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\leftline{MONT 105Q -- Mathematical Journeys}
\leftline{Midterm Exam, April 1, 2016}
\leftline{Your Name:  $\underline{\qquad\qquad\qquad\qquad\qquad\qquad}$ }
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\noindent
{\it Directions}
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Do all work on the sheets provided (if you use the back of a sheet,
please place a note telling me to look there).  
There is an extra sheet of scratch paper at the end.  There are 100 possible points,
distributed as indicated in the questions.
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\noindent
I.  The following data set has $n = 11$:
$$1.3,\ 2.2,\ 0.8,\ 2.3,\ 3.9,\ 4.1,\ 2.2,\ 1.7,\ 1.2,\ 3.1,\ 2.4$$
(Note:  These are \emph{not in numerical order}.)
\begin{enumerate}
\item[(A)]  (5) Find the ``5-number'' summary for this
data set (min, 1st quartile, median, 3rd quartile, max).
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\item[(B)]  (5) Construct a frequency histogram for the data using 5 equal bins
on the range $0.0$ to $5.0$.  
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\item[(C)]  (10) Compute the SD of the data set, showing all work to justify your answer. 
How many of the data values lie within one SD of the mean?  
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\item[(D)]  (5) If the numbers in this data set are written on slips
of paper and put in a box, the box is shaken, and two slips are drawn 
at random without replacement, what is the probability that both are $< 3.0$?
\end{enumerate} 
\vfil\eject
\noindent
II.  Answer each question with a sentence or two.  
Suppose that a researcher collects 40 tufted titmice 
(these are small North American songbirds) to measure their body weights.  
The collected birds have an average weight of $25$ g with an SD of $1$ g.  
Think of this as a sampling process.
\begin{enumerate}
\item[(A)] (5) What is the population?  What is the sample?
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\item[(B)] (5) What could we say about the distribution of 
sample mean weights of samples of size 100 of tufted titmice 
(even if the distribution of the individual weights was unknown)?  Explain. 
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\item[(C)] (5) The average lifespan of tufted titmice is 2.1 years but many birds 
die as nestlings (i.e. before reaching maturity, which takes about 4 months from the time they
hatch).  Would you expect the average age of the birds in a sample of adults collected in 
the wild to be larger or smaller than 2.1 years?  Explain.  
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\item[(D)] (5) If you knew that the researcher was only taking
birds from a particular location, is that a random sample
from the appropriate population?   Might that process 
introduce a bias in the measured weights?  
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\end{enumerate}
\noindent
III.  Suppose that a large data set of standardized raw test scores is \emph{normally 
distributed} with  $\mu = 75$ and $\sigma = 4$.
The ``$z$-score'' of a raw test score $x$ is computed by
$$z = \frac{x - \mu}{\sigma}.$$
\begin{enumerate}
\item[(A)]  (5) What would be the $z$-score of a raw test score of $80$?
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\item[(B)]  (5)   Based on this information, if a raw test score $x$ 
is selected at random from the data set, 
what is the probability that $74 \le x \le 80$?
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\item[(C)]  (10) Based on this information, if $n = 20$ raw test scores
are selected at random, what is the probability that \emph{none of them}
is in the range $74 \le x \le 80$.
\vskip 2.5truein
\end{enumerate}
\noindent
IV.  Essay.  (35) What is the central thesis of the article {\it The Dawning of 
the Age of Stochasticity} by David Mumford?  How does Mumford think mathematics
should develop in the current century?  How is his proposal related to the
outcomes of the various attempts to solve the ``crisis of foundations'' we
read about in {\it Logicomix}?  Would Mumford agree or disagree with this
statement:  ``what mathematicians should be doing is closer to the kind of 
science Darwin practiced on the voyage of the {\it Beagle} than it is to a 
Euclidean search for absolute certainty.''  Explain.
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$\qquad$
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