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\rightline{May 9, 2003}

\centerline{\bf Math 102 / Core 143 Section AX and BX --- Final Exam II}

\vspace*{0.5cm}
\noindent
Show all work clearly for partial credit --- an unevaluated
expression is worth more than the numerical answer.

\begin{enumerate}

\item[1.] {\it(24 points)} A survey is taken of 1600 households
in a large city; it is found that 900 of them heat with
electricity, and that the average monthly electric bill (for all
1600) is \$250, with an SD of \$120.
\begin{itemize}
\item[(a)] Assuming that the households surveyed were simple random
sample of the city, what is a 95\% confidence interval for the
average monthly electric bill in the city?
\item[(b)] How many households would the survey have had include to
get a 95\% confidence interval only half as wide (assuming similar
average and SD)?
\item[(c)] The power company claims that 60\% of the city's
households use electric heat.  Is the survey's result significant
evidence that the company's estimate is too high?
\item[(d)] Suppose the survey sample was conducted by e-mail, using
the client list of an internet service provider.  How might that have
biased the results?
\end{itemize}

\item[2.] {\it(20 points)} A ``Magic 8-Ball'' is a toy that is almost
spherical but with one flat window into its interior.  A dodecahedron
(12-sided solid) floats in the black fluid inside the 8-ball; when the
8-ball is held with the window up, one side of the dodecahedron floats
up to the window, and the user can read what is printed on that side.
Four sides of the dodecahedron say ``Yes'', four say ``No'', three say
``Maybe'', and one says ``Ask again later''.  The manufacturer tests the
the 8-ball by shaking it 240 times and noting results.  She gets 85
``Yes'', 70 ``No'', 63 ``Maybe'' and 22 ``Ask again later''.  Is the
8-ball fair?

\item[3.] {\it(13 points)} From a new batch at the Utica Club Brewery,
16 bottles of Erie Canal Hard Water are sampled for levels of duclamine
(an ingredient necessary for that authentic rock-strewn flavor).  The
specifications for ECHW say it should contain 3~mg of duclamine per ounce;
the 16 bottles averaged 2.8~mg per ounce, with an SD of 0.7~mg.  We are
to decide whether the sample's lower level of duclamine means (with 95\%
certainty) that the whole batch's duclamine level is too low.  Complete
as much of the decision process as you can.

\item[4.] {\it(16 points)} Recall that a ``straight'' deck of cards (i.e.,
one used for bridge or poker) has 52 cards, in 13 ranks and 4 suits.
\begin{itemize}
\item[(a)] If two cards are chosen at random \underbar{without}
replacement, what is the probability that both of them are clubs?
\item[(b)] If a card is chosen at random, what is the probability that
it is either a king or a club?
\item[(c)] What is the probability of getting a king at least once if
a card is drawn 8 times \underbar{with} replacement?
\item[(d)] What is the probability of getting a \underbar{club} exactly
5 times if a card is drawn 8 times \underbar{with} replacement?
\end{itemize}

\pagebreak

\item[5.] {\it(15 points)} At Turning Stone Casino, you decide to play
roulette 100 times, betting \$1 each time.
\begin{itemize}
\item[(a)] What are your expected \underbar{total} winnings, give or
take how much, if you bet splits (2 winners out of 38 numbers,
paying 17 to 1)?
\item[(b)] What are your expected \underbar{average} winnings, give or
take how much, if you bet sections (12 winners out of 38 numbers, paying
2 to 1)?
\item[(c)] With which game are you more likely to lose \underbar{more}
than \$10? Explain.
\end{itemize}

\item[6.] {\it(15 points)} The employment rate (fraction of the work
force that is employed) averages 87\% in New York villages, with a
SD of 10\%, and the welfare rate (fraction of the population on welfare)
averages 8\% with an SD of 6\%.  The correlation between the rates is
$-0.6$.
\begin{itemize}
\item[(a)] What should we guess is the welfare rate in a village with
an employment rate of 92\%?
\item[(b)] How far should we expect our guess in (a) to be off?
\item[(c)] If we found that the welfare rate in that village was
really 7\%, what is the corresponding residual?
\end{itemize}

\item[7.] {\it(12 points)} 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[8.] {\it(10 points)} Relative to the article ``Monitor after-school
programs carefully" by Megan Beckett:  Describe how this article calls for
methods from the current section of this course [significance tests] and
from the first section on experimental design. Why are both concepts needed?

