enumerate}
\probpart{-- Margin of error}
The chances that a sample will yield the true value in the whole population are very small. Furthermore, there may be
important differences between the percentages in different samples. This phenomenon is known as \textbf{sampling
fluctuation}.
To illustrate this, the results of one hundred surveys were collected, each one over a population of 900
people. The scatterplot below shows the percentage of potential Obama electors in each survey.
\begin{center}
\psset{xunit=0.1cm,yunit=50cm}
\begin{pspicture}(0,0.47)(101,0.58)
\psaxes[Dx=10,Dy=0.02,Oy=0.48](0,0.48)(101,0.581)
\psdots(1,0.53)(2,0.54)(3,0.49)(4,0.54)(5,0.5)(6,0.51)(7,0.5)(8,0.50)(9,0.54)(10,0.53)(11,0.52)(12,0.51)(13,0.5)(14,
0.5)(15,0.50)(16,0.51)(17,0.52)(18,0.51)(19,0.51)(20,0.51)(21,0.51)(22,0.54)(23,0.54)(24,0.51)(25,0.52)(26,0.53)(27,
0.53)(28,0.50)(29,0.50)(30,0.51)(31,0.52)(32,0.53)(33,0.53)(34,0.50)(35,0.52)(36,0.54)(37,0.52)(38,0.52)(39,0.52)(40,
0.51)(41,0.52)(42,0.53)(43,0.53)(44,0.53)(45,0.55)(46,0.53)(47,0.52)(48,0.53)(49,0.52)(50,0.54)(51,0.49)(52,0.51)(53,
0.54)(54,0.53)(55,0.52)(56,0.52)(57,0.54)(58,0.53)(59,0.52)(60,0.53)(61,0.51)(62,0.51)(63,0.52)(64,0.57)(65,0.52)(66,
0.54)(67,0.5)(68,0.51)(69,0.5)(70,0.52)(71,0.52)(72,0.51)(73,0.53)(74,0.54)(75,0.54)(76,0.52)(77,0.52)(78,0.53)(79,
0.52)(80,0.55)(81,0.55)(82,0.52)(83,0.52)(84,0.52)(85,0.55)(86,0.53)(87,0.52)(88,0.53)(89,0.48)(90,0.5)(91,0.52)(92,
0.5)(93, 0.53)(94,0.54)(95, 0.54)(96,0.5)(97,
0.56)(98,0.54)(99 ,0.51)(100,0.52)
\end{pspicture}
\end{center}
\begin{enumerate}
\item It turns out that, on Election day, Obama won with 53\%\ of the votes. On the scatterplot, show the proportion $p$
of Obama electors in the whole population with a horizontal red line. How many simulated surveys gave that exact value ?
\item \begin{enumerate}
\item The value $m=\frac{1}{\sqrt{n}}$, where $n$ is the size of a sample, is called the \textbf{margin of
error at 95\%\ confidence} for that sample. Compute this value to 3DP.
\item On the graph, show the values $p-m$ and $p+m$ with two horizontal blue lines.
\item How many surveys gave a percentage included in the interval $[p-m;p+m]$, called \textbf{fluctuation
interval at $95\%$ confidence} ?
\item Is the answer to the previous question consistent with the name of the interval ?
\end{enumerate}
\end{enumerate}
}
\exoc{{\bf\textsf{The French lottery and odd numbers}}
% Probabilities, simulation, margin of error, confidence interval (Aymar's add-on : Not very sure having confidence interval in this exercise)
% 2 hours
The principles of the French National Lottery (Loto) are fairly simple. Each player picks six numbers (plus one, that
we won't consider in this exercise) between 1 and 49. On lottery day, 6 over 49 balls with numbers from 1 to 49 are randomly
drawn from a machine. The balls are not put back in the machine, so the same number cannot appear twice in a drawing.
The order in which the balls are drawn is irrelevant.
Among the numbers from 1 to 49, there are 25 odd numbers and 24 even numbers.
\probpart{-- Drawing a single number}
In this part, we consider the random experiment that consists in drawing a single ball from the 49 in the machine.
\begin{enumerate}
\item What is the probability of the drawn number being odd ? Give the result as an irreducible fraction and as an
approximate value to 2DP.
\item Fifty samples, each made of $n=100$ independant drawings of a ball were simulated with a computer. For each
sample, the proportion of odd numbers was computed. The results of these fifty samples of size $100$ are given below.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
0.44 & 0.52 & 0.50 & 0.44 & 0.51 & 0.41 & 0.44 & 0.40 & 0.57 & 0.50 \\
0.51 & 0.43 & 0.59 & 0.46 & 0.55 & 0.35 & 0.55 & 0.43 & 0.53 & 0.53 \\
0.45 & 0.42 & 0.47 & 0.48 & 0.50 & 0.45 & 0.48 & 0.47 & 0.46 & 0.57 \\
0.52 & 0.55 & 0.53 & 0.46 & 0.45 & 0.44 & 0.45 & 0.48 & 0.51 & 0.46 \\
0.55 & 0.48 & 0.43 & 0.51 & 0.49 & 0.38 & 0.52 & 0.40 & 0.50 & 0.46 \\
\end{tabular}
\end{center}
\begin{enumerate}
\item How many samples showed a proportion equal to the theoretical value to 2DP ?
\item Compute the fluctuation interval at $95\%$ confidence.
\item How many samples showed a proportion inside the margin of error ?
\item Can you find a margin of error at $98\%$ confidence ?
\end{enumerate}
\end{enumerate}
\pagebreak
\probpart{-- Drawing six numbers}
In this second part, we consider the random experiment that consists of drawing successively six balls, without putting
them back in the machine. It can be proven that in each drawing of six numbers, there is an average of $3.0612$ odd
numbers, so a proportion $q=\frac{3.0612}{6}\approx 0.51$, or approximately $51\%$.
