\documentclass[11pt]{article}
\def\Z{\mathbb{Z}}
\def\Q{\mathbb{Q}}
\def\R{\mathbb{R}}
\def\C{\mathbb{C}}
\usepackage{latexsym,amsmath,amssymb}
\usepackage[all]{xy}
\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{0in}
\setlength{\textheight}{8.0in}
\begin{document}
\centerline{MONT 106N -- Identifying Patterns}
\centerline{Group Discussion 2}
\centerline{September 21, 2009}
\vskip 10pt
\noindent
{\it Background and Goals}
\vskip 10pt
\noindent
We have discussed the average and the SD of a list of numbers
as ways to measure where the ``middle'' is (the average) and
how ``spread out,'' or diverse, the numbers are (the SD).  
The goals for today's discussion are to practice using 
these ideas, and especially to understand their meaning
in a rough, ``back of the envelope'' way (that is, without
necessarily doing extensive computations).  
As on the last group day, you will be handing in one set of 
solutions from each group \emph{at the end of class}.
\vskip 10pt
\noindent
\begin{enumerate}
\item[A.] Which of the two lists below has the larger
average?  Explain -- this does not require any calculation.
\begin{enumerate}
\item[1.] $12, 11, 8, 7, 8, 10, 6, 4, 10$
\item[2.] $10,  9, 8, 7, 8,  9, 6, 10, 4$.
\end{enumerate}
\item[B.]  Each of the following lists has an average of 50.
For each one, is the SD probably closer to 1, to 5, or to 10?  Explain, and 
again \emph{no calculation is necessary}.
\begin{enumerate}
\item[1.] $59, 41, 48, 52, 53, 47, 50, 57, 43, 50$  
\item[2.] $60, 36, 31, 50, 48, 50, 54, 56, 72, 43$
\item[3.] $51, 49, 48, 48, 51, 52, 50, 51, 48, 52$
\end{enumerate}
\item[C.]  One investigator takes a sample of 100 men in 
a New York City, and a second investigator takes a 
sample of 1000 such men.  Both investigators make 
an effort to assure their samples are random and 
representative of the whole population.
\begin{enumerate}
\item[1.]  Which investigator will get a bigger average
for the heights in her sample?  Or should the averages
be about the same?
\item[2.]  Which investigator will get a bigger SD
for the heights in her sample?  Or should the SD's
be about the same?
\item[3.]  Which investigator will probably find
a greater \emph{range} of heights in the sample
(the range is the biggest height minus the smallest
height)?
\end{enumerate}
\item[D.]  Suppose you have a list of positive numbers.
\begin{enumerate}
\item[1.] If you add 10 to each number to make a new list, 
how do the average and SD of the new list compare with the 
average and SD of the original list?
\item[2.]  Same question if you increase each number in the 
list by 10\% to make a new list.
\end{enumerate}
\item[E.] 
\begin{enumerate}
\item[1.] Suppose you took a fair (evenly balanced) coin
and tossed it a large number of times recording the faces that come
up as numbers: heads = 1, tails = 0 for each toss.  Approximately 
what would you expect the average and SD of your list to be?
\item[2.] Try it with $50$ throws of an actual coin
and see how close you get to your guesses.
(You will need to do some calculations here, but you 
can reduce the work to a minimum by thinking before you
pull out the calculator!) 
\end{enumerate}
\end{enumerate}
\end{document}
