Practice
for Chapters 2, 3, 4
Chapter
2 Practice Questions
|
Also,
Be able to do a Binomial (not on practice test, but on test) see HW
solutions chapter 2-32, 2-33, 2-35 |
.
2.148
Fast Service Store has maintained daily sales records on the various size
"Cool Drink" sales. Assuming
that past performance is a good indicator of future sales, what is the
probability of a customer purchasing a $0.50 "Cool Drink?"
|
“Cool
Drink” Price |
Number Sold |
|
$0.25 |
75 |
|
$0.35 |
120 |
|
$0.50 |
125 |
|
$0.75 |
50 |
|
Total |
400 |
ANSWER:
125/400 = 0.3125
2.151
A market research study is being conducted to determine if a product
modification will be received well by the public.
A total of 1,000 consumers are questioned regarding this product.
The table below provides information regarding this sample.
|
|
Positive
Reaction |
Neutral
Reaction |
Negative
Reaction |
|
Male |
240 |
60 |
100 |
|
Female |
260 |
220 |
120 |
(a)
What is the probability that a randomly selected male would find this
change favorable (positive)?
(b)
What is the probability that a randomly selected person would be a female
who had a negative reaction?
(c)
If it is known that a person had a positive reaction to the study, what
is the probability that the person is female?
ANSWER:
(a) 240/400 = 0.60 (b) 120/1000 = 0.120 (c)
260/500 = 0.520
2.152 In a production run of 200 units,
there are exactly 10 defective items and 190 good items (200 total).
(a) What is the probability that a randomly selected item is defective?
(b) If two items are sampled without replacement, what is the probability
that both are good?
(c) If two items are randomly sampled without replacement, what is the
probability that the first is good but the second is defective?
ANSWER:
(a) 10/200 = 0.05 (b) (190/200)(189/199) = 0.902
(c) (190/200)(10/199) = 0.048
2.154
Last semester, the grade distribution in a quantitative methods course
had the following distribution: 10
percent A, 25 percent B, 35 percent C, 10 percent D, and 15 percent W
(withdrew).
(a) If this grade distribution does not change this semester, what is the
probability that a randomly selected student will make at least a D?
(b) If this grade distribution does not change this semester, what is the
probability that a randomly selected student will fail the course?
(c) If this grade distribution does not change this semester, what is the
probability that a randomly selected student who finished the course (did not
withdraw) made a grade of D or better?
ANSWER:
(a) 80 percent (b) 5 percent (c)
0.80/0.85 = 0.94
2.157 A southwestern tourist city has
records indicating that the average daily temperature in the summer is 82
degrees F, which is normally distributed with a standard deviation of 3 degrees
F. Based on these records, determine:
(a) the probability of a
daily temperature between 79 degrees F and 85 degrees F
(b) the probability that the
daily temperature exceeds 90 degrees F
(c) the probability that the
daily temperature is below 76 degrees F
ANSWER: (a) P(79<X<85)
= 0.68268 (b) P(X>90) = 0.00379
(c) P(X<76) = 0.02275
2.158 Using the table for finding the
areas under normal curves, find the area under a normal curve with a mean of 200
and a standard deviation of 10 between the values of:
(a) 200 to 205
ANSWER: (a) 0.19146 (b)
0.38292 (c) 0.43319 (d) 0.62465 (e) 0.86692
2.159
ABC Manufacturing has 6 machines that perform a particular task.
Breakdowns occur frequently for this machine.
Past records indicate that the number of breakdowns that occur each day
is described by the following probability distribution:
|
Number
of Breakdowns |
Probability |
|
0 |
0.4 |
|
1 |
0.3 |
|
2 |
0.2 |
|
3 |
0.1 |
|
More
than 3 |
0.0 |
(a) What is the expected
number of breakdowns in any given day?
(b) What is the variance for
this distribution?
(c) What is the probability
that there will be at least 2 breakdowns in a day?
ANSWER:
(a) expected value = 1.0 (b)
variance = 1.0 (c) P(2 or
more) = 0.2 + 0.1 = 0.3
2.160
Arrivals in a university advising office during the week of registration
are known to follow a Poisson distribution with an average of 4 people arriving
each hour.
(a) What is the probability
that exactly 4 people will arrive in the next hour?
(b) What is the probability
that exactly 5 people will arrive in the next hour?
ANSWER: (a) P(X = 4) = 0.1952 (b) P(X = 5) = 0.1563
2.161
The time required to complete a project is known to be normally
distributed with a mean of 46 weeks and a standard deviation of 4 weeks.