Fifty samples, each made of $n=100$ independant drawings of six succesive balls were simulated with a computer. For each
sample, the proportion of odd numbers was computed. The results of these fifty samples of size $100$ are given below.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
0.527 & 0.475 & 0.500 & 0.522 & 0.558 & 0.518 & 0.510 & 0.518 & 0.550 & 0.607 \\
0.515 & 0.468 & 0.517 & 0.485 & 0.498 & 0.473 & 0.505 & 0.507 & 0.492 & 0.498 \\
0.508 & 0.408 & 0.563 & 0.612 & 0.542 & 0.497 & 0.508 & 0.498 & 0.500 & 0.535 \\
0.498 & 0.508 & 0.525 & 0.478 & 0.517 & 0.528 & 0.492 & 0.487 & 0.535 & 0.523 \\
0.512 & 0.543 & 0.522 & 0.482 & 0.530 & 0.478 & 0.508 & 0.532 & 0.528 & 0.527 \\
\end{tabular}
\end{center}
\begin{enumerate}
\item Compute the fluctuation interval at $95\%$ confidence.
\item How many samples showed a proportion inside the interval ?
\end{enumerate}
\probpart{-- Probabilities on the number of odd numbers}
The table below shows the probabilities of drawing $k$ odd numbers among the six, for $k$ from $0$ to $6$. Values have
been rounded to 3DP.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
Odd numbers & $0$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ \\
\hline
Probability & $0.010$ & $0.076$ & $0.228$ & $0.333$ & $0.250$ & $0.091$ & $0.013$ \\
\hline
\end{tabular}
\end{center}
For each of the following sentences, say if it's true or false. Justify each answer with a computation or an
explanation.
\begin{enumerate}
\item There are more chances to draw 4 odd numbers or more than 2 odd numbers or less.
\item There are more than $90\%$ chances to draw at least 2 odd numbers.
\item There are as many chances to draw exactly 3 odd numbers than exactly 3 even numbers.
\item There are $50\%$ chances to draw as many odd numbers as even numbers.
\item There are more chances to draw no even number than to draw no odd number.
\item There are as many chances to draw at least 3 odd numbers than at least 3 even numbers.
\item It's a good strategy to play only odd numbers.
\end{enumerate}
% \probpart{-- A good strategy}
%
% We've seen in this exercise that the French lottery indeed favours slightly odd numbers over even numbers. Still,
}
\exoc{{\bf\textsf{A biaised four-sided die}}
A role-playing enthusiast has bought a new die with four sides.
She notices that there is a dent on the number four vertex and fears that it may make the die biased.
\begin{enumerate}
\item What should be the probability $p$ of getting a four if the die was really balanced ?
\item She throws the die 50 times and gets 11 times the number four.
\begin{enumerate}
\item Compute the proportion of occurences of the number four in the sample.
\item Compare the proportion in the sample to the value probability you gave in question 1. What do you conclude
about the die ? %arguement in the class expected
\item Compute the margin of error and the fluctuation interval at $95\%$ confidence for the probability $p$ and
a sample of 50 throws.
\item Is your previous conclusion still the same ?
\end{enumerate}
\item She still isn't convinced and throws the die 250 times. She gets 55 times the number four.\\
Answer the previous questions with this new sample.
\item While she's satisfied with the results of her expriment, a friend tells her that 250 throws are not enough to
decide if the die is biased. She then throws the die 2000 times and counts 440 occurences of the number four.\\
Answer the previous questions with this sample.
\item What do you notice about the margin of error when the size of the sample increases ? What impact can it have on a
test like this ?
\end{enumerate}}
\exoc{{\bf\textsf{Male-female parity or not ?}}
Two companies $A$ and $B$ are hiring people in a region where there are as many men as women. By law, they are bound to male-female parity. In company $A$, there are $100$ employees and $43$ of them are women. In company $B$ there are $2,500$ employees with $1,150$ women.
\begin{enumerate}
\item \begin{enumerate}
\item Compute the proportion of women for each company.
\item What do you think of the way each company respect parity ?
\end{enumerate}
\item \begin{enumerate}
\item If parity was respected, what should be the proportion of women ?
\item Compute the fluctuation interval at $95\%$ confidence for each company.
\item Does the previous result confirm your answer to question {\bf\sf{1.b}} ? Explain.
\end{enumerate}
\end{enumerate}
}
\exoc{{\bf\textsf{Using margins of error to make decisions}}
% Surveys, sampling, confidence intervals
% 1 hour
\probpart{-- Accepting or rejecting an assumption}
It is known that in the French population, $26\%$ are allergic to pollen. The sanitary services in a city suspect that
the proportion is more important in their town thand elsewhere in France. To check if this is true, they study a sample
of 400 people, and observe that 130 suffer from that allergy.
\begin{enumerate}
\item Compute the fluctuation interval at $95\%$ confidence.
\item What is the frequency of allergic individuals in this sample ?
\item Does this result confirm the suspicions of the sanitary services ?
\end{enumerate}
\probpart{-- Parity in French Region councils}
\begin{multicols}{2}
After the 2004 regional elections in France, the repartition between women and men in four regional councils was as
follows. We consider that these councils are random samples of the local politician population in each region.
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
& Men & Women & Total \\
\hline
Burgundy & 32 & 25 & 57 \\
\hline
Brittany & 38 & 47 & 85 \\
\hline
Rhône-Alpes & 81 & 76 & 157 \\
\hline
Île-de-France & 103 & 106 & 209\\
\hline
\end{tabular}
\end{center}
\end{multicols}
\begin{enumerate}
\item Supposing that parity between men and women is real in a regional council, what should be the percentage of women
in that council ?
