Consider
a Poisson distribution with μ = 5. If needed, round your
answer to four decimal digits.
(a)
Choose the appropriate Poisson probability mass function.
(i)
(ii)
(iii)
(iv)
(b)
Compute f(2).
(c)
Compute f(1).
(d)
Compute P(x ≥ 2).
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According
to a 2018 article in Esquire magazine, approximately 70% of
males over age 70 will develop cancerous cells in their prostate. Prostate
cancer is second only to skin cancer as the most common form of cancer for
males in the United States. One of the most common tests for the detection of
prostate cancer is the prostate-specific antigen (PSA) test. However, this
test is known to have a high false-positive rate (tests that come back
positive for cancer when no cancer is present). Suppose there is a 0.02
probability that a male patient has prostate cancer before testing. The
probability of a false-positive test is 0.75, and the probability of a
false-negative (no indication of cancer when cancer is actually present) is
0.20.
Let
event male patient has prostate cancer
positive PSA test for prostate cancer
negative PSA test for prostate cancer
(a)
What is the probability that the male patient has prostate cancer if
the PSA test comes back positive? Round your answer to four decimal places.
(b)
What is the probability that the male patient has prostate cancer if
the PSA test comes back negative? Round your answer to four decimal places.
(c)
For older men, the prior probability of having cancer increases.
Suppose that the prior probability of the male patient is 0.3 rather than
0.02. What is the probability that the male patient has prostate cancer if
the PSA test comes back positive? Round your answer to four decimal places.
What is the probability that the male patient has prostate cancer if
the PSA test comes back negative? Round your answer to four decimal places.
(d)
What can you infer about the PSA test from the results of parts (a),
(b), and (c)?
The difference between and in parts (a) and
(b) is than
the difference between and in part (c).
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Suppose
that Motorola uses the normal distribution to determine the probability of
defects and the number of defects in a particular production process. Assume
that the production process manufactures items with a mean weight of 10
ounces. Calculate the probability of a defect and the suspected number of
defects for a 1,000-unit production run in the following situations.
(a)
The process standard deviation is 0.24, and the process control is
set at plus or minus one standard deviation. Units with weights less than
9.76 or greater than 10.24 ounces will be classified as defects. If
required, round your answer for the probability of a defect to four decimal
places and for the number of defects to the nearest whole number.
Probability of a defect:
Number of defects:
(b)
Through process design improvements, the process standard deviation
can be reduced to 0.08. Assume that the process control remains the same,
with weights less than 9.76 or greater than 10.24 ounces being classified
as defects. If required, round your answer for the probability of a defect
to four decimal places and for the number of defects to the nearest whole
number.
Probability of a defect:
Number of defects:
(c)
What is the advantage of reducing process variation, thereby causing
process control limits to be at a greater number of standard deviations
from the mean?
Reducing the process standard deviation causes a in
the number of defects.
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Consider a Poisson distribution with μ =
5. If needed, round your answer to four decimal digits.
(a)
Choose the appropriate Poisson probability mass function.
(i)
(ii)
(iii)
(iv)
(b)
Compute f(2).
(c)
Compute f(1).
(d)
Compute P(x ≥ 2).
The
percent frequency distributions of job satisfaction scores for a sample of
information systems (IS) senior executives and middle managers are as
follows. The scores range from a low of 1 (very dissatisfied) to a high of 5
(very satisfied).
Job Satisfaction Score
IS Senior Executives (%)
IS Middle Managers (%)
1
5
4
2
9
10
3
30
11
4
42
46
5
14
29
If
required, round your answers to two decimal places.
(a)
Develop a probability distribution for the job satisfaction score of
a randomly selected senior executive.
x
f(x)
1
2
3
4
5
(b)
Develop a probability distribution for the job satisfaction score of
a randomly selected middle manager.
x
f(x)
1
2
3
4
5
(c)
What is the probability that a randomly selected senior executive
will report a job satisfaction score of 4 or 5?
(d)
What is the probability that a randomly selected middle manager is
very satisfied?
(e)
Compare the overall job satisfaction of senior executives and middle
managers.
Senior executives appear to be than
middle managers.
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Consider
the following exponential probability density function.
for x ≥
0
If
needed, round your answer to four decimal digits.
(a)
Choose the correct formula for P(x ≤ x0).
(i)
(ii)
(iii)
(iv)
(b)
Find P(x ≤ 2).
(c)
Find P(x ≥ 3).
(d)
Find P(x ≤ 5).
(e)
Find P(2 ≤ x ≤ 5).
