251x0622 3/4/06
ECO251 QBA1
SECOND HOUR EXAM
March 22 and 23, 2006
Name: _____________________
Student Number: _____________________
Hour of Class Registered: Circle MWF1, MWF2, TR12:30, TR2:00
There will be a penalty if you do not provide all this information!
Part I: (3 points) 2 point penalty for not trying.
The scores of a sample of statistics exams are (with extra credit included) as below. Find the sample
standard deviation. Show your work!
96 85 105 86 103 101 97 94 75 98
x
2
 89146
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Part II: (35 points+) Do all 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 any question, you should provide an alternative answer and explain why. You may receive credit
for this even if you are wrong.
1.
2.
3.
If events A and B are independent and their probabilities are not zero, the following cannot be true.
a) A and B are mutually exclusive.
b) A and B are complements.
c) Neither of the above can be true.
d) Both of the above can be true.
If events A and B are independent and P A  .5 and PB   .3 what is P A  B  ?
a) .80.
b) .15
c) .20
d) .65
e) None of the above.
f) Not enough information to tell.
 
If P A  .3, PB   .5 and P B A  .4 what is P A  B  ?
a)
b)
c)
d)
e)
.80.
.15
.12
.20
None of the above.
[6]
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Questions 4-7 are based on exhibit 1.
Exhibit 1: A fast food restaurant has its outlets distributed as follows. (Dummeldinger). A restaurant is to
be chosen at random to test market a new menu item.
Region
NE
SE
SW
NW
Below
.05
.06
.03
.00
10000
A
City
10000.15
.13
.05
Population 100000
B
Above
.20
.03
.10
.05
100000
C
4.
Fill in the missing number (1)
5.
What is the probability that, if we select a restaurant at random, we pick a restaurant that is in a
city of above 100000 and is in the Northeast. (2.5)
a) PC  NE 
b) PC  NE 
c)
PC NE 
PNE C 
e) None of the above.
And the probability is _________________.
d)
6.
What is the probability that, if we select a restaurant at random, we pick a restaurant that is in a
city of above 100000 or one that is in the Northeast. (2.5)
a) PC  NE 
b) PC  NE 
c)
d)
PC NE 
PNE C 
e) None of the above.
And the probability is _________________.
7.
What is the probability that, if we select a restaurant in a city in the northeast at random, we pick a
restaurant that is in a city above 100000. (2.5)
a) PC  NE 
b)
c)
PC  NE 
PC NE 
PNE C 
e) None of the above.
And the probability is _________________.
d)
[14.5]
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8.
You and two of your friends are shooting arrows at a target. Each of you has an independent
probability of hitting the target of 80%. What is the probability that all three of you will hit the
target?
9.
You and two of your friends are shooting arrows at a target. Each of you has an independent
probability of hitting the target of 80%. What is the probability that at least one of you will hit the
target?
[18.5]
10. You and two of your friends are shooting arrows at a target. Each of you has an independent
probability of hitting the target of 80%. Let x be the number of individuals that hit the target. Do
the following:
Find the distribution of x . (Show what values x can take, what the probability of each
value is, and demonstrate that we have a valid distribution.) (6) If you cannot do this
problem you may make up a reasonable distribution to do b), c) and d). Warning – this
problem could take a while to do. If time is a constraint for you, skip it and come
back to it later.
b) Find F 2 , the cumulative probability ( Px  2 ) (2)
c) Find the expected value and variance of x . (3)
d) Using the mean and variance that you found in c) , If you and your buddies as a group
paid $8 to shoot and the operator of the game gives your group $3 for each hit on the
target, what is the expected value and variance of your winnings? (3) [32.5]
a)
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11. You are dealt 7 cards from a deck of 53 cards containing 4 each of 2s, 3s, 4s, 5s, 6s, 7s, 8s, 9s,
10s, jacks, queens kings and aces, 13 each of hearts, diamonds, spades and clubs and one joker.
What is the probability that your hand contains the joker? (4) You must work this out all the way
to get full credit and at least show how many possible hands there are.
12. You are dealt 7 cards from a deck of 53 cards containing 4 each of 1s, 2s, 3s, 4s, 5s, 6s, 7s, 8s, 9s,
10s, jacks, queens kings and aces, 13 each of hearts, diamonds, spades and clubs and one joker.
What is the probability that your hand contains three hearts? (2.5) This can be left in factorial
form.
