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\centerline{MONT 105Q -- Mathematical Journeys}
\centerline{Problem Set 6}
\centerline{{\bf Due:} in class, Friday, April 15}
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1.  Doctors have determined that about $2.29\%$ of all human pregnancies 
result in children who have major birth defects.  A study of the effect 
of folic acid (reported in the {\it New England Journal of Medicine}) found 
that out of a random sample of $n = 2701$ women given folic acid as a dietary 
supplement during pregnancy, $35$ of the children had major birth defects.  
(For simplicity, assume there were no twins or higher multiples.)
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\item{A)} Identify an appropriate test to decide whether taking folic
acid reduces the incidence of major birth defects.   Give
the null and alternative hypotheses, describe how the test statistic would be 
computed, and state the rejection region for a test with $\alpha = .01$.  
\item{B)}  Now carry out your test.  Is there sufficient evidence to 
conclude that women who take folic acid have a reduced risk of having children
with major birth defects?  Explain.  
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{\it Comment:}  This is a simplified version of the actual study.  That involved
randomly assigning women to a treatment group (getting folic acid)
and a control group (getting only a placebo) and comparing the 
rates of birth defects.
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2.  (Assuming it is possible to generate ``truly random'' samples of likely 
Republican voters -- a BIG assumption, 
of course!), how many people would you need to include in a poll in order
to predict the percentage of people who will vote for Donald Trump in next
week's New York primary to within a margin of error of $\pm 3\%$?  
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3.  Cadmium, a heavy metal, is toxic to humans.  Vegetables and fungi
are able to absorb and accumulate cadmium at high concentrations from
the soil they grow in.  A safety limit of $0.5$ ppm
cadmium in mushrooms has been proposed and a study published in the {\it Journal 
of Environmental Science and Health} measured the cadmium
levels in a random sample of $n = 12$ edible mushrooms of the species 
{\it Boletus pinicola}.  The published data was as follows (all cadmium levels 
in parts per million) 
$$0.24, 0.59, 0.62, 0.16, 0.77, 1.33, 0.92, 0.19, 0.33, 0.25, 0.59, 0.32$$ 
\item{A)}  What are the average cadmium level and the SD
of the cadium level in this mushroom sample?
\item{B)}  Describe an appropriate test to decide whether the 
cadmium level in these mushrooms exceeds the safety limit of $.5$ ppm.  Give
the null and alternative hypotheses, describe how the test statistic would be 
computed, and state the rejection region for a test with $\alpha = .05$.
(Note:  You will need to use the $t$-table here appropriately; this is not
a large sample since $n < 30$.)  
\item{C)}  Carry out your test and explain the results.  
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\bye