\end{enumerate}

\pagebreak

{\bf Math 102 / Core 143 AX and BX--- Solutions to Final Exam II}

\begin{enumerate}

\item[1.] \begin{itemize}
\item[(a)] $\$250\pm 2(\$120/\sqrt{1600}) = \$250\pm\$6$.
\item[(b)] Four times as many households, or 6400.
\item[(c)] Taking the power company's estimate as the null
hypothesis, we perform a $z$-test: Because the SE for percent is
(using the figure from the null hypothesis)
$$
(1-0)\sqrt{(.6)(.4)}/\sqrt{1600} \approx 1.2\%\ ,
$$
we have
$$
P(\%\leq900/1600\approx56\%)
   = P(z\leq(56\%-60\%)/1.2\% \approx -3.1) < 5\%\ ;
$$
so we reject the NH: the power company's estimate is too high.
\item[(d)] One possible answer: People with e-mail addresses are more
likely to be technologically inclined, and hence to prefer electric
heat to, say, oil heat.
\end{itemize}

\item[2.] We expect counts of 80,80,60,20 of a fair 8-ball, and we
got 85,70,63,22; so we can do a ${\chi}^2$-test to decide whether there
is a significant difference between the two.
\begin{eqnarray*}
{\chi}^2 &=& \frac{(85-80)^2}{80}+\frac{(70-80)^2}{80}
     +\frac{(63-60)^2}{60}+\frac{(22-20)^2}{20} \\
   &=& \frac{25}{80}+\frac{100}{80}+\frac9{60}+\frac4{20}
     = \frac{25+100+12+16}{80} \approx 1.91
\end{eqnarray*}
and with $4-1=3$ degrees of freedom, we see from the table that a
${\chi}^2$-value that high occurs by chance between 70\% and 50\% of
the time, not less than 5\%; so we do not reject the null hypothesis: The
8-ball is fair.

\item[3.] With the small sample, we conduct a $t$-test, with the null
hypothesis that the batch's duclamine level is the required 3~mg per ounce,
and using the SD$^+$ of the sample to estimate the SD of the batch: The SE
is $.7(\sqrt{\frac{16}{15}})/\sqrt{16} \approx .18$, so $t = (2.8-3)/.18$, which
is slightly less than $-1$.  The degrees of freedom is $16-1=15$.
We don't have a $t$-table, but it would probably show
that the probability of a $t$-value that low or lower is not less than
5\%, so we would not reject the null hypothesis and would accept the batch.

\item[4.] \begin{itemize}
\item[(a)] $(13/52)(12/51) = 1/17 \approx 6\%$.
\item[(b)] $(4/52)+(13/52)-(1/52) = 16/52 = 4/13 \approx 31\%$.
\item[(c)] $1-(12/13)^8 \approx 47\%$
\item[(d)] $C(8,5)(1/4)^5(3/4)^3 \approx 2\%$
\end{itemize}

\item[5.] \begin{itemize}
\item[(a)] The EV of the sum is $(\frac2{38}(17)+\frac{36}{38}(-1))(100)
= \frac{-100}{19} \approx -5.3$ dollars, and the SE of the sum is
$(17-(-1))\sqrt{\frac2{38}\cdot\frac{36}{38}}(\sqrt{100}) \approx 40.2$ dollars.
\item[(b)] The EV of the average is $(\frac{12}{38}(2)+\frac{26}{38}(-1))
= \frac{-1}{19} \approx -0.053$ dollars, and the SE of the average is
$(2-(-1))\sqrt{\frac{12}{38}\cdot\frac{26}{38}}/\sqrt{100} \approx .139$ dollars.
\item[(c)] Multiplying our answers to (b) by 100, to compare sums of winnings,
we see that with both games we expect to lose 5.3~dollars, but the greater
variability of playing splits (40 vs. 14 dollars) makes it more likely to
lose more than 10~dollars with that game rather than playing sections.
\end{itemize}

\item[6.] \begin{itemize}
\item[(a)] The employment rate in standard units is $(92\%-87\%)/10\% = .5$,
so we should expect the welfare rate in standard units to be $(-.6)(.5) = -.3$,
or in percent, $8\%+(-.3)(6\%) = 6.2\%$.
\item[(b)] By the RMS error for regression: $\sqrt{1-(-.6)^2}(6\%) = 4.8\%$.
\item[(c)] $7\%- 6.2\%= .8\%$.  (I won't take off points for the sign,
because we haven't talked about it much.)
\end{itemize}

\item[7.] \begin{itemize}
\item[(a)] Someone in the restaurant, perhaps even an employee,
might have spread the cold to those around him/her.
\item[(b)] Cold sufferers might not feel up to cooking for themselves
and decide to go out to eat.
\item[(c)] One possibility: If it's a college town, students who
are studying for finals late into the night are more susceptible
to colds and less likely to feel like cooking for themselves.
\end{itemize}

\item[8.] Beckett points out that, although California schools have
been given a large amount of money (\$550 million, because I have
the article beside me) for after-school programs, little research
has been done on which ones give the best results.  She urges testing,
using random assignment to test and control groups (which she calls
the ``gold standard'' of experimental design), to look for
statistically significant improvements in measurements of the
qualities that (she says) everyone wants such programs to have.
For instance, one such quality is "a safe place to stay for youngsters
who don't have a parent at home when school lets out"; so one might
compare the accident rates for students in one such program versus
another --- probably \underbar{not} versus students who are not in any
program, because that would not tell whether one program is better
than another, but only whether a program was better than no program
at all (though of course a badly run program may allow more accidents
than a student would experience who simply goes home to watch TV).
Both parts are needed because the test must be fair (requiring
good experimental design) and the results interpreted unambiguously
(in the standard scientific manner) for the results to gain general
acceptance and be trustworthy bases for establishment of new programs.


\end{enumerate}

\end{document}


