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ECO251 QBA1
THIRD EXAM
Apr 18, 2007
Name: _____________________
Student Number: _____________________
Part I. (16 points) Do all the following (2 points each unless noted otherwise). Make Diagrams! Show
your work!
z has the standardized Normal distribution z ~ N 0, 1 for the first four problems.
1. Pz  1.43 
2. P0  z  3.21
3. P1.45  z  1.45 
4. z .14
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x ~ N 3, 7.2 for problems 5 through 8.
5. Px  1.43 
6. P0  x  3.21
7. P1.45  x  1.45 
8. x.14
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Part II: (15+ points) Do all the following: All questions are 2 points each except as marked. Exam is
normed on 50 points including take-home. (Showing your work can give partial credit on some
problems! In open-ended questions it is expected. Please indicate clearly what sections of the
problem you are answering and what formulas you are using. Neatness counts!) Remember that you
may not be able to finish this section, so ration your time on each problem. [Numbers in brackets are a
cumulative total]
1. Which of the following are correct for a discrete distribution? (You may circle more than one.)
a) P7  x  20   Px  20   Px  7
b) Px  0  1  Px  0 . This should have read Px  1  1  Px  0 .
c) Px  2  1  Px  2
d) Px  2  1  Px  2
e) Px  6  Px  5
f) None of the above is correct.
2. Which of the following are correct for a continuous distribution? (You may circle more than one.)
a) P7  x  20   Px  20   Px  7
b) Px  0  1  Px  0 . This should have read Px  1  1  Px  0 .
c) Px  2  1  Px  2
d) Px  2  1  Px  2
e) Px  6  Px  5
f) None of the above is correct.
Both of the above questions will have points taken off for wrong choices, though no question will be
given a score below zero.
3. Which of the following is not a requirement of a discrete probability distribution?
a) Equally likely probability of a success.
b) Sum of the possible outcomes is 1.00.
c) The outcomes are mutually exclusive.
d) The probability of each outcome is between 0 and 1.
e) All of the above are required.
f) None of the above is required.
4. A company that sells annuities must base the annual payout on the probability distribution of the length
of life of the participants in the plan. Suppose the probability distribution of the lifetimes of the participants
is approximately a normal distribution with a mean of 68 years and a standard deviation of 3.5 years. What
proportion of the plan recipients would receive payments beyond age 75? (2.5) [8.5]
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5. What type of probability distribution will the consulting firm most likely employ to analyze the
insurance claims in the following problem?
An insurance company has called a consulting firm to determine if the company has an unusually
high number of false insurance claims. It is known that the industry proportion for false claims is
3%. The consulting firm has decided to randomly and independently sample 100 of the company’s
insurance claims. They believe the number of these 100 that are false will yield the information the
company desires.
a) Binomial distribution.
b) Poisson distribution.
c) Normal distribution.
d) Hypergeometric distribution.
e) Geometric distribution
6. If n = 10 and p = 0.70, then the standard deviation of the binomial distribution of x is
a) 0.021
b) 0.07
c) 0.145
d) 1.45
e) 2.10
f) 7.00
g) 14.29
7. If n = 10 and p = 0.70, then the standard deviation of the binomial distribution of p 
a)
b)
c)
d)
e)
f)
g)
x
is
n
[14.5]
0.021
0.07
0.145
1.45
2.10
7.00
14.29
8. What type of probability distribution will most likely be used to analyze warranty repair needs on new
cars in the following problem?
The service manager for a new automobile dealership reviewed dealership records of the past 20
sales of new cars to determine the number of warranty repairs he will be called on to perform in the
next 90 days. Corporate reports indicate that the probability any one of their new cars needs a
warranty repair in the first 90 days is 0.05. The manager assumes that calls for warranty repair are
independent of one another and is interested in predicting the number of warranty repairs he will be
called on to perform in the next 90 days for this batch of 20 new cars sold.
[16.5]
a) Binomial distribution.
b) Poisson distribution.
c) Normal distribution.
d) Hypergeometric distribution.
e) Geometric distribution
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9. On the average, 3 customers per minute arrive at any one of the checkout counters of a convenience
store. Checkout time takes 2 minutes. How many clerks are needed to be sure that the chance of someone
having to wait is no more than 10%? Your answer should briefly explain what distribution you are using
and how you got the number you give. (2.5)
[19]
10. 13 of a shipment of 15 hard disks are defective. 4 disks are inspected. What is the chance of finding at
least one defective?
