MATH 32 – Midterm 1 Fall Semester 2008
Duration: 50 minutes
Instructions: Answer all questions, without the use of notes, books or calculators. Partial credit
will be awarded for correct work, unless otherwise specified. The total number of points is 100.
1. (20 points: 4 each) Exactly one of the following statements is true. Choose which one is the
correct one and explain why the others are false. You may do so by either presenting the
correct version or give a counterexample.
(a) Chebyshev’s Inequality asserts that the proportion of data values that lie within kstan-
dard deviations from the mean is approximately 1−1/k2.
(b) For a normal data set, about 95% of data lies within plus and minus two standard de-
viations of the mean.
(c) If events Eand Fare mutually exclusive, then they are
independent.
(d) The mean of a data set almost always exceeds its median.
(e) The correlation of the data in the scatter plot on the right is
about 0.9.
2
TRUE FALSE E[X + Y] = E[X] + E[Y] for all random variables X and Y.
TRUE FALSE Var[X + Y] = Var[X] + Var[Y] for all random variables X and Y.
TRUE FALSE The correlation of the data in the scatter plot below is about 0.9
TRUE FALSE Approximately 95% of the observations of a normal data set will be within one
standard deviation of the mean.
2. (10 pts) The table below gives the number of students in different departments of a certain school. Use
the data in the table to find the probability that a randomly chosen student from this school is part of the
Natural Sciences Department given that the student was female.
Natural Sciences
Other
Total
Female
73
152
225
Male
191
203
394
Total
264
355
619
3. (5 pts each) Let the events below come from the sample space S = {1,2,3,4,5,6}. Write each set in parts
(a) – (c) in set notation (similar to how the events themselves are written below. Be sure to show work.
E = {1,2,3,4} F = {2,4,6} G = {1,4,5} H = {2,3,4,5}
a)
GHEF U
2. (20 points total) Consider the data set with eight values:
4,7,9,0,9,4,8,0
(a) (7 points) Find the sample mean and sample median.
(b) (8 points) Draw a box plot for the data set.
(c) (5 points) Write down the expression to calculate the standard deviation.
3. (20 points: 10 each) The joint probability density function of random variables Xand Yis
given by
f(x, y) = (Csin(x+y),0<x< π
2,0< y < π
2
0,otherwise .
(a) Find the constant C.
(b) What is the probability that X > Y ?
4. (20 points) A fair coin is flipped 5 times. Let Zbe the difference between the number of
heads flipped and the number of tails flipped. (Note that the difference can be negative.)
(a) (5 points) Find all values that Zmay take on.
(b) (10 points) Find the probability mass function of Z.
(c) (5 points) What is the expectation of Z?
5. (20 points) Three different airlines, called AM, UN, and SW, fly out of Ontario. AM airline
has 70 flights per day, of which 10% are late departures. UN airline has 50 flights per day,
of which 8% are late. SW has 65 flights per day, of which 13% are late. You randomly hear
someone at the airport complaining about their late flight, but do not hear them say which
airline. What is the probability that they are traveling on an SW flight?
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