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\rightline{October 18, 2005}

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

\vspace*{0.5cm}
\noindent
Show all work clearly for partial credit.  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.  If your calculator
has buttons for statistical functions like average or SD, do not
use them for the computations below.
\begin{enumerate}

\item[1.] \textit{(20 points)}
For a list of numbers consisting of 6 ones, 3 twos, and 1 eighteen,
find the
\vskip 0 pt
\centerline{
\begin{tabular}{l@{\qquad\qquad}l@{\qquad\qquad}l}
(a) average & (b) SD & (c) median \\
(d) 20-th percentile & (e) IQR (interquartile range)
\end{tabular}
}

\item[2.]  \textit{(16 points)}
Which of these histograms represent each of
the following lists of data?  You may use any histogram more than
once or not at all.  If you are not sure, or if you cannot find
one that looks right, feel free to sketch your own and explain why
it is better than the given histograms.
\begin{itemize}
\item[(a)] jersey numbers of Colgate football players in the 2005 season
\item[(b)] heights of people at a party of Big Brothers/Sisters and their
Little Brothers/Sisters
\item[(c)] the numbers of shirts owned by various men
\item[(d)] scores on an easy statistics exam
\end{itemize}

\centerline{\includegraphics{e1a.png}}

\item[3.] \textit{(15 points)} The coordinates of a data point are described.
If many data points of each kind are collected, would the correlation
coefficient be closest to $-1$, $-0.5$, $0$, $0.5$, or $1$?
\begin{itemize}
\item[(a)] A person's birthdate (number of days into the year)
and his/her IQ.
\item[(b)] The average number of hours per day a person spends
watching TV and the hours he/she spends reading.
\item[(c)] A child's height measured in inches and the same
child's height measured in centimeters.
\end{itemize}

\newpage

\item[4.] \textit{(30 points)} A side effect
of a certain drug is to change blood pressure.  Dosage of the
drug and systolic blood pressure are recorded for a large
number of patients, and the following ``summary data'' is
found:\newline
\centerline{\begin{tabular}{lccr}
& dose (mg) & pressure (mmHg) \\
AV & 40 & 120 & $r=0.8$ \\
SD & 20 & 15
\end{tabular}
}\newline
On the basis of this data, estimate:
\begin{itemize}
\item[(a)] the systolic blood pressure of a randomly chosen patient
(with no information on his dosage).
\item[(b)] the likely error in the estimate found in (a)
\item[(c)] the systolic blood pressure of a patient whose dosage
is 60~mg.
\item[(d)] the dosage of a patience whose systolic blood pressure
is measured as 110~mmHg.
\item[(e)] the likely error in the estimate found in (d).
\end{itemize}


\item[5.] \textit{(10 points)} Relative to the article, ``A place for 
race in medicine?" by Gregory M Lamb: On the basis of the arguments 
in the article, is ``race" a genetic or a social construct? Explain. 

\item[6.] \textit{(9 points)} Relative to the article, ``Hardwiring 
of child's brain linked to reading ability, researchers find: Study 
backs evidence of neurological cause of weak performance" by Sarah 
Schmidt: What ``association/causation" question, left after an earlier 
study done at Stanford, was the Alberta study meant to address? 

\end{enumerate}

\vskip 12 pt
\hrule
\vskip 12 pt

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

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

\begin{enumerate}

\item[1.]\begin{itemize}
\item[(a)] $(6(1)+3(2)+18)/(6+3+1) = 3$
\item[(b)] $\sqrt{(6(1-3)^2+3(2-3)^2+(18-3)^2)/10} \approx 5.02$
\item[(c)] 1, because over half the values are less than or equal to it.
\item[(d)] Also 1.
\item[(e)] $2-1 = 1$
\end{itemize}

