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ECO251 QBA1
SECOND HOUR EXAM
March 27, 2008
Name: _____________________
Student Number: _____________________
Hour of Class Registered: _______________
Part I: (3 points) 2 point penalty for not trying question 1. (Problem modified from Webster)
1. A professor wishes to see if the number of absences of his students is related to the number of miles the
students live from the classroom. (Is it true that a miss is as good as a mile?) He takes a sample of eight
students and gets the results below. Compute the sample standard deviation of ‘Miles.’ Show your
work! (3)
Row
Row
1
2
3
4
5
6
7
8
y
x
Miles
5
6
2
0
9
12
16
5
Misses
2
2
4
5
4
2
5
2
7
For your convenience the sum of the first seven numbers in x is
x
 50 and the sum of the first
i 1
7
seven numbers squared is
x
2
 546 .
i 1
2. Compute

7
x y . For your convenience the sum of the first seven numbers in xy is
 xy
 170 . (1)
i 1
Exhibit 0: According to Minitab y  3.250 and s 2y  1.92857 .
3. Compute the sample covariance. What does this show about the relationship between Miles and Misses?
(2.5)
4. Compute the sample correlation between Misses and Miles. What does this show about the strength of
the relationship between Miles and Misses? (2.5)
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5. The amount of time in minutes t  it takes in minutes to get to campus can be approximated by the
equation t  20  3x , where x is the number of miles. Using only the results that you have so far find:
(i) The sample variance of the time
(ii) The covariance of time with ‘misses’
(iii) The correlation of time with misses.
Please do not waste our time by computing a column with the 8 values that time would take and then
recomputing the values of these three statistics. There is a much easier way. (4.5)
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Part II: (24.5+ points) Do most of the following: All questions are 2 points each except as marked. Exam
is normed on 50 points including take-home. [Bracketed numbers are a point total.] If you answer ‘None of
the above’ in most questions, you should provide an alternative answer and explain why. You may receive
credit for this even if you are wrong. This section is long and few people will finish. You only need 35
points for a perfect score, so look them over! You only need 24.5 points for a perfect score in this section,
so look them over! As always any extra points in this section wrap around to other sections. Don’t let
similarity with last year’s questions fool you!
1. If both events A and B are possible and P A  B   0 , the following cannot be true.
a) A and B are complements.
b) A and B are mutually exclusive.
c) A and B are independent
d) All of the above could be true
e) None of the above can be true
f) Not enough information
Exhibit 1: (Webster) Consider the Following table. It divides the 500 employees of a corporation by their
gender and their job classification. The events are: A1 'Male', A2 'Female', B1 'Staff Job', B2 'Line Job' and
B3 'Auxiliary Job'.
Staff Line Aux
Male  120
150
30 


Female  50 140
10 
Hint: Before doing anything on the following four questions, you may want to make these into proportions
or probabilities. Be sure that you look at both sides of the answer you choose. Guessing is very unlikely to
get you anywhere.
3. Find the probability that a woman is an Auxiliary employee.
a. PA2  B3  .02
b. PA2  B3  .46
c. PA2  B3  .25
d. PA2  B3  .05
e. PA2  B3  .02
f. PA2  B3  .46
g. PA2  B3  .25
h. PA2  B3  .05


 
k. PB3 A2  .25
l. PB3 A2  .05
m. PA2 B3  .02
n. PA2 B3  .46
o. PA2 B3  .25
p. PA2 B3  .05
j. P B3 A2  .46
i. P B3 A2  .02
q. None of the above – fill in the correct
answer
4. Find the probability that an employee is both a woman and an Auxiliary employee.
a. PA2  B3  .02
j. P B3 A2  .46
b. PA2  B3  .46
k. P B3 A2  .25
c. PA2  B3  .25
d. PA2  B3  .05
e. PA2  B3  .02
f. PA2  B3  .46
g. PA2  B3  .25
h. PA2  B3  .05


i. P B3 A2  .02


 
l. PB3 A2  .05
m. PA2 B3  .02
n. PA2 B3  .46
o. PA2 B3  .25
p. PA2 B3  .05
q. None of the above – fill in the correct
answer.
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5. Find the probability that an employee is either a woman or an Auxiliary employee (or both).
a. PA2  B3  .02
j. P B3 A2  .46
b. PA2  B3  .46
k. P B3 A2  .25
c. PA2  B3  .25
l. P B3 A2  .05
d. PA2  B3  .05
m. P A2 B3  .02
e. PA2  B3  .02
n. P A2 B3  .46
f. PA2  B3  .46
 




 
 
o. PA2 B3  .25
p. PA2 B3  .05
g. PA2  B3  .25
h. PA2  B3  .05


q. None of the above – fill in the correct
answer.
i. P B3 A2  .02
6. On the basis of what you found in the last three questions, A2 and B3 are:
a) complements.
b) mutually exclusive.
c) independent.
d) collectively exhaustive.
e) All of the above could be true.
f) None of the above can be true.
g) Not enough information
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7. z is a standardized random variable. (What do you know about the mean and variance of a standardized
variable?) If y  13z 100 , the standard deviation of y  y is:
 
