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\leftline{MONT 105Q -- Mathematical Journeys}
\leftline{Midterm Exam Solutions, April 1, 2016}
%\leftline{Your Name:  $\underline{\qquad\qquad\qquad\qquad\qquad\qquad}$ }
\vskip 10pt
\noindent
I.  The following data set has $n = 11$:
$$1.3,\ 2.1,\ 0.8,\ 2.3,\ 3.9,\ 4.1,\ 2.2,\ 1.7,\ 1.2,\ 3.1,\ 2.4$$
(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).
\item[] {\it Solution:}  After reordering smallest to largest we have:
$$0.8,\ 1.2,\ 1.3,\ 1.7,\ 2.1,\ 2.2,\ 2.3,\ 2.4,\ 3.1,\ 3.9,\ 4.1.$$
Then 
\begin{eqnarray*}
{\rm minimum} &=& 0.8\\
{\rm first\ quartile} &=& 1.3\\ 
{\rm median} &=& 2.2\\
{\rm third\ quartile} &=& 3.1\\
{\rm max} &=& 4.1
\end{eqnarray*}
\item[(B)]  (5) Construct a frequency histogram for the data using 5 equal bins
on the range $0.0$ to $5.0$.  
\item[] {\it Solution:}  The bins and counts ($=$ heights of the boxes) are 
$$ [0,1) \rightarrow 1, [1,2) \rightarrow 3, [2,3) \rightarrow 4, [3,4) \rightarrow 2, 
[4,5] \rightarrow 1$$
\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?  
\item[] {\it Solution:} We compute the mean $\overline{X}$ first:
$$\overline{X} = \frac{1.3+ 2.1+0.8+2.3+3.9+4.1+2.2+1.7+1.2+3.1+2.4}{11} \doteq 2.2818$$
Then the SD is 
$$\sqrt{\frac{1.3^2+2.1^2+0.8^2+2.3^2+3.9^2+4.1^2+2.2^2+1.7^2+1.2^2+3.1^2+2.4^2}{11} - (2.2818)^2} \doteq 1.0142.$$
The range within 1 SD of the mean is $2.2818 - 1.0143 = 1.2675$ to $2.2818 + 1.0143 = 3.2961$.
Of the 11 numbers, 7 are in this range.  (Comment:  $7/11 \doteq .64$, which fits well with
the ``empirical rule'' based on the normal distribution.)
\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 $< 3.0$?
\item[] {\it Solution:}  There are $8$ numbers $< 3.0$ to begin with.  After a 
first number $< 3.0$  is chosen, $7$ of the remaining $10$ numbers are $< 3.0$.  So 
this probability is $\frac{8}{11}\cdot \frac{7}{10} = \frac{56}{110} = \frac{28}{55} \doteq .51$.  
(This could also be computed as the ratio $\frac{\binom{8}{2}}{\binom{11}{2}}$.)
\end{enumerate} 
\vskip 10pt
\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?
\item[] {\it Solution:}  The population would be the weights of all tufted titmice.
The sample is the 40 weights of the collected birds.
\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. 
\item[] {\it Solution:}  By the Central Limit Theorem, the sample mean weights 
would be approximately normally distributed with mean equal to the population mean
weight and SD equal to the population SD divided by $\sqrt{100}$.  This last
quantity could be estimated by $\frac{1}{\sqrt{100}} = .1$.
\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 birds in a sample of adults collected in 
the wild to be larger or smaller than 2.1 years?  Explain.  
\item[] {\it Solution:}  Given that we are observing living, adult birds, they 
have survived the nestling stage, so the average age should be \emph{greater than} 2.1
years.  (Another way to say this:  the birds that die as nestlings pull the average
age down.  So the ages of the living adult birds we collect would be larger.)
\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?  
\item[] {\it Solution:}  It is probably not a random sample, and it could bias
the measured weights if food is particularly plentiful (or not) in the area
where the sample is taken.
\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 $80$?
\item[] {\it Solution:}  $z = \frac{80 - 75}{4} = 1.25$.
\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 $74 \le x \le 80$?
\item[] {\it Solution:}  $P(74 \le x \le 80) = P(-.25 \le z \le 1.25)$.
Using the normal table, this is: $(.5 - .4013) + (.5 - .1056) = .4931$.
\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 $74 \le x \le 80$.
\item[] {\it Solution:}  This is 
$$(1 - .4931)^{20} = \binom{20}{0}(.4931)^0 (1 - .4931)^{20} \doteq .000001$$
(i.e. virtually no chance of this happening!)
\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.
\vskip 10pt
\noindent
{\it Model Response:}  The central thesis of Mumford's article is that stochastic
(that is, probability-based) models and statistical thinking will be more important in
the mathematics of the 21st century than exact models, logical argumentation and proof.  They
are more relevant to the real world, to many parts of mathematics, and in particular
to understanding how our minds work.  In a way, we might say this suggestion is at
least in part a reaction to the fact that none of the various attempts to put 
mathematics on a completely secure footing and solve the ``crisis in foundations'' 
really got off the ground.  They never successfully addressed the basic concerns about
the certainty of mathematical knowledge.  Mumford points out (rather scornfully, too)
how the logicist approach taken in 
Russell and Whitehead's monumental {\it Principia Mathematica} only succeeded in 
developing the most basic of mathematical facts (e.g. $1 + 1 = 2$), at the cost
of hundreds of pages of very hard work and almost impenetrable symbolism.  
Similarly, Hilbert's dream of axiomatic
systems powerful enough to allow all true statements to be proved and all false
statements to be proved false was effectively dashed by G\"odel's incompleteness
theorems.  The constructivist approaches to mathematics have been marginally more
successful.  But very few mathematicians work that way (restricting themselves to 
finite sets and things that can be constructed in finite terms).  Constructive
methods still require some very hard work to prove very basic facts, especially
in areas like calculus.  If Mumford's ideas become widely accepted, the net effect of 
his polemic in favor of stochastic
models and statistical thinking would definitely be to make mathematics \emph{closer
to the other sciences}, and closer to what Darwin was doing in his observations.
That would give a greater role to the sort of inductive reasoning that scientists
also do and a smaller role to deductive reasoning and logical argumentation.  
This sort of work will of necessity lack the sort of absolute certainty that people ascribed 
to the results of Euclidean geometry.  So he \emph{would agree with 
this statement}.    
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