\item Compute the fluctuation interval at $95\%$ confidence for the proportion of women in each council.
\item What do you think of the parity between men and women in the local politician population of each of these
regions ?
\end{enumerate}
\probpart{-- A car factory}
In a car factory, a control is done for flaws of the type ``grainy spots on the hood''. Normally,
$20\%$ of the vehicles present this kind of flaws. While controlling a random sample of 50 vehicles produced in the
same week, it is seen that 13 vehicles have it. Should it be a matter of concern ?
\pagebreak
\probpart{-- Rodrigo Partida's case}
In 1970, the Mexican-American Rodrigo Partida was sentenced to eight years of prison. He appealed to the judgment
contending that he was denied due process and equal protection of law because the grand jury of Hidalgo County, Texas,
which indicted him, was unconstitutionally underrepresented by Mexican-Americans. He introduced evidence that in
1970, the total population of Hidalgo County was 181,535 persons of which 143,611, or approximately 79.2\%\, were
persons of Spanish language or Spanish surname. Next, he presented evidence showing the composition of the
grand jury lists over a period of ten years prior to and including the term of court in which the indictment against
him was returned. Of the 870 persons selected for grand jury duty, only 39.0\%\ were Mexican-Americans. If you were a
judge in the court of appeals, how would you react to these allegations ?
}
\exoc{{\bf\textsf{Lime or orange Tic Tac}}
In this exercise, we will try to answer to an existential question :
\begin{quote}
Is there the same proportion of each flavour in a box of lime or orange Tic Tac ?
\end{quote}
To do so, each pupil in the class will be given a box of candies and use it as a sample of the whole lime or orange Tic
Tac production. To avoid biasing the experiment, it's important not to eat a single candy before the end of the
exercise.
\begin{enumerate}
\item Count the number of candies in your box. What does it tell you about your sample ?
\item Assume that the proportions of each flavour are the same. Compute the fluctuation interval at $95\%$ confidence
for the proportion of lime candies.
\item Count the number of lime candies in your box and compute the observed proportion.
\item What can you conclude from your sample ?
\item How many pupils in the class rejected the hypothesis that the proportions of each flavour are the same ?
\item Put the candies back in the box and/or eat them.
\end{enumerate}
}
\exoc{Write an algorithm where the input are a proportion to test and a sample size, and ouput are the boundaries of the
fluctuation interval. Implement it with your calculator.}
%\exocpart{Simulations}
\exoc{{\bf\textsf{Random walks on an axis}}
% Simulation with the calculator, probabilities, algorithm, margin of error, confidence interval
% 2 hours
\begin{center}
\psset{xunit=2cm,yunit=1cm}
\begin{pspicture}(-3.5,0)(3.5,1)
\psaxes(0,0)(-3.5,0)(3.5,0)
\uput[u](0,0.5){\includegraphics[width=0.7cm]{Images/flea.eps}}
\end{pspicture}
\end{center}
A flea is moving along an axis. It starts from the origin and, after each jump, lands one unit to the right or one unit
to the left, randomly and with the same probability.
A sequence of jumps is called a \emph{walk}. For example, if the flea is always jumping to the right, the walk will be
noted RRRR. If it alternates between right and left, the walk will be noted RLRL.
\probpart{-- Simulations of 4-jumps walks}
The ``Random'' or ``Alea'' function on your calculator delivers a random decimal number between $0$ and $1$.
\begin{enumerate}
\item Devise a method to simulate a 4-jumps walk using the ``Random'' function.
\item Simulate 25 walks and note the final position of the flea at the end of each walk.
\item What are the possible final positions on the axis ? Explain why some are impossible.
\item Count the number of walks for each final position and show the counts in a table.
\item Add a row to the previous table with the absolute frequencies for the whole class.
\item Compute the relative frequencies for the whole class.
\item Compute the average final position of the flea at the end of a 4-jumps walk.
\end{enumerate}
\pagebreak
\probpart{-- An algorithm}
\begin{multicols}{2}
A random walk can be described by the algorithm shown on the right-hand side, where the alea function delivers a random
number in the interval $[0,1[$. Parts of the algorithm have been omitted on purpose.
\begin{enumerate}
\item Explain the functions of the integers $x$ and $i$ in this algorithm.
\item Fill the two incomplete lines.
\item Here are the results of applying the algorithm once. What is the final position of the flea at
the end of this walk ?
\begin{center}
\begin{tabular}{c|c|c|c|c|c}
$i$ & & $1$ & $2$ & $3$ & $4$ \\
\hline
alea & & $0.37$ & $0.01$ & $0.93$ & $0.11$ \\
\hline
$x$ & $0$ & $1$ & $2$ & $1$ & $2$ \\
\hline
\end{tabular}
\end{center}
\end{enumerate}
\begin{center}
\begin{minipage}{7cm}\IncMargin{1em}
\begin{algorithm}[H]
\Begin{$0\rightarrow x$ \;
$1\rightarrow i$ \;
\While{$i\leq 4$}{
\eIf{alea < $0.5$}{
$\ldots\ldots\ldots\rule{0pt}{15pt} \rightarrow x$ \;
}{
$\ldots\ldots\ldots\rule{0pt}{15pt} \rightarrow x$ \;
}
$i+1\rightarrow i$\;
}
\KwOut{$x$}}
\end{algorithm}\DecMargin{1em}
\end{minipage}
\end{center}
\end{multicols}
\begin{enumerate}\setcounter{enumi}{3}
\item Apply the algorithm to create five new walks, using the random function of your calculator and displaying all the
steps of the algorithm like in the example of the previous question.
\item How would you change the algorithm to simulate a $30$-jumps walk ?
\end{enumerate}
\probpart{-- Probabilistic study}
In this part, we will use probabilities to study the situation and compare the theoretical results to the frequencies
we found in part A.
\begin{enumerate}
\item Draw a tree to show all the possible 4-jumps walks. At the end of each branch, write the final position of the
flea.
