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\centerline{MONT 105N -- Analyzing Environmental Data}
\centerline{Problem Set 5}
\centerline{{\bf Due:} in class, Friday, April 26}
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{\it Information}
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Because of the closeness of the due dates for the final projects, I have decided to 
make this an {\it optional, extra credit assignment}.  
If you submit solutions for some or all of these questions, 
I will add what ever points you get to your total score on 
the homework portion of the course grade.  So it can only help your final course 
grade.  
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1.  Cadmium, a heavy metal, is toxic to animals.  Mushrooms
are able to absorb and accumulate cadmium at high concentrations from
the soil they grow in.
The Czech and Slovak authorities set a safety limit of $0.5$ ppm
cadmium in mushrooms.  M. Melgar, et. al. measured the cadmium
levels in a random sample of $n = 12$ examples of the edible mushroom {\it Boletus
pinicola} and published the results in the {\it Journal 
of Environmental Science and Health}, vol. B33(4), p. 439-455.
Their data was as follows (all cadmium levels in ppm) 
$$0.24, 0.59, 0.62, 0.16, 0.77, 1.33, 0.92, 0.19, 0.33, 0.25, 0.59, 0.32$$ 
\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 government safety limit.  Give
the null and alternative hypotheses, describe how the test statistic would be 
computed, and state the rejection region for a test with $\alpha = .05$.  
\item{C)}  Carry out your test and explain the results.  
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2.  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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3.  The seagrass {\it Thalassia testudinum} is an integral part
of the Texas coastal ecosystem.  Essential to the growth of
this plant is ammonium in the sediment on the sea floor where the 
plants are rooted.
In a study reported in the {\it Marine Ecology Progress Series},
vol 196, p. 39-48, the following data on sediment ammonium
concentrations was collected at two locations: Corpus Christi
Bay (CCB) and Lower Laguna Madre (LLM).  The $y_i$ are
the average ammonium concentrations (in micromoles) , the $SD_i$ are the sample SD's,
and the $n_i$ are the number of measurements made at each site:
$$
\matrix{{\rm CCB} & {\rm LLM}\cr
         \overline{y_1} = 115.1 & \overline{y_2} = 24.3\cr
         SD_1 = 79.4 & SD_2 = 10.5\cr
         n_1 = 51 & n_2 = 19\cr}$$
\item{A)}  At the $\alpha = .05$ level, is there sufficient 
evidence to say that the ammonium level is different at CCB and LLM?  (Note: Since
the total number of samples is large, you can use the large-sample
formulas even though the $n_2 = 19$ would not ``qualify'' by itself.  Also, the 
way the question is phrased indicates you want to carry out a 2-tail test!)
\item{B)}  Estimate the $p$-value of your test in part A.
\item{C)}  Based on your answer to part A and the information given 
above, would you expect the seagrass beds to be taller and thicker at CCB or
LLM? 

\bye
