\documentclass[11pt]{article}
\def\Z{\mathbb{Z}}
\def\Q{\mathbb{Q}}
\def\R{\mathbb{R}}
\def\C{\mathbb{C}}
\usepackage{latexsym,amsmath,amssymb,epsfig}
\usepackage[all]{xy}
\newenvironment{packed_enum}{
\begin{enumerate}
  \setlength{\itemsep}{1pt}
  \setlength{\parskip}{0pt}
  \setlength{\parsep}{0pt}
}{\end{enumerate}}
\newenvironment{packed_item}{
\begin{itemize}
  \setlength{\itemsep}{1pt}
  \setlength{\parskip}{0pt}
  \setlength{\parsep}{0pt}
}{\end{itemize}}
\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{0in}
\setlength{\textheight}{8.0in}
\begin{document}
\centerline{MONT 105Q -- Mathematical Journeys}
\centerline{Problem Set 1 -- Basic Descriptive Statistics}
\centerline{Due: Friday, February 19}
\vskip 10pt
\noindent
{\it Directions}
\vskip 10pt
Do the following problems and prepare individual writeups to hand in at 
the start of class on February 19.
\vskip 10pt
\noindent
I.  Consider the following data set (list of numbers):
\begin{eqnarray*}
&&0.7,\ 1.6,\ 9.8,\ 3.2,\ 5.4,\ 0.8,\ 7.7,\ 6.3,\ 2.2,\ 4.1\\
&&8.1,\ 6.5,\ 3.7,\ 0.6,\ 6.9,\ 9.9,\ 8.8,\ 3.1,\ 5.7,\ 9.1
\end{eqnarray*}
\begin{enumerate}
\item[(A)]  What are the mean ($\mu$) and median of this data set?  
\item[(B)]  Suppose the last number $9.1$ was altered to $15.3$.  \emph{Without
doing any computations}, say what would happen to the mean and the median.  
Does each of them increase, decrease, or stay the same?  
\item[(C)]  What is the SD (standard deviation, or $\sigma$) of this data set?  
\item[(D)] To what extent is
the pattern from the diagram on page 26 of \emph{Naked Statistics} true for 
this data set?  Count how many of the numbers in the list are in each of the 
intervals $[\mu - 2\sigma,\mu - \sigma)$, $[\mu - \sigma, \mu)$, 
$[\mu,\mu + \sigma)$, $[\mu + \sigma, \mu + 2\sigma]$ and compute the percent of the
numbers in the whole list that lie in each interval.  We never expect
exact agreement with something like the percentages from page 26, but are
the actual percentages even in the right ballpark for this data set?  
Does this look like a random sample from a population described by 
a normal distribution?  Explain.  
\item[(E)]  Another way to understand the distribution of a data set
like this is to draw a diagram called a \emph{histogram}.  There
are several possible choices statisticians might make to do this. Follow
these directions to produce first a \emph{frequency histogram}, and then
a \emph{standardized histogram} with total area 1:  
\begin{enumerate}
\item[(1)]  The numbers in the data set are all contained in the range $[0,10]$.
Suppose we divide this interval into 5 ``bins'' $[0,2),[2,4),[4,6),[6,8),[8,10]$.
How many numbers from the data set fall into each of the bins?  (By the way, 
there's nothing magic about using 5 bins.  Any number is possible, but larger
numbers of bins tend to spread out the data too much and produce less informative
histograms, so 5 or 6 is a common choice!  Also, to avoid having to make awkward
choices, it's usually good to choose bin boundaries so that no numbers from 
the data set cooincide with them.  Note that that worked out fine for us here.)
\item[(2)]  Draw perpendicular $x$- and $y$-axes to make a coordinate system 
and plot the 5 ``bins'' along the $x$-axis.  Then make rectangular boxes over each of those
bins where the height gives the number of values from the data set in the 
corresponding bin.  This is the \emph{frequency histogram}.
\item[(3)]  Then determine a vertical scaling factor $\lambda$ (a number) with the 
property that multiplying each height from the \emph{frequency histogram} by $\lambda$
will yield a collection of boxes whose \emph{areas add up to 1}.  Redraw the histogram
with the new scale on the $y$-axis showing this standardized version of the histogram.
(The shape should be exactly the same, only the vertical scale changes!)
\end{enumerate}
\end{enumerate}
\noindent
II.  A study on college students showed that the 150 men studied had an average mass 
of 66kg, with an SD of 9kg.  The 200 women studied had an average mass of 55kg, with an
SD of 9kg.
\begin{enumerate}
\item[(A)]  About how many of the 150 men would we expect to have masses
between 57kg and 75kg?  
\item[(B)]  If you took the men and women together would the SD for the whole
group of 350 students be exactly 9kg, larger than 9kg,  or smaller than 9kg?  Why?
\end{enumerate}
\end{document}


  
\end{document}