\item Use the tree to compute the probability of each final position and give the results in a probability table.
\item Compute the margin of error at $95\%$ confidence for your sample of 25 random walks.
\item For each probability, count in the class how many samples of 25 walks gave a frequency within the margin of error.
\end{enumerate}
}
%\pagebreak
\exoc{{\bf\textsf{A birth policy}}
% Simulation, algorithm, sampling, margin of error, confidence interval
% 2 hours
\vspace*{-10pt}
\begin{multicols}{2}
A government has decided to impose a strict birth policy. Births in a family must stop as soon as a boy is born or
after the birth of the fourth child.
We consider in this exercise that the probabilities of giving birth to a girl or a boy are equal and that each birth is
independant from the previous births in the same family.
This birth policy can be represented as a tree, where the possible families are boxed.\par
\begin{center}
\begin{tabular}{c@{\hskip 0.1cm}c@{\hskip 0.1cm}c@{\hskip 0.1cm}c@{\hskip 0.1cm}c@{\hskip 0.1cm}c}
& \rnode{n1}{$\emptyset$} & & & & \\[0.5cm]
\rnode{n21}{\fbox{B}} & & \rnode{n22}{G} & & & \\[0.5cm]
& \rnode{n31}{\fbox{GB}} & & \rnode{n32}{GG} & & \\[0.5cm]
& & \rnode{n41}{\fbox{GGB}} & & \rnode{n42}{GGG} & \\[0.5cm]
& & & \rnode{n51}{\fbox{GGGB}} & & \rnode{n52}{\fbox{GGGG}}\\
\end{tabular}
\end{center}
\psset{nodesep=2pt} \ncline{->}{n1}{n21} \ncline{->}{n1}{n22}
\ncline{->}{n22}{n31} \ncline{->}{n22}{n32} \ncline{->}{n32}{n41}
\ncline{->}{n32}{n42} \ncline{->}{n42}{n51} \ncline{->}{n42}{n52}
\end{multicols}
\pagebreak
\probpart{-- Simulation and statistical approach}
\begin{enumerate}
\item Discuss in the class to conjecture a value for the percentage of girls generated by this policy.
\item Devise a method to simulate the composition of a family with the calculator.
\item Simulate and write down the composition of 100 families. Count the number of children per family and show the
results in a table with absolute and relative frequencies.
\item Compute the arithmetic mean $m_4$ and the median $d_4$ for the number of children per family in your sample of
$100$ families.
\item Compute the arithmetic mean $M_4$ and the median $D_4$ for all the families in the class.
\end{enumerate}
\probpart{-- An algorithm}
\begin{multicols}{2}
This process can be described as an algorithm. The output is then a list of digits, with $0$ representing a girl and $1$
representing a boy.
\begin{enumerate}
\item Explain the functions of the whole numbers $x$ and $i$ in this algorithm.
\item Explain the condition ``$x\neq 1$ and $i\leq 4$''. Does it ensure that the algorithm will always stop ?
\item What is the function of the list $L$ in this algorithm ?
\item Explain the notation $L(i)$.
\end{enumerate}
\begin{center}
\begin{minipage}{7cm}\IncMargin{1em}
\begin{algorithm}[H]
\Begin{
Clear list $L$ ; $0\rightarrow x$ ; $1\rightarrow i$ \;
\While{$x\neq 1$ and $i\leq 4$}{
\eIf{alea < $0.5$}{
$0\rightarrow x$\;
}{
$1\rightarrow x$\;
}
$x\rightarrow L(i)$\;
$i+1\rightarrow i$\;
}
\KwOut{$L$}}
\end{algorithm}\DecMargin{1em}
\end{minipage}
\end{center}
\end{multicols}
\begin{multicols}{2}
\begin{enumerate}\setcounter{enumi}{4}
\item Here are the results of applying the algorithm once.
Apply the algorithm to get 5 families, displaying all the
steps of the algorithm like in the example.
\end{enumerate}
\begin{center}
\begin{tabular}{c|c|c|c|c}
$i$ & & $1$ & $2$ & $3$ \\
\hline
alea & & $0.37$ & $0.01$ & $0.93$ \\
\hline
$x$ & $0$ & $0$ & $0$ & $1$ \\
\hline
$L$ & $()$ & $(0)$ & $(0,0)$ & $(0,0,1)$ \\
\hline
\end{tabular}
\end{center}
\end{multicols}
\probpart{-- The percentage of girls}
The aim of this part is to study the percentage of girls $g$ induced by this birth policy, and therefore check the
answer to the first question of part A. To do so, we will first use the simulations of part A, and then the
probabilities of part C.
\begin{enumerate}
\item Use the value conjectured by the class at the beginning of the exercise to compute the fluctuation interval at
$95\%$ confidence in a sample of 100 families.
\item Compute the percentage of girls in your 100 simulated families. According to this result, would you reject the
hypothesis formulated by the class ?
\item Answer the previous questions with the sample made of all the families simulated in the class. Is the conclusion
the same ?
\end{enumerate}
\probpart{-- Probabilistic study}
\begin{enumerate}
\item Copy the tree at the beginning of the exercise and add the probabilities.
\item Compute the probability of each type of family.
\item Show in a table the possible numbers of children and their probabilities. Are these probabilities consistent
with the frequencies found at the end of part A ?
\item Use the table to compute the expected value for the number of children in a family.
\item Compute the expected values of the numbers of girls and boys in a family. Deduce the theoretical proportion of
girls. Was your initial hypothesis correct ?