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According
to a 2018 article in Esquire magazine, approximately 70% of
males over age 70 will develop cancerous cells in their prostate. Prostate
cancer is second only to skin cancer as the most common form of cancer for
males in the United States. One of the most common tests for the detection of
prostate cancer is the prostate-specific antigen (PSA) test. However, this
test is known to have a high false-positive rate (tests that come back
positive for cancer when no cancer is present). Suppose there is a 0.02
probability that a male patient has prostate cancer before testing. The
probability of a false-positive test is 0.75, and the probability of a
false-negative (no indication of cancer when cancer is actually present) is
0.20.
Let
event male patient has prostate cancer
positive PSA test for prostate cancer
negative PSA test for prostate cancer
(a)
What is the probability that the male patient has prostate cancer if
the PSA test comes back positive? Round your answer to four decimal places.
(b)
What is the probability that the male patient has prostate cancer if
the PSA test comes back negative? Round your answer to four decimal places.
(c)
For older men, the prior probability of having cancer increases.
Suppose that the prior probability of the male patient is 0.3 rather than
0.02. What is the probability that the male patient has prostate cancer if
the PSA test comes back positive? Round your answer to four decimal places.
What is the probability that the male patient has prostate cancer if
the PSA test comes back negative? Round your answer to four decimal places.
(d)
What can you infer about the PSA test from the results of parts (a),
(b), and (c)?
The difference between and in parts (a) and
(b) is than
the difference between and in part (c).
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Suppose
that Motorola uses the normal distribution to determine the probability of
defects and the number of defects in a particular production process. Assume
that the production process manufactures items with a mean weight of 10
ounces. Calculate the probability of a defect and the suspected number of
defects for a 1,000-unit production run in the following situations.
(a)
The process standard deviation is 0.25, and the process control is
set at plus or minus one standard deviation. Units with weights less than
9.75 or greater than 10.25 ounces will be classified as defects. If
required, round your answer for the probability of a defect to four decimal
places and for the number of defects to the nearest whole number.
Probability of a defect:
Number of defects:
(b)
Through process design improvements, the process standard deviation
can be reduced to 0.10. Assume that the process control remains the same,
with weights less than 9.75 or greater than 10.25 ounces being classified
as defects. If required, round your answer for the probability of a defect
to four decimal places and for the number of defects to the nearest whole
number.
Probability of a defect:
Number of defects:
(c)
What is the advantage of reducing process variation, thereby causing
process control limits to be at a greater number of standard deviations
from the mean?
Reducing the process standard deviation causes a in
the number of defects.
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Consider
a Poisson distribution with μ = 3. If needed, round your
answer to four decimal digits.
(a)
Choose the appropriate Poisson probability mass function.
(i)
(ii)
(iii)
(iv)
(b)
Compute f(2).
(c)
Compute f(1).
(d)
Compute P(x ≥ 2).
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The
percent frequency distributions of job satisfaction scores for a sample of
information systems (IS) senior executives and middle managers are as
follows. The scores range from a low of 1 (very dissatisfied) to a high of 5
(very satisfied).
Job Satisfaction Score
IS Senior Executives (%)
IS Middle Managers (%)
1
5
4
2
9
10
3
40
8
4
42
46
5
4
32
If
required, round your answers to two decimal places.
(a)
Develop a probability distribution for the job satisfaction score of
a randomly selected senior executive.
x
f(x)
1
2
3
4
5
(b)
Develop a probability distribution for the job satisfaction score of
a randomly selected middle manager.
x
f(x)
1
2
3
4
5
(c)
What is the probability that a randomly selected senior executive
will report a job satisfaction score of 4 or 5?
(d)
What is the probability that a randomly selected middle manager is
very satisfied?
(e)
Compare the overall job satisfaction of senior executives and middle
managers.
Senior executives appear to be than
middle managers.
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Consider
the following exponential probability density function.
for x ≥
0
If
needed, round your answer to four decimal digits.
(a)
Choose the correct formula for P(x ≤ x0).
(i)
(ii)
(iii)
(iv)
(b)
Find P(x ≤ 2).
(c)
Find P(x ≥ 3).
(d)
Find P(x ≤ 5).
(e)
Find P(2 ≤ x ≤ 5).
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Suppose that Motorola uses the normal distribution to determine the probability of defects and the number of defects in a particular production process. Assume that the production process manufactures items with a mean weight of 10 ounces. Calculate the probability of a defect and the suspected number of defects for a 1,000-unit production run in the following situations.
(a)
The process standard deviation is 0.20, and the process control is set at plus or minus one standard deviation. Units with weights less than 9.80 or greater than 10.20 ounces will be classified as defects. If required, round your answer for the probability of a defect to four decimal places and for the number of defects to the nearest whole number.
