\documentclass[11pt]{article}
\usepackage{graphicx}
\textwidth=6.5truein
\textheight=9truein
\oddsidemargin=-0.0truein
%\def\blank{\mbox{\rule[-1pt]{.6truein}{1pt}}}

\begin{document}
\pagestyle{empty}

\voffset -.7 true in
\hoffset -.2 true in

\rightline{February 23, 2010}

\centerline{\bf Math 102 / Core 143 BX --- Exam I}

\vskip 12 pt
\noindent
An unsimplified answer like $12\sqrt{3.51}+6/7$ is usually worth more
than 23.3, because it is easier to understand where it came from.

\begin{enumerate}

\item[1.] \textit{(20 points)} Consider the histogram of number of years
of service of employees at a firm (in whole numbers of years):\newline
\centerline{\includegraphics[scale=.5]{e1s07n1.png}}
\begin{itemize}
\item[(a)] What is the median of the data?
\item[(b)] Approximately what is the average?
\item[(c)] Which direction is the skew, if any?
\item[(d)] Using the ``density scale'' in which the total area of the
bars is 100\%, what value should go on the lowest hash mark (above 0)
on the vertical axis?
\end{itemize}

\item[2.] \textit{(25 points)} A botanist studying a certain type of
plant has many samples in pots in her greenhouse.  She records the
heights in centimeters, measuring from the level of soil in the pots.
She finds that three are 8~cm tall, one is 9~cm, two are 11~cm, and
one each 12 and 13~cm.
\begin{itemize}
\item[(a)] What is the average height?
\item[(b)] What is the standard deviation of the heights?
\item[(c)] What percent of the data is within one standard deviation
of the average? (There are two equally acceptable answers, depending
on how you interpret ``within''.)
\item[(d)] In view of your answer to (c), would you describe the heights
as normally distributed?
\item[(e)] She decides to record the heights in decimeters (10~cm =
1~dm) measuring from the tabletop rather than the top of the soil.
The soil level in each pot is 15~cm (i.e., 1.5~dm) above the tabletop.
What are the new average and standard deviation?
\end{itemize}

\item[3.] \textit{(7 points)} Heights of female college students are
normally distributed, with an average of 67~inches and a standard
deviation of 4~inches.  Approximate the 80-th percentile for height.

\newpage

\item[4.] \textit{(20 points)} The botanist of an earlier problem also
has plants of another species in her lab and is testing the effect of
a certain fertilizer on the height of this species.  She finds the
following summary data:\newline
\centerline{\begin{tabular}{lccr}
& fertilizer (cc) & height (cm) \\
mean & 50 & 30 & $r=0.8$ \\
std dev & 16 & 8
\end{tabular}
}\newline
\begin{itemize}
\item[(a)] Use linear regression to project the height of a plant that
receives 58~cc of fertilizer.
\item[(b)] Use linear regression to estimate the amount of fertilizer
given to a plant 22~cm high.
\item[(c)] How far is your estimate in (b) likely to be off?
\item[(d)] Among the scatter diagrams below, which is most likely to
be the graph of fertilizer amounts vs. heights?\newline
\centerline{\includegraphics[width=4in]{e1s07n4.png}}
\end{itemize}

\item[5.] \textit{(8 points)} Classify the following as categorical
or numerical variables.
\begin{itemize}
\item[(a)] Heights of plants
\item[(b)] Sex, coded as 0 for males and 1 for females
\item[(c)] Responses of ``agree strongly'', ``agree'', ``neutral'',
``disagree'', ``disagree strongly'' to a statement of opinion
\item[(d)] Diastolic blood pressure
\end{itemize}

\item[6.] \textit{(10 points)} [The point of this exercise is to explore
the mantra ``Correlation does not imply causation'' --- what causes what?]
In a small community, an unusually large
number of people seem to have colds, and many of the sufferers are
found to frequent the community's only restaurant. Answer each of the
following in a sentence or two:\begin{itemize}
\item[(a)] How might going to the restaurant have caused its customers
to have colds?
\item[(b)] How might the fact that they have colds caused people
to eat at the restaurant?
\item[(c)] What other factor may have caused people both to get
colds and to eat at the restaurant?
\end{itemize}

\item[7.] \textit{(10 points)} Related to the articles ``The Sins
of the Fathers, Take~2'' and ``What Alters Our Genes'' by Sharon Begley:
Is epigenetics science? Is it a denial of Darwinism?