\end{enumerate}
}
\pagebreak
\exoc{{\bf\textsf{Random walks on a tetrahedron}}
% Simulation, probabilities, margin of error, confidence interval
% 2h
\begin{minipage}{10cm}
An ant is walking on the edges of a tetrahedron $ABCD$, starting from vertex $A$. When it gets to a vertex, it chooses
randomly the next edge it will walk on. The aim of this exercise is to study the time it will take for the ant to go
back to vertex $A$, assuming that it walks along one edge in exactly 1 minute.
\end{minipage}
\begin{minipage}{5.6cm}
\begin{center}
\psset{unit=0.6cm}
\pspicture(0,0.5)(7,5.5)
\pspolygon(1,2)(5,1)(6,3)(3,5)
\psline(5,1)(3,5)
\psline[linestyle=dashed](1,2)(6,3)
\cput(0.5,2){$A$}
\uput[d](5,1){$B$}
\uput[r](6,3){$C$}
\uput[u](3,5){$D$}
\endpspicture
\end{center}
\end{minipage}
A walk will be noted as a succession of vertices, as in the example below :
$$A\rightarrow D\rightarrow B\rightarrow C\rightarrow A.$$
A walk will always start from $A$ and stop as soon as the ant comes back to $A$.
\probpart{-- Simulations}
\begin{enumerate}
\item Devise a method to simulate a random walk.
\item Simulate 25 random walks and count the duration of each one. Gather the data in a table with the absolute
frequency of each duration (from 1 to 20 minutes).
\item Explain the value in the column for 1 minute.
\item Is the duration necessarily less than 20 minutes ?
\item Find out the minimum, maximum, range, mean and median of this data.
\item Carry out the previous computations for all the simulated walks in the class.
\end{enumerate}
\probpart{-- Probabilistic study}
We can illustrate the situation with a probabilistic graph. The vertices B, C, D, for which
the walk doesn't end, have been gathered as a single vertex noted BCD.
From vertex A, the only possibility is to go to BCD, while from BCD it's possible to go to A or stay in BCD.
\begin{center}
\psset{xunit=1.0cm,yunit=1.0cm,algebraic=true,dotstyle=*,dotsize=3pt 0,linewidth=0.8pt,arrowsize=3pt
2,arrowinset=0.25}
\begin{pspicture*}(-0.72,-0.4)(7.66,2.68)
\pscircle(0.22,1.04){0.46}
\rput{-0.73}(4.71,1.03){\psellipse(0,0)(0.92,0.47)}
\rput[tl](0.08,1.22){A}
\rput[tl](4.32,1.20){BCD}
\parametricplot{1.0305922595526105}{2.1616224923228797}{
1*2.95*cos(t)+0*2.95*sin(t)+2.36|0*2.95*cos(t)+1*2.95*sin(t)+-0.97}
\parametricplot{4.19943498692105}{5.199866644868201}{1*3.27*cos(t)+0*3.27*sin(t)+2.33|0*3.27*cos(t)+1*3.27*sin(t)+3.47}
\parametricplot{-2.7287925292108506}{2.603954909218935}{1*0.7*cos(t)+0*0.7*sin(t)+6.34|0*0.7*cos(t)+1*0.7*sin(t)+0.92}
\psline{->}(0.97,1.63)(0.68,1.4)
\psline{->}(3.66,0.48)(3.94,0.64)
\psline{->}(5.79,0.49)(5.62,0.76)
\end{pspicture*}
\end{center}
\begin{enumerate}
\item Compute the probabilities to go to A and to stay in BCD when you are in BCD. Write theses probabilities on the
edges of the graph.
\item Build a probability tree to illustrate a four-steps random walk.
\item Compute the probabilities of a 2-minutes, a 3-minutes and a 4-minutes walk.
\item Without adding a level to the tree, conjecture a value for the probability of a 5-minutes walk. Deduce the
probability of a walk lasting 5 minutes or less.
\end{enumerate}
\exoc{{\bf\textsf{Estimation in the 2008 US election}}
In this exercise, we look again at the US 2008 election. We will introduce a better method of
estimation, based
on the concept of margin of error. Instead of a simple point estimate, we will build for each sample a \emph{confidence
interval} whose diameter depends on the margin of error we allow.
We still note $p$ the percentage of Obama electors in the whole population (so $p=0.53$). Now, consider a sample fo
size $n$ yielding a point estimate $f$ of $p$. We've seen in the previous part that the margin of error at 95\%\
confidence is $m=\frac{1}{\sqrt{n}}$. Indeed, the probability of the point estimate $f$ being in the interval
$\left[p-\frac{1}{\sqrt{n}},p+\frac{1}{\sqrt{n}}\right]$ is approximately equal to $95\%$.
\begin{enumerate}
\item Translate the fact that $f$ belongs to that interval with two inequalities.
\item Prove that the fact that $f$ belongs to that interval is equivalent to the fact that $p$ belongs to the interval
$\left[f-\frac{1}{\sqrt{n}},f+\frac{1}{\sqrt{n}}\right]$.
\end{enumerate}
The interval $\left[f-\frac{1}{\sqrt{n}},f+\frac{1}{\sqrt{n}}\right]$ is called a \emph{$95\%$ confidence interval}.
Intuitively, this means that, knowing $f$ and not $p$, we have a $5\%$ risk of being wrong if we consider that $p$ is in
the interval. But, as $p$ is fixed, it's not really correct to talk about probability. Once the confidence interval is
determined, $p$ is either in it or not !
\begin{enumerate}\setcounter{enumi}{2}
\item Find the $95\%$ confidence intervals for the surveys of exercise 1 part A.
\item How many surveys gave a confidence interval including the real value ?
\end{enumerate}
}
\exoc{{\bf\textsf{The referendum on the European constitution}}
\begin{enumerate}
\item The French referendum on the Treaty establishing a Constitution for Europe was held on 29 May 2005 to decide
whether France should ratify the proposed Constitution of the European Union. The question put to voters was: ``Do you
approve the bill authorising the ratification of the treaty establishing a Constitution for Europe?''\\
Below are given the results of some surveys carried out before the referendum.