11. (Extra Credit) A catalog company that receives the majority of its orders by telephone conducted a
study to determine how long customers were willing to wait on hold before ordering a product. The length
of time was found to be a random variable best approximated by an exponential distribution with a mean
equal to 3 minutes. What proportion of customers will not be on the line after a wait of 4.5 minutes?
a) 0.22313
b) 0.48658
c) 0.51342
d) 0.77687
e) The answer is nowhere near any of the above. Here is my answer.
12. Assume that there are an average of 6 chocolate chips per cookie. Now assume that an inspector will
discard any cookies with fewer than 4 chips. About how many will be discarded from a batch of 100.[23]
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ECO251 QBA1
THIRD EXAM
Apr 18, 2005
TAKE HOME SECTION
Name: _________________________
Student Number: _________________________
Throughout this exam show your work! Please indicate clearly what sections of the problem you are
answering and what formulas you are using.
Part III. Do all the Following (19+ Points) Show your work! Neatness counts!
1. (Webster) Identify the distribution that you are using in each problem. If you have a number like
n  20  g , make it very clear what value of n you are using. Look at the solved problems for Section L,
the solution to Grass3 and ‘Great Distributions’ (especially the hints on the 3 rd page) before you
start.
a) The average number of calls that come into a switchboard is 1 each minute.
Let g be the last digit of your student number. Assume that the operator is away from her desk
for 10  g 
minutes (so that the time that you use will be between 5 and 9.5 minutes). What is
2
the average number of calls that will come into the switchboard in that time? (0)
(i) What is the probability that at least one call is missed while the operator is away? (1)
(ii) What is the probability that between 10 and 20 (inclusive) calls are missed while the
operator is away? (1)
b) You purchase parts from a supplier who has an average defect rate of 1  g  units per lot of
100. (Your average for a lot of 100 will be between 1 and 10.)
(i) You order 150 units (1.5 lots) , but say that you will not buy from this supplier again if
more than 4 units are defective. What is the probability that you will not buy from this
supplier again? (1) (Note: If g  9, you order 200 units.)
(ii) The next time you order (if there is a next time), you resolve to take a sample of ten
items from the thousand in this order. You will return the whole mess if more than two
items are defective. If an average of 1  g  units per lot of 100 means that the probability
of an item being defective is 1  g % , what is the probability that you will return the
order? (2) Note: To keep us on a level playing field, you must work out this answer by
hand. Though it is strongly recommended that you check your answer against Table 14
and/or Table 15, no answer using tables will be accepted!
[5]
c) Lawn fertilizer comes in bags whose weight follows the continuous uniform distribution with a
range between 23.8 and 26 0.2g pounds. You need 24.8 pounds to cover your lawn and buy only
one bag. (Your upper number will be between 26 and 28.9 pounds.)
(i) What is the probability that you will not have enough fertilizer in the bag? (1)
(ii) If you buy two bags, what is the mean and standard deviation of their total weight?
(1)
[7]
(iii) If the bags are distributed with a mean of 25.5 pounds and a range of 2  0.1g 
pounds, what is the probability that a bag will be between 20 and 24 pounds? (Your
range will be between 2 and 2.9 pounds) (1)
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d) There are only 10  2g officers in your firm that are eligible to be on the Compensation
Committee. Of these half make over $500,000.00 a year.
(i) If you pick 4 officers at random, what is the probability that exactly three make over
$500,000? (2)
(ii) The shareholders will be very unhappy if they see that more than half of the members
of the Compensation Committee make over $500,000. If you pick 4 officers at random,
what is the probability that more than half make over $500,000? (Note that the answer is
not the same as the answer to (i).) (1)
[11]
e) Only 65% of the many workers in a high security scientific installation seem to be capable of
remembering to carry their badges. If they are among the 35% that forget they will be sent home
to get them!