13. You are dealt 7 cards from a deck of 53 cards containing 4 each of 1s, 2s, 3s, 4s, 5s, 6s, 7s, 8s, 9s,
10s, jacks, queens kings and aces, 13 each of hearts, diamonds, spades and clubs and one joker.
What is the probability that your hand contains at least one heart? (2.5) This can be left in factorial
form.
[41.5]
14. Assume that z is a standardized variable and that y  3z  17 . The standard deviation of y is:
(2.5)
[44]
a) 14.
b) -3
c) 3
d) 9
e) 51
f) Not enough information.
g) None of the above.
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(Blank)
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ECO251 QBA1
SECOND EXAM
March 22 and 23, 2006
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. (5 points)
row
x Qx 
1
2
3
4
5
6
7
row
1
2
3
4
5
6
7
-4
-3
-2
-1
0
1
2
x Qx 
-4
-3
-2
-1
0
1
2
Make yourself an individualized distribution as
follows. Take the six digits of your student
number and add them to the numbers in Qx  .
Divide the result by 100 to get a column of
numbers and fill in the missing number to get a
valid distribution.
5
5
10
..
10
5
5
5
5
10
..
10
5
5
 stu 
  Q' x 
no 
9
9
9
.
9
9
9
14
14
19
..
19
14
14
P x 
0.14
0.14
0.19
....
0.19
0.14
0.14
For example, assume that my student number is
999999. I have written it in the column labeled
‘stu no’ and then added it to Qx  to get the
column Q' x  . Then I divided the result by 100
and filled in the missing number in the Px 
column.
a) Use the numbers in the x column and the probabilities in Px  to compute the population standard
deviation of x . You will not get credit for this if you do not show your work.
b) I got the numbers below, which I will call y , by multiplying by -9 (negative 9) and subtracting 3 from the
result.
33
24
15
6
-3
-12
-21
y
Without using the numbers in y , find the expected value and standard deviation of y . You will not get
credit for this unless you show what formulas you use to get the results.
2. (Bassett et. al.). (10 Points) The stock of a warehouse consists of bulbs of Type A, Type B and Type C in
separate unlabeled boxes. Let the last three digits of your student number be a, b, and c. The proportions of
the bulbs in our stock and the proportion that are defective are given by the table below.
Type
Proportion in stock
Proportion of this type that is defective.
A
.2
(0 + a)%
B
.4
(10 + b)%
C
.4
(20 + c)%
So if my student number is 123456, the proportions defective will be (0 + 4)% = .04, (10 + 5)% = .15 and
(20 + 6)% = .26. Hint: You might want to start with 1000 bulbs taken from 100 boxes of bulbs. Your events
should be (types) A , B , C . Let the event that the bulb is defective be D. As above, you must show your
work if you want any credit.
a) What proportion of 1000 bulbs would be defective?
b) If I open a box and find a defective bulb, what is the probability that it comes from a box of type A?
Type B? Type C?
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c) If I open a box and find a satisfactory bulb, what is the probability that it comes from a box of type A?
Type B? Type C?
d) Actually this problem as originally written asked about the probabilities above in the event we found two
satisfactory bulbs in a given box. So assume that the bulbs’ satisfactoriness are independent of one another,
what is the probability of finding two satisfactory bulbs of Type A, of type B and of Type C.
e) Now, assuming that we open a box and find two satisfactory bulbs, what is the probability that they come
from a box of type A?
[15]
3. (6 points extra Credit) (Bassett et. al.). This is a stinker but there is a way to do it fairly rapidly.
Three quarters of the members of a club are adults. Three quarters of the adults and three fifths of the
children are male. Half of the adult males, one third of the adult females, four fifths of the boys and four
fifths of the girls use the pool. Find the following Probabilities:
a) A member of the club uses the swimming pool.
b) A user of the swimming pool is male.
c) A member of the club is female.
d) A user of the swimming pool is female.
e) A male user of the swimming pool is a child.
f) A non-user of the swimming pool is either female or an adult.
Hint: Start out with the following events: A adult; C child; F female; M male; S user of the swimming
pool. Assume that the club has 800 members. Identify the probabilities in terms of these events.
A C
total
M
..... .....
Fill in the following table.
Use the results of the table to fill in a second table.
F
..... .....
total
800
A  M A  F C  M C  F total
S
.....
.....
.....
.....
You should now have the numbers you need to solve the
S
.....
.....
.....
.....
total
800
problem.
8