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
Dates & Institute & Size & Proportion of <<~no~>> \\
\hline
18 and 19 March 2005 & Ipsos & 860 & 0.52 \\
\hline
25 and 26 March 2005 & Ipsos & 944 & 0.54 \\
\hline
1er and 2 April 2005 & Ipsos & 947 & 0.52 \\
\hline
16 and 17 March 2005 & CSA & 802 & 0.51 \\
\hline
23 March 2005 & CSA & 856 & 0.55 \\
\hline
1 and 2 April 2005 & Louis Harris & 1004 & 0.54 \\
\hline
31 March and 1 April 2005 & IFOP & 868 & 0.55 \\
\hline
24 March 2005 & IFOP & 817 & 0.53 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}
\item Find the $95\%$ confidence interval for each survey.
\item The result was a victory for the "No" campaign, with $54.67\%$. A commentator then said that not many surveys
had anticipated such a decisive result. What do you think of that opinion ?
\end{enumerate}
\item The United Kingdom referendum was expected to take place in 2006. Following the rejection of the Constitution by
voters in France in May 2005 and in the Netherlands in June 2005, the referendum was postponed indefinitely.\\
ICM research asked 1,000 voters in the third week of May 2005 ``If there were a referendum tomorrow, would you vote for
Britain to sign up to the European Constitution or not?'' : 57\%\ said no. Find the $95\%$ confidence interval for this
survey. If you were a politician, what would you deduce from this ?
\end{enumerate}}
%
% \probpart{-- Estimation of a percentage}
%
% In this part, we will find an estimation of the percentage $p$ of walks lasting 5 minutes or less.
%
% \begin{enumerate}
% \item Compute the percentage of walks lasting 5 minutes or less in your 25 simulated walks.
% \item Compute the $95\%$ confidence interval for this percentage in your sample.
% \item Does this interval validate the value conjectured in part B ?
% \item Compute the $95\%$ confidence interval for this percentage in the sample made of all the simulated walks in the
% class.
% \item Does this interval validate the value conjectured in part B ?
% \end{enumerate}
}
\pagebreak
\partie{Homework \#9}
\en
\raz
\begin{minipage}{11cm}
Pass the Pigs is a commercial version of the dice game Pig, that you studied in a previous homework.
Each turn involves one player throwing two model pigs, each of which has a dot on one side only. The player will have
points either given or taken away, based on the way the pigs land (see below). Each turn lasts until the player throwing
either rolls the pigs in a way that wipes out their current turn score or decides to stop their turn, add their turn
score to their total score and pass the pigs to the next player.
\end{minipage}
\begin{minipage}{5cm}
\begin{center}
\includegraphics[width=4cm]{Images/passthepigs.eps}
\end{center}
\end{minipage}
\smallskip
The winner is the first player to score a total of 100. You can play a virtual version of the game on the following
webpage :
\begin{center}
http://www.toptrumps.com/play/pigs/pigs.html
\end{center}
There are 6 main positions, each of them worth a certain number of points.
\begin{itemize}
\item The pig is lying on its side, with the dot visible, 0 points.
\item The pig is lying on its side, with the dot not visible, 0 points.
\item Razorback: The pig is lying on its back, 5 points.
\item Trotter: The pig is standing upright, 5 points.
\item Snouter: The pig is leaning on its snout, 10 points.
\item Leaning Jowler: The pig is resting on its snout and ear, 15 points.
\end{itemize}
\bigskip
As the game is played with two pig dice, it's the combinations that really count. The number of points for each
combination is given below :
\begin{itemize}
\item Sider: The pigs are on their sides, either both with the spot facing upward or both with the spot
facing downward, 1 Point.
\item Double Razorback: The pigs are both lying on their backs, 20 points.
\item Double Trotter: The pigs are both standing upright, 20 points.
\item Double Snouter: The pigs are both leaning on their snouts, 40 points.
\item Double Leaning Jowler: The pigs are both resting between snouts and ears, 60 points.
\item Mixed Combo: A combination not mentioned above is the sum of the single pigs score.
\item Pig Out : If both pigs are lying on their sides, one with the spot facing upwards and one with the spot
facing downwards the score for that turn is reset to 0 and the turn changes to the next player.
\end{itemize}
There are in fact two other combinations. ``Making bacon'' is when the two pigs touch other. Then the total score of
the player is reduced to 0. For the sake of decency, the last combination can't be described in this homework. Anyway,
we won't consider these two combinations, are they are very unlikely.
\probpart{-- Single pig frequencies}
It is almost impossible to know the probability of each position for one pig. The shape of the pig is so complicated
that it's not even easy to answer the simple question : do the two sides of the pig have the same probability to appear
?
The best approach is therefore to use statistics. A few statistical studies have been carried over a large number of
throws of a single pig die. One of these studies, using a standardized surface and trap-door rolling device and a
sample size of 11,954 gives the following absolute frequencies :
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Position & Side (no dot) & Side (dot) & Razorback & Trotter & Snouter & Leaning Jowler \\
\hline
Frequency & $4177$ & $3615$ & $2678$ & $1052$ & $359$ & $73$ \\
\hline
\end{tabular}
\end{center}
As the margin of error at $95\%$ confidence for this sample is $\frac{1}{\sqrt{11,954}}\approx0.0091$, we will consider in the next questions that the probabilities are equal to the relative frequencies in this large sample.
% As the true probabilities are unknown, we will consider in the next questions that they are equal to the relative
% frequencies in this large sample.
\begin{enumerate}
\item Compute the relative frequencies to 2DP and show them in a frequency table.