(i) If the installation opens at 6am, what is the probability that the second employee to
arrive is the first one sent home? (1)
(ii) On the average, if we number the employees by their arrival order, what is the
average number of the first employee to be sent home? (The third? The 4 th? The 3.74th?)
(1)
(iii) If a group of 6  g  employees arrive at 9am, what is the probability that at least one
will not be sent home? (1)
(iv) In the same group, what is the probability that at least one will be sent home? (1)
(v) In the same group, what is the probability that more than half will not be sent home?
(1)
[16]
(vi) (Extra Credit) By late morning, the employees are trickling in at only 12 per hour.
The guard decides that if the probability of someone arriving in the next 5 minutes after
you come in is less than 50%, he will hide behind the warning sign and take a few nips
from his bottle. He asks you whether the probability is less than 50%. Do not use the
Poisson distribution for this problem. (2)
2. As everyone knows, a jorcillator has two components, a phillinx and a flubberall. It seems that the
jorcillator only works as long as both components work (so that it fails in the first month if either
component fails). Note change.
The probability of the phillinx failing is given by a Normal distribution with   2  .1h and
  0.5  .01h , where h is the second-to-last digit of your student number. For example, if the life of
the phillinx is represented by x1 , the chance of the phillinx failing in the 3rd month is P2  x1  3
and the probability of it failing after the third month is Px1  3 . For example: Ima Badrisk has the
number 375292, so her distribution has a mean of   2  .1h  2.9 and a standard deviation of
  0.5  .019  0.59 . Your mean will be between 2.0 and 2.9 and your standard deviation will be
between 0.50 and 0.59.
The probability of the flubberall failing is: in the first month 40%; in the second month 30%; in the
third month 20% and after the third month 10%. , We will divide time into four periods with the last
period being ‘beyond the third month.’ The jorcillator is guaranteed to fail in one of the four periods.
Failure of components is assumed to be independent, so if the probability of the phillinx failing in the
first month is .1, and the probability of the flubberall failing in the first month is .4, the probability of
both components failing in the first month is (.1) (.4) =.04 (This is not the necessarily the probability
that the jorcillator will fail in the first month!)
In order to maintain my sanity, use the following events.
Failure of the phillinx in period 1, 2, 3, 4 are events A1, A2 , A3 , and A4 .
Failure of the flubberall in period 1, 2, 3, 4 are events B1, B2 , B3 , and B4 .
Failure of the jorcillator in period 1, 2, 3, 4 are events C1, C2 , C3 , and C 4 .
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a) What is the probability that the phillinx will fail in month 1? Month 2? Month 3? After Month
3? (2)
b) What is the probability that the jorcillator will fail in the first month? (2)
c) What is the probability that the jorcillator will fail in the second month? (1)
d) What is the probability that the jorcillator will fail in the third month? (1)
e) What is the probability that the jorcillator will last beyond 3 months? (1)
[21]
If you haven’t figured it out already, one of the easiest ways to do this is to make a joint probability
table. Put the A events across the top. Put the B events down the side. Figure out what the probability
of the joint events must be if they are independent. Now make a similar table. This time, instead of
probabilities, fill in the period in which the jorcillator fails.
f) (Extra Credit) Find the probability that the jorcillator and the Phillinx both fail in the third
month (1)
g) (Extra Credit) Find the probability that the phillinx fails in the third month, given that the
jorcillator fails in the third month i.e. P A3 C3 (1)


h) (Extra Credit) Demonstrate Bayes’ rule by finding showing how to get the probability that the
jorcillator fails in the third month, given that the phillinx fails in the third month, two ways. The first
would be directly by working with joint probabilities that both the jorcillator and the phillinx fail. The
second would be by using Bayes’ rule and your result in g). (1)
3. (Extra Credit) Do the following Binomial problems. If you substitute another distribution for the
Binomial, justify the substitution by showing that a substitution condition is satisfied. (6)
a) 45% of our many management employees have MBAs. If we select 150 at random, what is the
probability that exactly 72 have college degrees?
b) 45% of our many management employees have MBAs. If we select 150 at random, what is the
probability that more than half have college degrees?
c) 5% of our many management employees have MBAs. If we select 90 at random, what is the
probability that more than 6 have college degrees?