\end{enumerate}

\newpage

%\vskip 12 pt
%\hrule
%\vskip 12 pt

\noindent
{\bf Some possibly useful formulas:}
$$
\sqrt{{\rm average\ of\ }(x-AV_x)^2} \qquad\qquad SD_y\cdot\sqrt{1-r^2}
$$
$$
r = {\rm average\ of\ }((x{\rm\ in\ std\ units)}\cdot(y{\rm\ in\ std\ units)})
$$
$$
z = \displaystyle\frac{x-AV_x}{SD_x} \qquad\qquad
   x = z\cdot SD_x + AV_x
$$
$$
y - AV_y = {\rm(sign\ of\ }r)\displaystyle\frac{SD_y}{SD_x}(x-AV_x) \qquad\qquad
   y - AV_y = r\,\displaystyle\frac{SD_y}{SD_x}(x-AV_x) \\
$$

\vskip 24 pt

\centerline{\begin{tabular}{|cc|cc|cc|cc|cc|}
\hline\multicolumn{10}{|c|}{Normal table (Area between $-z$ and $z$)}\\\hline
$z$ & Area(\%) & $z$ & Area(\%) & $z$ & Area(\%) & $z$ & Area(\%) & $z$ & Area(\%)\\
\hline
0.0 & 0.0 & 0.9 & 63.19 & 1.8 & 92.81 & 2.7 & 99.31 & 3.6 & 99.968 \\
0.05 & 3.99 & 0.95 & 65.79 & 1.85 & 93.57 & 2.75 & 99.4 & 3.65 & 99.974 \\
0.1 & 7.97 & 1 & 68.27 & 1.9 & 94.26 & 2.8 & 99.49 & 3.7 & 99.978 \\
0.15 & 11.92 & 1.05 & 70.63 & 1.95 & 94.88 & 2.85 & 99.56 & 3.75 & 99.982 \\
0.2 & 15.85 & 1.1 & 72.87 & 2 & 95.45 & 2.9 & 99.63 & 3.8 & 99.986 \\
0.25 & 19.74 & 1.15 & 74.99 & 2.05 & 95.96 & 2.95 & 99.68 & 3.85 & 99.988 \\
0.3 & 23.58 & 1.2 & 76.99 & 2.1 & 96.43 & 3 & 99.73 & 3.9 & 99.99 \\
0.35 & 27.37 & 1.25 & 78.87 & 2.15 & 96.84 & 3.05 & 99.771 & 3.95 & 99.992 \\
0.4 & 31.08 & 1.3 & 80.64 & 2.2 & 97.22 & 3.1 & 99.806 & 4 & 99.9937 \\
0.45 & 34.73 & 1.35 & 82.3 & 2.25 & 97.56 & 3.15 & 99.837 & 4.05 & 99.9949 \\
0.5 & 38.29 & 1.4 & 83.85 & 2.3 & 97.86 & 3.2 & 99.863 & 4.1 & 99.9959 \\
0.55 & 41.77 & 1.45 & 85.29 & 2.35 & 98.12 & 3.25 & 99.885 & 4.15 & 99.9967 \\
0.6 & 45.15 & 1.5 & 86.64 & 2.4 & 98.36 & 3.3 & 99.903 & 4.2 & 99.9973 \\
0.65 & 48.43 & 1.55 & 87.89 & 2.45 & 98.57 & 3.35 & 99.919 & 4.25 & 99.9979 \\
0.7 & 51.61 & 1.6 & 89.04 & 2.5 & 98.76 & 3.4 & 99.933 & 4.3 & 99.9983 \\
0.75 & 54.67 & 1.65 & 90.11 & 2.55 & 98.92 & 3.45 & 99.944 & 4.35 & 99.9986 \\
0.8 & 57.63 & 1.7 & 91.09 & 2.6 & 99.07 & 3.5 & 99.953 & 4.4 & 99.9989 \\
0.85 & 60.47 & 1.75 & 91.99 & 2.65 & 99.2 & 3.55 & 99.961 & 4.45 & 99.9991 \\
\hline
\end{tabular}}

%\newpage 
%\strut 
%\newpage

%{
%\Large
%\baselineskip 24 pt

%\noindent
%Answers: \hfill {\bf Name:} {\mbox{\rule[-1pt]{3truein}{1pt}}}

%\begin{itemize}
%\item[1.] \begin{itemize}
%\item[(a)]
%\item[(b)]
%\item[(c)]
%\item[(d)]
%\end{itemize}

%\item[2.] \begin{itemize}
%\item[(a)]
%\item[(b)]
%\item[(c)]
%\item[(d)]
%\item[(e)]
%\end{itemize}

%\item[3.]