\item[2.] \begin{itemize}
\item[(a)] (iii) --- No jersey number is used more than once.
\item[(b)] (ii) --- The Big Brothers/Sisters have heights in the
right high point, the Little Brothers/Sisters in the left one (though
that one should really be broader and lower, because there is probably
more variability in the children's heights).
\item[(c)] (i) --- Most men have a few shirts, where ``few'' can cover
a broad range, say 10 to 30.  But a few men have many more. 
\item[(d)] (iv) --- Exam scores are usually between 70 and 90,
with a few at the extremes.  But there is less room for a right
tail than for a left one, because scores cannot exceed 100\%.
\end{itemize}

\item[3.] \begin{itemize}
\item[(a)] 0 --- There is no reason to assume an association between
birthday and IQ.
\item[(b)] $-.5$ --- More time spent on TV leaves less time for
reading; but the correlation is far from perfect, because there are
so many other things on which time could be spent.
\item[(c)] $1$ --- Except for small roundoff errors, the two
measurements should give identical lists of numbers in standard units.
\end{itemize}


\item[4.] \begin{itemize}
\item[(a)] 120 mmHg --- guess the average.
\item[(b)] The SD of pressure, 15 mmHg.
\item[(c)] Because his dosage is $(60-40)/20 = 1$ SD above average,
we project his blood pressure as $0.8(1)$ SD above average:
$120+(0.8)(15) = 132$~mmHg.
\item[(d)] Because his blood pressure is $(110-120)/15 = -2/3$ SD
above average (i.e., two-thirds of an SD below average), we project
his dosage as $0.8(-2/3)$ SD's above average, i.e.,
$40+(0.8)(-2/3)(20) \approx 29.3$~mg.
\item[(e)] The RMS error for regression: $20 \sqrt{1-(0.8)^2} = 12$~mg.
\end{itemize}

\item[5.] Most of the article, for example the bulleted points near the 
beginning, presents evidence on the side of the issue that race is a 
genetic construct; but some of the arguments presented say it is essentially
social, at least from a medical point of view, as in these paragraphs: 
`` `Much of what we're seeing in human illness can be explained 
by issues as prosaic and mundane as access to healthy water and good nutrition,' 
says Professor Duster, who teaches at New York University and the University of 
California at Berkeley. For example: A 2002 report by the Institute of Medicine 
showed that racial and ethnic minorities tend to receive lower-quality 
healthcare than whites, even if factors such as insurance status, income, age, 
and severity of condition are similar.

Many in the field are calling for broader international studies to make sure 
the bigger picture emerges. Studies comparing white and black Americans, for 
example, have shown that blacks have higher rates of hypertension, suggesting 
a genetic difference. But earlier this year research by Dr. Cooper and a team 
at Loyola showed that although African-Americans do show higher levels than 
North American whites, whites have higher levels than Nigerians and Jamaicans, 
who are `ethnically' black. Overall, the range of levels of hypertension among 
blacks in the study ranged from 14 to 44 percent while in whites it was higher, 
27 to 55 percent."

Whichever answer you give is fine, as long as you support it by some reference 
to the article.

\item[6.]  Here are the relevant paragraphs: ``The Alberta study builds on the 
findings of an early project by Stanford University researchers, whose study of 
dyslexic adults demonstrated a correlation between the micro-structural integrity 
of the left temporo-parietal white matter and their reading ability.

Since the imaging findings involving dyslexic adults could be a result of a 
lifetime of limited reading, Mr. Beaulieu said a lingering question had to be 
addressed --- whether the neural differences are present at an earlier stage 
during neurodevelopment, suggesting a causative effect, or whether the 
differences are a result of years of limited reading by struggling adults.''

Of course you are not expected to quote ``micro-structural integrity of the 
left temporo-parietal white matter".  Your answer should say something like, 
``The Stanford study found a correlation between a certain brain structure 
and reading disability in adults.  The Alberta study was of children, to 
determine whether the brain structure formed early and therefore may have
damaged reading ability, or whether years of struggling to read had caused 
the brain structure to develop.''

\end{enumerate}
\end{document}