Probability of a defect:
Number of defects:
(b)
Through process design improvements, the process standard deviation can be reduced to 0.08. Assume that the process control remains the same, with weights less than 9.80 or greater than 10.20 ounces being classified as defects. If required, round your answer for the probability of a defect to four decimal places and for the number of defects to the nearest whole number.
Probability of a defect:
Number of defects:
(c)
What is the advantage of reducing process variation, thereby causing process control limits to be at a greater number of standard deviations from the mean?
Reducing the process standard deviation causes a in the number of defects.
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The percent frequency distributions of job satisfaction scores for a sample of information systems (IS) senior executives and middle managers are as follows. The scores range from a low of 1 (very dissatisfied) to a high of 5 (very satisfied).
Job Satisfaction Score
IS Senior Executives (%)
IS Middle Managers (%)
1
5
4
2
9
10
3
34
16
4
42
46
5
10
24
If required, round your answers to two decimal places.
(a)
Develop a probability distribution for the job satisfaction score of a randomly selected senior executive.
x
f(x)
1
2
3
4
5
(b)
Develop a probability distribution for the job satisfaction score of a randomly selected middle manager.
x
f(x)
1
2
3
4
5
(c)
What is the probability that a randomly selected senior executive will report a job satisfaction score of 4 or 5?
(d)
What is the probability that a randomly selected middle manager is very satisfied?
(e)
Compare the overall job satisfaction of senior executives and middle managers.
Senior executives appear to be than middle managers.
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Consider a Poisson distribution with μ = 2. If needed, round your answer to four decimal digits.
(a)
Choose the appropriate Poisson probability mass function.
(i)
(ii)
(iii)
(iv)
(b)
Compute f(2).
(c)
Compute f(1).
(d)
Compute P(x ≥ 2).
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Correct
Consider the following exponential probability density function.
for x ≥ 0
If needed, round your answer to four decimal digits.
(a)
Choose the correct formula for P(x ≤ x0).
(i)
(ii)
(iii)
(iv)
(b)
Find P(x ≤ 2).
(c)
Find P(x ≥ 3).
(d)
Find P(x ≤ 5).
(e)
Find P(2 ≤ x ≤ 5).
Suppose that Motorola uses the normal distribution to determine the probability of defects and the number of defects in a particular production process. Assume that the production process manufactures items with a mean weight of 10 ounces. Calculate the probability of a defect and the suspected number of defects for a 1,000-unit production run in the following situations.
(a)
The process standard deviation is 0.20, and the process control is set at plus or minus one standard deviation. Units with weights less than 9.80 or greater than 10.20 ounces will be classified as defects. If required, round your answer for the probability of a defect to four decimal places and for the number of defects to the nearest whole number.
Probability of a defect:
Number of defects:
(b)
Through process design improvements, the process standard deviation can be reduced to 0.08. Assume that the process control remains the same, with weights less than 9.80 or greater than 10.20 ounces being classified as defects. If required, round your answer for the probability of a defect to four decimal places and for the number of defects to the nearest whole number.
Probability of a defect:
Number of defects:
(c)
What is the advantage of reducing process variation, thereby causing process control limits to be at a greater number of standard deviations from the mean?
Reducing the process standard deviation causes a in the number of defects.
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Suppose that Motorola uses the normal distribution to determine the probability of defects and the number of defects in a particular production process. Assume that the production process manufactures items with a mean weight of 10 ounces. Calculate the probability of a defect and the suspected number of defects for a 1,000-unit production run in the following situations.
(a)
The process standard deviation is 0.30, and the process control is set at plus or minus one standard deviation. Units with weights less than 9.70 or greater than 10.30 ounces will be classified as defects. If required, round your answer for the probability of a defect to four decimal places and for the number of defects to the nearest whole number.
Probability of a defect:
Number of defects:
(b)
Through process design improvements, the process standard deviation can be reduced to 0.10. Assume that the process control remains the same, with weights less than 9.70 or greater than 10.30 ounces being classified as defects. If required, round your answer for the probability of a defect to four decimal places and for the number of defects to the nearest whole number.
Probability of a defect:
Number of defects:
(c)
What is the advantage of reducing process variation, thereby causing process control limits to be at a greater number of standard deviations from the mean?
Reducing the process standard deviation causes a in the number of defects.
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Freelance reporter Irwin Fletcher is examining the historical voting records of members of the U.S. Congress. For 175 representatives, Irwin has collected the voting record (yes or no) on 16 pieces of legislation. To examine the relationship between representatives' votes on different issues, Irwin has conducted an association rules analysis with a minimum support of 40% and a minimum confidence of 90%.
