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\centerline{MONT 106N -- Identifying Patterns}
\centerline{Final Exam -- December 17, 2009}
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{\it Directions}
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Do all work in the blue exam booklet or on the graph paper
provided.  Be sure to provide explanations for your answers if that is asked for.
There are 200 possible points.
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I.  The City University of New York has about 200,000 students
on 21 campuses.  Their ages are distributed as in the following
table:
$$\matrix{{\rm Ages} & \qquad & {\rm Percent}\cr
           & \qquad &\cr
           \le 19 & \qquad & 21.1\cr
           20 {\rm\ to\ } 22 & \qquad & 26.2\cr
           23 {\rm\ to\ } 24 & \qquad & 10.8\cr
           25 {\rm\ to\ } 29 & \qquad & 15.5\cr
           30 {\rm\ to\ } 44 & \qquad & 20.2\cr
           45 {\rm\ to\ } 64 & \qquad & 4.6\cr
            \ge 65 & \qquad & 1.6\cr}$$
\item{A)}  (10)  On the graph paper supplied, construct
a histogram for this data.  To do this in a standard way, think of
each year of age as corresponding to the interval starting at that year
and going all the way to the next whole number of years.  Thus, for instance, the 
age range 20 to 22 will correspond to a histogram bar with base 
from 20 to 23.  Make the first bar extend from 15 to 20 and the last
one extend from 65 to 75. 
\item{B)}  (10)  Is the {\it average} age or the {\it median} age larger
for the CUNY student population?
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II.  Suppose you are going to design a study to test the effectiveness of
teaching elementary German by two different methods:  (a) in 
a traditional classroom, with spoken lessons, text readings, etc.; 
(b) in a media lab with language tapes, music videos,
cultural films, etc.  There are 50 students who will be starting
elementary German at your school next year and one section will
be taught by each method.  
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\item{A)}  (10) {\it From the point of view of the reliability of 
the results of the study}, 
would it be a good idea to have the same professor assigned to teach 
both sections?  Explain.
\item{B)}  (10) {\it From the point of view of the reliability of 
the results of the study}, is there a problem with letting
the students select which section they register for?  Explain. 
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III.  The scatter plot for the first- and second-year
GPA's of a large number of university students is football-shaped, with
a positive $r$-value.
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\item{A)} (10)  True or False:  On average, students who had GPAs above
average in their first year also had GPAs above average in their second year.
Explain, {\it briefly}.   
\item{B)} (10)  True or False:  A student who was at the 40th percentile 
of the first year GPAs is also likely to be at the 40th percentile
of the second year GPAs.  Explain, {\it briefly}.
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IV. (Hypothetical) The grades in a large university statistics class were normally
distributed with an average of 65 (out of 100) and an SD of 10.
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\item{A)} (10)  Draw a smoothed histogram for the data, showing
the locations of the average and ${\rm ave} \pm SD$ on your plot.
\item{B)} (10)  If 250 students took the test, estimate how
many of them scored between 70 and 75.  
\item{C)} (10)  What exam score was at the 90th percentile
of the distribution?
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V.  All parts of this problem deal with the following data set:
$$\matrix{ x & \qquad & y\cr
             & \qquad & \cr
           1 &\qquad & 1\cr
           8 &\qquad & 4\cr
           10 &\qquad & 6\cr
           10 &\qquad & 12\cr
           14 &\qquad & 12\cr
           17 &\qquad & 7\cr}$$
\item{A)} (10)  On the graph paper provided, draw the scatter plot.
\item{B)} (10)  Compute the averages and SD's of the $x$ and $y$ data.
\item{C)} (10)  Compute the correlation coefficient, $r$.
\item{D)} (10)  On your scatter plot, draw in the regression line
for estimating values of $y$, given values of $x$.  
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VI. (Hypothetical) At a college, a large group of incoming 
students took language placement tests in {\it both} Spanish and French.
The average on the Spanish test was 70 (out of 100) with an SD of 10.
The average on the French test as 75, with an SD of 12.  
The correlation coefficient between the Spanish and French test scores
was $r = .6$.  The scatter plot was football-shaped (homoscedastic).
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\item{A)} (10)  Estimate the average French score for the students who scored
80 on the Spanish exam.  
\item{B)} (10)  For the students who scored 80 on the Spanish test,
estimate the SD of their scores on the French exam.
\item{C)} (10)  Using your answers for parts A and B, among
the students who scored an 80 on the Spanish test, estimate
the percentage who scored above 85 on the French test.  
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VII.  Three cards are dealt (without replacement) off
the top of a well-shuffled standard deck (with 52 cards -- 
ace through king in each of the four suits: clubs, diamonds,
hearts, spades).  
\item{A)} (10)  What is the chance that you get only kings?
\item{B)} (10)  What is the chance that you get no kings?
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VIII.  A box contains 8 red marbles and 3 green marbles.
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\item{A)} (10)  If you draw six marbles at random, record
the color, replace the marble each time and mix up the box,
what is the chance of getting 4 red and 2 green marbles?
\item{B)} (10) Is the chance the same if you set aside
each marble as it is drawn (without replacing it in the box)?   
Explain why or why not.
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\centerline{\it Have a peaceful and joyous holiday season!}  
\bye