(a) What is the probability
that the project is finished in 40 weeks or less?
(b) What is the probability
that the project is finished in 52 weeks or less?
(c)
There is an 80 percent chance that the project will be finished in less than how
many weeks?
ANSWER: (a) 0.06881
(b) 0.93319 (c) 46 +
0.84(4) = 49.36
2.163
An urn contains 7 blue and 3 yellow chips.
If the drawing of two chips (or
3 in part a) in succession is done with replacement,
determine the probability of:
(a)
drawing three yellow chips
(b) drawing a blue chip on the first draw and a yellow chip on the second
draw
(c) drawing a blue chip on the second draw given that a yellow chip was drawn
on the first draw
(d)
drawing a yellow chip on the second draw given that a blue chip was drawn
on the first draw
(e)
drawing a yellow chip on the second draw given that a yellow chip was
drawn on the first draw
ANSWER:
(a) 0.027 (b) 0.210
(c) 0.700 (d) 0.300
(e) 0.300
2.167
A new television program was viewed by 200 people (120 females and 80
males). Of the females, 60 liked
the program and 60 did not. Of the
males, 60 of the 80 liked the program.
(a)
What is the probability that a randomly selected individual disliked the
program?
(b)
If a male in this group is selected, what is the probability that he
disliked the program?
(c)
What is the probability that a randomly selected individual is a female
and disliked the program?
ANSWER:
(a) 80/200 = 0.40 (b) 20/80 = 0.25 (c)
60/200 = 0.30
Chapter
3 Practice Questions
3.97
A concessionaire for the local ballpark has developed a table of
conditional values for the various alternatives (stocking decision) and states
of nature (size of crowd).
|
|
STATES
OF NATURE |
||
|
|
(size
of crowd) |
||
|
Alternatives
|
Large |
Average |
Small |
|
Large
Inventory |
$22,000 |
$12,000 |
-$2,000 |
|
Average
Inventory |
$15,000 |
$12,000 |
$6,000 |
|
Small
Inventory |
$
9,000 |
$
6,000 |
$5,000 |
If the probabilities
associated with the states of nature are 0.30 for a large crowd, 0.50 for an
average crowd, and 0.20 for a small crowd, determine:
(a) the alternative that provides the greatest expected
monetary value (EMV)
(b) the expected value of perfect information (EVPI)
ANSWERS:
(a) maximum EMV = $12,200
3.98
A concessionaire for the local ballpark has developed a table of
conditional values for the various alternatives (stocking decision) and states
of nature (size of crowd).
|
|
States
of Nature |
||
|
|
(size
of crowd) |
||
|
Alternatives |
Large
|
Average |
Small |
|
Large
Inventory |
$22,000 |
$12,000 |
-$2,000 |
|
Average
Inventory |
$15,000 |
$12,000 |
$6,000 |
|
Small
Inventory |
$
9,000 |
$
6,000 |
$5,000 |
If the probabilities
associated with the states of nature are 0.30 for a large crowd, 0.50 for an
average crowd, and 0.20 for a small crowd, determine:
(a) the opportunity loss table
ANSWERS:
(a)
Opportunity Loss Table
|
|
States of Nature |
||
|
Alternatives |
Large |
Average |
Small |
|
Large |
0 |
0 |
8,000 |
|
Average |
7,000 |
0 |
0 |
|
Small |
13,000 |
6,000 |
1,000 |
(b) minimum EOL = $1,600
3.99
Given the following conditional value table, determine the appropriate
decision under uncertainty using:
(a) maximax
|
|
States
of Nature |
||
|
Alternatives |
Very
Favorable Market |
Average
Market |
Unfavorable
Market |
|
Large
Plant |
$275,000 |
$100,000 |
-$150,000 |
|
Small
Plant |
$200,000 |
60,000 |
-$
10,000 |
|
Overtime |
$100,000 |
$
40,000 |
-$
1,000 |
|
Do
Nothing |
0 |
0 |
0 |
ANSWERS:
(a) Large Plant
3.101
The ABC Co. is considering a new consumer product.
They have no idea whether or not the XYZ Co. will come out with a
competitive product. If ABC adds an
assembly line for the product and XYZ does not follow with a competitive
product, their expected profit is $40,000; if they add an assembly line and XYZ
does follow, they still expect $10,000 profit.
If ABC adds a new plant addition and XYZ does not produce a competitive
product, they expect a profit of $600,000; if XYZ does compete for this market,
ABC expects a loss of $100,000.
Calculate Hurwicz’s
criterion of realism using a’s of 0.7, 0.3, and 0.1.
