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\textbf{College of the Holy Cross, Spring Semester 2019} \\
\textbf{MONT 105N -- Analyzing Environmental Data} \\
\textbf{Makeup Midterm Exam, March 27 and 28}

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\noindent\textbf{Your Name}: \underline{\phantom{Name Name Name Name Name}}

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\paragraph{Instructions} Please write your answers in the spaces
provided on the following pages, and show work on the test itself. \textbf{For possible
partial 
credit, you must show work}. Use the back of the preceding
page if you need 
more space for scratch work.



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\noindent\textbf{Please do not write in the space below}


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Problem & Points/Poss    \strut \\

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I  & \hfil/\  30\hfil \strut \\

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II  & \hfil/\  25\hfil \strut \\

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III  & \hfil/\  25\hfil \strut \\

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Essay  & \hfil/\  20\hfil \strut \\


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Total   & \hfil/100\hfil \strut \\

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Do all work on these sheets.  You may use the back sides if necessary. 
There are 100 possible points (distributed as indicated in the questions).

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\noindent
I. Both parts of this question deal with the data set:
$$\matrix{11 & 9 & 8 & 11 & 13 & 10 & 1 & 8 & 7\cr}$$
\begin{enumerate}
\item[A)]  (15) Find the 5-number summary, and draw the ``box and whisker plot.''
(Use the inclusive method for the quartiles.)
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\item[B)]  (15)  Given that the SD of this data set is $S \doteq 3.43$, how many data values
are within two SD's of the mean?  Is this consistent with Chebyshev's Theorem?
\end{enumerate}
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II.  10 balls are drawn from an urn containing 53 red and 47 black balls, 
{\it with replacement}.
\begin{enumerate}    
\item[A)] (10) What is the probability of getting
$4$ red and $6$ black balls in the $n = 10$ draws?
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\item[B)] (10)  Is your answer in A the same as the probability of 
tossing $4$ heads and $6$ tails in $n = 10$ tosses, if the coin being
used is weighted so that the probability of a head is $.53$ and the probability
of a tail is $.47$?  Explain.  
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\item[C)] (5)  {\it Given that the first 4 balls come up red}, 
what is the probability that the first black ball comes up on the 8th draw?  
(Note:  This is a conditional probability.)  
\end{enumerate}
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III.   Females over 20 years old in the U.S. at present have an average
height of $65$ in, with $SD = 3.5$ in.  The corresponding figures for males
over 20 years old are $70$ in and $SD = 4$ in.  
\begin{enumerate}
\item[A)] (10)  If a male over 20 years old is selected at random, 
what is the probability that he has height between $70$ and $75$ in?
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\item[B)] (10)  What is the probability that a female over 20 years old, selected at random, 
has a height between $63$ and $68$ in?
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\item[C)]  (5) (True/False and Explain): If a man and an unrelated woman were selected 
at random from the over 20 years old populations, it would be reasonable to compute
the probability the male is between $70$ and $75$ in and the female is
between $63$ and $68$ in {\it by multiplying your answers} from A and B. 
\end{enumerate} 
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\noindent
{\it Essay} (20)
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One very common everyday situation where we are given information phrased
in probabilistic terms is when weather forecasts for a given geographical
area include an estimated {\it probability of precipitation} for some
time period in the future.   But what does one of those probabilities mean, really?  What is the 
exact formula used to calculate a probability of precipitation?  Explain by 
indicating exactly what it means, for instance, if a weather forecast says 
there is an 80\% chance of rain in Worcester tomorrow.  Can it mean more than 
one thing?  What does the National Oceanographic and Atmospheric 
Administration (NOAA) say about this and what is the justification they give 
for doing it this way?  Do you think just the number gives an adequate indication 
to an average person?  Would it help to educate people better about how these 
probabilities are actually computed?  Is there a better way to indicate how likely 
certain weather conditions might be?  
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