\item According to these values, do the two sides of the pig have the same probability to appear ? If not, which side
is more likely ?
\item Compute the average score for a single pig.
\item What would be the total score expected when throwing 300 times a pig ?
\end{enumerate}
Elliot, a four years old boy, has been playing around with a pig-shaped die. He threw it 300 times and got the
following absolute frequencies :
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Position & Side (no dot) & Side (dot) & Razorback & Trotter & Snouter & Leaning Jowler \\
\hline
Frequency & $93$ & $107$ & $63$ & $31$ & $5$ & $1$ \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}\setcounter{enumi}{4}
\item Compute the relative frequencies to 3DP and show them in a frequency table.
\item Compute, to 3DP, the margin of error for a sample of 300 throws.
\item For each position, build the fluctuation interval at $95\%$ around the relative frequencies in the biggest
sample, the ones used as probabilities.
\item In Elliot's sample, there were more dot sides than no dot sides. Use the fluctuation intervals to decide if this
is just due to randomness or if it means that the pig die used was not regular.
\item Compute Elliot's total score for the 300 throws. Compare it to a previous result to answer the following question
: Was he lucky or not ?
\end{enumerate}
\probpart{-- Two pigs probabilities}
The game is in fact played with two pigs. As the two pigs are independent, to compute the probability of each double
figure, we just need to multiply the probabilities of the two single figures. For example, as the probability of a
razorback is $0.224$, the probability of a double razorback is $0.224\times 0.224=0.05019$.
\begin{enumerate}
\item Copy and fill the double-entry table, that gives the probability of each possible double
figure. Give values to 5DP. Notice that the table is symmetric around one of the diagonals.
\begin{center}
\hspace*{-1cm}\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
& Side (no dot) & Side (dot) & Razorback & Trotter & Snouter & Leaning Jowler \\
\hline
Side (no dot) & & & & & & \\
\hline
Side (dot) & & & & & & \\
\hline
Razorback & & & $0.05019$ & & & \\
\hline
Trotter & & & & & & \\
\hline
Snouter & & & & & & \\
\hline
Leaning Jowler & & & & & & \\
\hline
\end{tabular}
\end{center}
\item Fill out a similar table with the scores of each double figure. For example, according to the rules, a double
razorback is worth 20 points.
\item Use the two tables to compute the average score for two pigs (to 5DP). Explain your method but don't show the
details of your computation on your paper.
\item Is the average score for two pigs equal to the double of the average for one pig ? If not equal, is it higher or
lower ? Explain why.
\end{enumerate}
\pagebreak
\raz
\partie{Last year's test}
% \en
% \exods{10}{%Tableaux de signe et fonctions
%
% \begin{center}\psset{xunit=2cm,yunit=0.5cm,algebraic=true,arrowsize=5pt}
% \def\xmin{-3.3}\def\ymin{-6.3}\def\xmax{2.3}\def\ymax{9}
% \begin{pspicture*}(\xmin,\ymin)(\xmax,\ymax)
% \psaxes[labels=all,labelsep=1pt, Dx=1,Dy=1]{->}(0,0)(\xmin,\ymin)(\xmax,\ymax)
% \psgrid[gridlabels=0pt,gridwidth=.3pt, gridcolor=black, subgridwidth=.3pt, subgridcolor=gray,
% subgriddiv=2,ticklinestyle=dashed](\xmin,\ymin)(\xmax,\ymax)
% \psplot[linecolor=rouge]{-3}{2}{(x-1)*x*(x+2.5)}
% \end{pspicture*}\end{center}
%
%
% \begin{enumerate}
% \item Above is given the curve of a function $f$ whose expression is unknown.
% \begin{enumerate}
% \item Read the image of $2$ under the function $f$.
% \item Read the preimage of $3$ under function $f$.
% \item Draw the variation table of the function $f$.
% \item Solve graphically the inequations $f(x)>3$ and $-1\leq f(x)\leq 1$/
% \item Give the maximum of the function $f$ over the interval $[-2 ; 0]$.
% \end{enumerate}
% \item A second function $g$ is defined by $g(x)=\sqrt{x+3}$.
% \begin{enumerate}
% \item Compute the image of $6$ under $g$.
% \item Compute the preimage of $5$ under $g$.
% \item Draw the table of values of $g$ over $[-3;2]$ (with approximates to 1 DP if necessary).
% \item Plot the function $g$ in the same coordinate graph than $f$.
% \item Solve graphically the equation $f(x)=g(x)$ and the inequation $f(x)\geq g(x)$.
% \end{enumerate}
% \end{enumerate}
% }
%
% \en
% \exods{10}{The aim of this exercise is to prove a theorem, first in a special case, then in a general situation.\\
%
% Let $ABC$ be a scalene triangle, with $I$ and $J$ the midpoints of the sides $AB$ and $BC$.
%
% \probpart{ -- A special case}
% Let consider $A(1,3)$, $B(5,1)$ and $C(-1,-1$) in an orthonormal coordinate graph.
% \begin{enumerate}
% \item Compute the coordinates of $I$, then of $J$.
% \item Compute the coordinates of $\vect{IJ}$ and $\vect{AC}$.
% \item Deduce that the straight lines $IJ$ and $AC$ are parallel.
% \item Compute the length $IJ$ and $AC$.
% \item Write down the theorem illustrated by this part. The whole theorem is expected, not just its name.
% \end{enumerate}
%
% \pagebreak
% \probpart{ -- General situation}
% \begin{enumerate}
% \item What relation between the vectors $\vect{IB}$ and $\vect{AB}$ comes from the definition of the point $I$ ? Same
% question for the vectors $\vect{BJ}$ and $\vect{BC}$.
% \item Decompose the vectors $\vect{IJ}$ through point $B$.