ANSWERS:
|
|
Criterion
of Realism |
||
|
Decision |
a
= 0.7 |
a
= 0.3 |
a
= 0.1 |
|
add
assembly line |
$31,000 |
$19,000 |
$13,000 |
|
plant
addition |
$390,000 |
$110,000 |
$30,000 |
|
do
nothing |
$0 |
$0 |
$0 |
3.104
The following payoff table provides profits based on various possible
stocking decisions and various demand situations.
|
|
States
of Nature |
||
|
|
Demand |
||
|
Alternatives |
Low |
Medium |
High |
|
Stock
12 |
800 |
800 |
800 |
|
Stock
13 |
700 |
900 |
900 |
|
Stock
14 |
600 |
800 |
1000 |
Based on current
information, it is believed that the probabilities of the three demand states
are each 1/3. If you wished to
minimize the expected opportunity loss, what decision should be made and what
would the minimum expected opportunity loss be?
ANSWER:
Opportunity loss table:
|
|
States
of Nature |
||
|
|
Demand |
||
|
Alternatives |
Low |
Medium
|
High |
|
Stock
12 |
0 |
100 |
200 |
|
Stock
13 |
100 |
0 |
100 |
|
Stock
14 |
200 |
100 |
0 |
EOL (12) = 100
EOL (13) = 66.7
====> Therefore,
stock 13.
EOL (14) = 100
3.105
The following payoff table provides profits based on various possible
decision alternatives and various levels of demand.
|
|
States
of Nature |
||
|
|
Demand |
||
|
Alternatives |
Low |
Medium |
High |
|
Alternative
1 |
80 |
120 |
140 |
|
Alternative
2 |
90 |
90 |
90 |
|
Alternative
3 |
50 |
70 |
150 |
The
probability of a low demand is 0.4, while the probability of a medium and high
demand is each 0.3.
(a) What decision would an
optimist make?
(b) What decision would a
pessimist make?
(c) What is the highest
possible expected monetary value?
(d) Calculate the expected
value of perfect information for this situation.
ANSWER:
(a)
Alternative 3
3.106
Norman L. Flowers holds the exclusive university contract for donut
sales. The demand (based on
historical records) appears to follow the following distribution:
|
Daily
Demand (Dozens) |
Probability |
|
4 |
0.15 |
|
5 |
0.25 |
|
6 |
0.30 |
|
7 |
0.25 |
|
8 |
0.05 |
The cost of producing these
is $1.20 per dozen while the selling price is $4.20 per dozen.
Based on a marginal analysis of this situation, how many donuts should
Norman produce each day?
ANSWER: P > 1.20/4.20 =
0.286. Therefore, Norman should
produce seven dozen.
3.107
Orders for clothing from a particular manufacturer for this year’s
Christmas shopping season must be placed in February.
The cost per unit for a particular dress is $20 while the anticipated
selling price is $50. Anything not
sold during the season can be sold for $15 to a discount store.
Demand is projected to be either 50, 60, or 70 units.
There is a 40 percent chance that demand will be 50 units, a 50 percent
chance that demand will be 60 units, and a 10 percent chance that demand will be
70 units. If the company decides to
use the EMV criterion, how many units should be ordered in February?
ANSWER:
Payoff table:
|
|
States
of Nature |
|
||
|
|
Demand |
|
||
|
Alternatives |
50 |
60 |
70 |
EMV |
|
Order
50 |
$1,500 |
$1,500 |
$1,500 |
$1,500 |
|
Order
60 |
$1,450 |
$1,800 |
$1,800 |
$1,660 |
|
Order
70 |
$1,400 |
$1,750 |
$2,100 |
$1,645 |
|
|
0.4 |
0.5 |
0.1 |
|
Therefore, order 60.
Chapter
4 Practice Questions
4.94 Bakery Products is considering the introduction of a new line of products. In order to produce the new line, the bakery is considering either a major or minor renovation of the current plant. The following conditional values table has been developed by the bakery.
|
Alternatives |
Favorable Market $ |
Unfavorable Market $ |
|
Major Renovation |
100,000 |
-90,000 |
|
Minor Renovation |
40,000 |
-20,000 |
|
Do Nothing |
1,000 |
0 |
Under the assumption that the probability of a favorable market is equal to the probability of an unfavorable market, determine:
(a) the EMV of a major renovation. (b) the EMV of a minor renovation.
(c) the EMV of the do nothing option. (d) the best alternative using EMV.
ANSWER:
(a) 5000 (b) 10000 (c) 500 (d) choose the minor renovation alternative