%\item[4.]\begin{itemize}
%\item[(a)]
%\item[(b)]
%\item[(c)]
%\item[(d)]
%\end{itemize}

%\item[5.]\begin{itemize}
%\item[(a)]
%\item[(b)]
%\item[(c)]
%\item[(d)]
%\end{itemize}

%\end{itemize}

%Answer Questions~6 and 7 on the back of this page.

%\newpage
%\baselineskip=36pt

%\begin{itemize}

%\item[6.]
%\begin{itemize}
%\item[(a)]
%\item[(b)]
%\item[(c)]
%\end{itemize}

%\item[7.]

%\end{itemize}

%}

\newpage

\noindent
{\bf Math 102 / Core 143 --- Solutions to Exam I}

\begin{enumerate}

\item[1.] \begin{itemize}
\item[(a)] The area in the bars can be divided into 20 equal
blocks (one horizontal unit by one vertical unit --- so the right
bar has 6 blocks, the next has 5, and so on).
Over half are to the left of or on 3, so 3 is the median.
\item[(b)] The point of balance seems to be about 4.  (In fact, if
we assume that the fractions of the list are as suggested by the
bars, the average is 4.1.)
\item[(c)] The skew (i.e., the tail) is to the right.  (This is also
shown by the fact that the average is larger than the median.)
\item[(d)] Each of the 20 blocks represents 5\% of the area, so the
first hash mark represents 5\% per year on the density scale, to make
the total area (height times years) under the graph come out to 100\%.
\end{itemize}

\item[2.] \begin{itemize}
\item[(a)] $(3\cdot8 + 9 + 2\cdot11 + 12 + 13)/8 = 10$~cm
\item[(b)] $\sqrt{[3(8-10)^2 + (9-10)^2 + 2(11-10)^2 + (12-10)^2
+ (13-10)^2]/8}\approx 1.87$~cm
\item[(c)] If ``within'' means \underbar{less than} 1.87~cm, 3 out of 8,
or 37.5\% of the 8 heights are within one standard deviation of the average.
If ``within'' means 1.87~cm \underbar{or less}, rounded up to 2, 7 out of 8,
or 87.5\%, are within one standard deviation of the average.
\item[(d)] Neither 37.5\% nor 87.5\% is really close to 68\%, so the
data does not seem to follow the Empirical Rule, and hence is not
really normally distributed.  (Most small data sets are not.)
\item[(e)] The average is divided by 10 and increased by 1.5,
becoming 2.5~dm.  The standard deviation is divided by 10,
becoming 0.187~dm.
\end{itemize}

\item[3.] We want an area of 60\% (the given 80\% minus the mirror
image on the right of the 20\% on the left), and the normal table
says that comes at standard units value of about 0.85, or in terms of
height, $67+.85(4) = 70.4$~inches.

\item[4.] \begin{itemize}
\item[(a)] 58~cc of fertilizer is 0.5~SDs above average, so we
should project that the height is $0.8(0.5) = 0.4$~SDs above
average, or $30+0.4(8) = 33.2$~cm.
\item[(b)] 22~cm is 1~SD below average, so we should project that
the fertilizer level is $0.8(1) = .8$~SD below average, or
$50-.8(16) = 37.2$~cc.
\item[(c)] The RMS error for regression: $16\sqrt{1-.8^2} = 9.6$~cc.
\item[(d)] II.  The correlation of I is too low to be 0.8, that of
III is too high, and that of IV is negative.
\end{itemize}

\item[5.] (a)~Numerical.  (b)~Categorical. (c)~Categorical (``ordinal''
would be possible, but ``numerical'' would be misleading).  (d)~Numerical.

\item[6.] \begin{itemize}
\item[(a)] Maybe someone at the restaurant --- an employee or a
frequent patron --- has a cold and is contagious.
\item[(b)] Maybe cold sufferers feel too ill to cook for themselves.
\item[(c)] Maybe those who go out and socialize get colds from those
they socialize with, and they also go out to eat.
\end{itemize}

\item[7.] In the answers to these questions, I'm looking for evidence
that you have read and thought about the articles, not for agreement
with my opinions.  In this case, my sense is that epigenetics (``on
top of genetics'') is science, the study of how, even though organisms
may have the same genes, they may be expressed differently; it can be
studied at the level of the whole organism or at the level of molecular
biology.  It isn't a denial of Darwinism; rather, it is a refinement of
Darwin's theory --- and Darwinism, like any scientific theory, is
always subject to being refined in similar ways.

\end{enumerate}
\end{document}

