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\centerline{MONT 107N -- Understanding Randomness}
\centerline{Group Discussion 2}
\centerline{February 22, 2010}
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\noindent
{\it Background and Goals}
\vskip 10pt
We have seen how to estimate percentages by sampling
a population, using
the insight provided by 0/1 box models.  Because
of the Central Limit Theorem, we can use the normal
curve to estimate the chances of certain results
of the sampling process.  The goal for today is 
to practice on several problems of this type, in order
to ``solidify'' our understanding of the ideas and
processes involved.
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\noindent
{\it Discussion Questions}
\begin{enumerate}
\item[A.]  The point of this question is to reinforce
the idea that in finding estimates of percentages by 
sampling, we actually use {\it an estimate, or approximation}
to the standard error -- this leads to what Freedman, Pisani,
and Purves call the ``bootstrap.''  
Suppose there is a 0/1 box with 10,000
tickets, and exactly 2000 of them are 1's. 
\begin{enumerate}
\item[1.]  What is the actual exact value of the standard
error for 400 draws from the box?
\item[2.]  Now a person (who does not know the true
percentage of 1's in the box) draws a random sample
of 400 tickets, and finds $75$ $1$'s in that sample.  
How would that person estimate the
percentage of tickets in the box and the standard error?
\item[3.]  Same question as in part 2, but for a 
second person who draws a different random sample
containing $87$ $1$'s.
\item[4.]  Is the true value of the percentage of 
1's within 2 (estimated) SE's of the estimate of
the percentage in parts 2 and 3?
\end{enumerate}
\item[B.]  
\begin{enumerate}
\item[1.] A random sample of size $400$ was taken from
the population of all manufacturing companies in a big
state.  Of the 400, 11 had more than 100 employees.  Estimate
the percentage of manufacturing companies with more than
100 employees. 
\item[2.]  In the situtation of part 1. estimate the standard
error for this percentage estimate.  
\item[3.]  How likely is it that the true percentage of 
companies with more than 100 employees was greater than
$4\%$?
\item[4.]  In the same state as in the previous parts, 
a random sample of 400 was taken from the population of 
all individuals employed by manufacturing companies.  A 
total of 187 individuals in this sample worked for companies 
with more than 100 employees.  Estimate the percentage of  
of workers employed by companies with more than 100 employees.
\item[5.]  Estimate the standard error for the percentage
from part 4.
\item[6.]  What is the chance that fewer than $40\%$
of the individuals employed by manufacturing companies
are employed by companies with more than 100 employees?
\item[7.]  Is the difference between your answers to 
parts 1 and 4 due to chance error?
\end{enumerate}
\end{enumerate}
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