a) -113.
b) -1.
c) -13.
d) 13.
e) 169.
f) 113
g) None of the above. Supply the correct answer.
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Exhibit 2: A deck of cards has 52 cards divided equally into hearts, diamonds, spades and clubs. There are
4 of each kind of card (2s, 3s, 4s, 5s, 6s, 7s, 8s, 9s, 10s, jacks, queens, kings and aces. 26 of the cards are
black, 26 of the cards are red and 12 of the cards are face cards.
8. You are dealt 4 cards. What is the probability that all are aces? You may present this answer as a
decimal, a per cent or a fraction. (3)
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9. What is the probability that you will get 2 aces? You may leave this answer as a fraction involving Cs or
Ps if you wish. (2)
10. What is the probability that you will get at least 1 ace? Please don’t waste our time by giving me the
probability of exactly one ace. You may leave this answer as a fraction involving Cs or Ps if you wish. (3)
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11. Find the probability that you will get 5 Aces. (1)
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Exhibit 3: As everyone knows a Jorcillator has two components. These are the Flubberall and the Phillinx.
Let us assume that our company owns a Jorcillator that only works as long as both components work. The
life of a Flubberall is described by a continuous uniform distribution where nonzero frequencies are
between c  0.5 and d  2.0 . The life of a Phillinx is described by a continuous uniform distribution
where c  0.5 and d  2.5 . For example, the probability that the Phillinx fails in year 1 is P0  x  1 and
the probability that the Phillinx fails in year 2 is P2  x  3 . Let A1 be the event that the Flubberall fails
in year 1, A2 be the event that the Flubberall fails in year 2, A3 be the event that the Flubberall lasts
beyond year 2, B1 be the event that the Phillinx fails in year 1, B2 be the event that the Phillinx fails in year
2, B3 be the event that the Phillinx lasts beyond year 2. The lives of the two components can be considered
independent.
12. Find the following. (Make diagrams.) (1 each)
(a) P A 2 , the probability that the Flubberall fails in year 2.
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(b) P A1 , the probability that the Flubberall fails in year 1.
(c) P A 3 , the probability that the Flubberall fails after year 2.
(d) PB 1 , the probability that the Phillinx fails in year 1.
(e) PB 2 , the probability that the Phillinx fails in year 2.
(f) PB 3 , the probability that the Phillinx fails after year 2.
13. Assuming that your probabilities are correct, find the probability that both components fail in year 2.
(2)
14. Find the probability that the Jorcillator fails in the first year. (2)
15. Find the probability that the Jorcillator fails in the second year. (2)
16. Find the probability that the Jorcillator fails after the second year. (1)
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Exhibit 4: A coin is tossed 5 times. Define the following events.
A : Exactly one head
17. Find P A
B : Exactly two heads
C : At least one head
18. Find PB 
19. Find PC 
20. Let x be the number of heads. List all the values that x can take and their probabilities. You have done
most of this already if you realize that the probability of 2 heads and the probability of 2 tails are equal.
Show that this is a valid distribution. (4)
21. Find the expected value (mean) and standard deviation using the values of x and the probabilities
found in question 20. Do not guess! Show your work. (3)
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(Blank page for calculations)
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ECO251 QBA1
SECOND EXAM
March 27, 2008
TAKE HOME SECTION
Name: _________________________
Student Number: _________________________
Throughout this exam show your work! Neatness counts! Please indicate clearly what sections of the
problem you are answering and what formulas you are using.
Part III. Do all the Following (12+ Points) Show your work!
1. Seymour Butz’s student number is 976543
Take the following set of numbers: and the probabilities below
x
P x 
10
.16
20
.17
30
.18
42
.19
53
.20
64
__
75
__
and use the second to last digit of your Student Number to provide the first digit of all seven numbers so
that the become 3 digit numbers the last six numbers, so that, for Seymour, the numbers would become
410, 420, 430, 442, 453, 464 and 475. Now compute the last two probabilities by putting the last digit of
your student number divided by 100 in the first blank. For Seymour, this would mean that he will write in
.03. In the second blank, put in whatever you need to make the distribution valid and explain how you got
that number. Compute a population standard deviation using your probabilities. (2) Find the probability that
x is an even number and find the median of the distribution. Again show your work. (1)
2. Compute a sample standard deviation for your x column. Ignore the probabilities. (2)
[5]
3. Assume that we roll a pair of dice. Use x for the amount on top of the first die, y for the amount on top
of the second die and w for their sum. Let zo be the value of the last digit of your student number plus 2.
Seymour Butz’s student number is 976543. For Seymour this means zo  5 .
Let event A be w  z0  . Let event B be y is an even number . Let event C be w  z0  . Make sure that
I know what z0 is and show your work. Find (1 each): [12]
a) P A
b) PC 
 
c) P B C
d) PB  C 
e) PB  C 
 
f) P C B

g) P B  C

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4. (Moore et al) 5% of male high school athletes (in basketball, football or baseball) compete in college. Of
those who compete in college, 1.7% go on to major league sports. 40% of the athletes who compete in both
high school and college and enter major leagues last more than three years. The probability that a high
school athlete who does not compete in college goes on to professional play is .0001.
Although the numbers given above are probably correct, modify the probabilities as follows: Seymour
Butz’s student number is 976543. Take the last digit of your student number and add it to 5%, so, for
example, Seymour will say that (5+3) % = 6% of male high school athletes compete in college; then take
the same number and add it to the 1 in .0001, so, for example, Seymour will say that .0001 + .0003 = .0004
is the probability that a high school athlete who does not compete in college goes on to professional play.
Let A be the event that a high school athlete competes in college. Let B be the event that an athlete
competes professionally. Let C be the event that an athlete has a professional career of more than 3 years.
Use these in your answer. (2 points extra credit for each part)
a) Identify your four probabilities as given the problem and modified by you in terms of events A , B and
C . That is to say probabilities like PB  C  , P B  C , P B C or P A  B  C  . (I do not want to

  
imply that any of these appear in the problem)
b) What is the probability that a high school athlete will play in college and then have a major league career
of more than three years?
c) What is the probability that a high school athlete will play professional sports?
d) What proportion of professional athletes competed in college?
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