% \item Deduce from the two previous questions a simple relation between the vectors $\vect{IJ}$ and $\vect{AC}$.
% \item What can you conclude about the straight lines $IJ$ and $AC$, then for the lengths.
% \end{enumerate}
% }
%
\raz
\fr
%\exods{20}{
\probpart{-- Étude d'un jeu}
Dans un casino, un jeu est proposé aux clients. Ceux-ci peuvent gagner entre 0 et 10 euros à chaque partie.
Avant de se décider à jouer, Roger a noté les résultats de 1000 parties. Les résultats sont donnés dans le
tableau ci-dessous.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Gain du joueur & 0 & 2 & 4 & 6 & 8 & 10 \\
\hline
Effectif & 129 & 327 & 331 & 158 & 47 & 8 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}
\item Établir la distribution des fréquences.
\item \begin{enumerate}
\item Calculer l'étendue, la médiane et les quartiles de cette série statistique.
\item Interpréter la valeur de la médiane dans le contexte de l'exercice.
\end{enumerate}
\item \begin{enumerate}
\item Calculer la moyenne de cette série statistique.
\item Le prix d'une partie étant de 5 euros, le jeu semble-t-il intéressant pour
Roger ?
\end{enumerate}
\item Suite à une indiscrétion du teneur de table, Roger sait que la probabilité de gagner 4 euros est de $35\%$.
\begin{enumerate}
\item Déterminer l'intervalle de fluctuation à $95\%$ pour un échantillon de 1000 parties.
\item Que doit penser Roger pour son relevé de 1000 parties ? on attend ici une réponse argumentée,
éventuellement nuancée.
\end{enumerate}
\end{enumerate}
\probpart{-- Étude des joueurs}
Sur les 1000 participants au jeu, Roger a constaté que :
\begin{list}{$\bullet$}{}
\item 60\% des joueurs sont de sexe masculin ;
\item 35\% des joueurs ont moins de 25 ans et, parmi ceux-ci, 80\% sont des garçons ;
\item 30\% des joueurs ont plus de 50 ans et, parmi celles-ci, 85\% sont des femmes.
\end{list}
\begin{enumerate}
\item Recopier et compléter le tableau suivant :
\begin{center}
\hspace*{-1cm}\begin{tabular}{|c|c|c|c|}
\hline
& Hommes & Femmes & Total \tabularnewline\hline
Moins de 25 ans & & & \tabularnewline\hline
De 25 à 50 ans & & & \tabularnewline\hline
Plus de 50 ans & & & \tabularnewline\hline
Total & & & 1\,000 \tabularnewline\hline
\end{tabular}
\end{center}
\item On choisit au hasard une personne parmi les 1000 joueurs.
On suppose que toutes les personnes ont la même probabilité d'être choisies.\\
On considère les événements :
$A$ : \og la personne interrogée est un homme \fg
$B$ : \og la personne interrogée a moins de 25 ans \fg.
\begin{enumerate}
\item Lire dans le tableau les probabilités $p(A)$ et $p(B)$.
\item Définir par une phrase l'événement $A\cap B$, puis lire dans le tableau $p(A\cap B)$.
\item Définir par une phrase l'événement $\bar{B}$, puis calculer sa probabilité.
\item Définir par une phrase l'événement $A\cup B$, puis calculer sa probabilité.
\item On sait maintenant que la personne interrogée n'est pas un garçon. Quelle est la probabilité qu'elle ait moins de
50 ans ?
\end{enumerate}
\end{enumerate}
%}
%
\newpage\thispagestyle{empty}\null\newpage
\null\vfill\thispagestyle{empty}
\begin{center}
{\Large\bf\textsf{Glossary}}
\end{center}
\begin{center}
\begin{tabularx}{15cm}{|c|c|X|}
\hline
\textbf{English} & \textbf{French} & \textbf{Explanation} \\
\hline
Survey & Sondage & A method for collecting quantitative information about items in a population. \\\hline
Sample & Échantillon & A subset of a population selected for measurement, observation or
questioning, to provide statistical information about the population. \\\hline
Sampling & Échantillonage & The process or technique of obtaining a representative sample. \\\hline
Margin of error & Marge d'erreur & An expression of the lack of precision in the results obtained
from a sample. \\\hline
Fluctuation interval & Intervalle de fluctuation & For a certain proportion of samples, the interval where the
parameter studied should be.\\\hline
Estimate (verb) & Estimer & To calculate roughly, often from imperfect data. \\\hline
Estimate & Estimation & A rough calculation or guess. \\\hline
Estimation & Estimation & The process of making an estimate. \\\hline
Point estimate & Estimation ponctuelle & A single value computed from sample data, used as a "best guess" for an
unknown population parameter. \\\hline
Confidence interval & Intervalle de confiance & A particular kind of interval estimate of a
population parameter. \\\hline
Simulate (verb) & Simuler & To model, replicate, duplicate the behavior, appearance or properties of a system or
environment\\\hline
Simulation & Simulation & Something which simulates a system or environment in order to predict
actual behaviour. \\
\hline
\end{tabularx}
\end{center}
\bigskip
\begin{center}
\begin{quote}
{\em Aw, people can come up with statistics to prove anything, Kent. Forfty percent of all people know that.}
\hfill (Homer Simpson)
\end{quote}
\bigskip
\begin{quote}
{\em Lottery: A tax on people who are bad at math.}
\hfill (Anonymous)
\end{quote}
\bigskip
\begin{quote}
{\em Do not put your faith in what statistics say until you have carefully considered what they do not say.}
\hfill (William W. Watt)
\end{quote}
\bigskip
\begin{quote}
{\em He uses statistics as a drunken man uses lampposts - for support rather than for illumination.}
\hfill (Andrew Lang)
\end{quote}
\end{center}
\vfill
\end{document}