The data included the following bills:
Budget: approve federal budget resolution
Contras: aid for Nicaraguan contra rebels
El_Salvador: aid to El Salvador
Missile: funding for M-X missile program
Physician: freeze physician fees
Religious: equal access to all religious groups at schools
Satellite: ban on anti-satellite weapons testing
The following table shows the top five rules with respect to lift ratio. The table displays representatives' decisions in a "bill-vote" format. For example, "Contras-y" indicates that the representative voted yes on a bill to support the Nicaraguan Contra rebels and "Physician-n" indicates a no vote on a bill to freeze physician fees.
Antecedent
Consequent
Support for A and C
Confidence
Lift Ratio
Contras-y, Physician-n, Satellite-y
El_Salvador-n
0.40
0.95
1.98
Contras-y, Missile-y
El_Salvador-n
0.40
0.91
1.90
Contras-y, Physician-n
El_Salvador-n
0.44
0.91
1.90
Missile-n, Religious-y
El_Salvador-y
0.40
0.93
1.79
Budget-y, Contras-y, Physician-n
El_Salvador-n
0.41
0.90
1.89
(a)
Interpret the lift ratio of the first rule in the table.
The lift ratio of the first rule in the table means that it is % likely that a representative votes no on the El Salvador bill given that he or she votes yes on the Contras bill, no on the Physician bill, and yes on the Satellite bill than a randomly selected representative.
(b)
What is the probability that a representative votes no on El Salvador aid given that they vote yes to aid to Nicaraguan Contra rebels and yes to the M-X missile program? Round your answer to two decimal places.
(c)
What is the probability that a representative votes no on El Salvador aid given that they vote no to the M-X missile program and yes to equal access to religious groups in schools? Round your answer to two decimal places.
(d)
What is the probability that a randomly selected representative votes yes on El Salvador aid? Round your answer to three decimal places.
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In an effort to inform political leaders and economists discussing the deregulation of electric and gas utilities, data on eight numerical variables from utility companies have been grouped using hierarchical clustering based on Euclidean distance to measure dissimilarity between observations and complete linkage as the agglomeration method.
(a)
Based on the following dendrogram, what is the most appropriate number of clusters to organize these utility companies?
clusters are appropriate based on complete linkage.
(b)
Using the following data on the Observations 10, 13, 4, and 20, confirm that the complete linkage distance between the cluster containing {10, 13} and the cluster containing {4, 20} is 2.577 units as displayed in the dendrogram.
If required, round your answers to three decimal places. Do not round intermediate calculations.
Distance from Observation 10 and Observation 4:
Distance from Observation 10 and Observation 20:
Distance from Observation 13 and Observation 4:
Distance from Observation 13 and Observation 20:
Based on complete linkage, the merger of cluster {10, 13} and cluster {4, 20} is of these four distances. Thus, the increase in within-cluster dissimilarity of the new merged cluster based on the distance between Observation 13 and Observation 4.
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Jay Gatsby categorizes wines into one of three clusters. The centroids of these clusters, describing the average characteristics of a wine in each cluster, are listed in the following table.
Jay has recently discovered a new wine from the Piedmont region of Italy with the following characteristics. In which cluster of wines should he place this new wine? Justify your choice with appropriate calculations.
If required, round your answers to three decimal places. Do not round intermediate calculations.
Distance from new wine and Cluster 1:
Distance from new wine and Cluster 2:
Distance from new wine and Cluster 3:
is the most appropriate for the new wine observation.
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Amanda Boleyn, an entrepreneur who recently sold her start-up for a multi-million-dollar sum, is looking for alternate investments for her newfound fortune. She is considering an investment in wine, similar to how some people invest in rare coins and fine art. To educate herself on the properties of fine wine, she has collected data on 13 different characteristics of 178 wines. Amanda has applied k-means clustering to this data for k = 1, ... , 10 and generated the following plot of total sums of squared deviations. After analyzing this plot, Amanda generates summaries for k = 2, 3, and 4. Which value of k is the most appropriate to categorize these wines? Justify your choice with calculations.
Do not round intermediate calculations. If required, round your answers to two decimal places.
k = 2
Cluster 1 to Cluster 2 Distance / Cluster 1 Average Distance =
Cluster 2 to Cluster 1 Distance / Cluster 2 Average Distance =
Average =
k = 3
Cluster 1 to Cluster 2 Distance / Cluster 1 Average Distance =
Cluster 2 to Cluster 1 Distance / Cluster 2 Average Distance =
Cluster 1 to Cluster 3 Distance / Cluster 1 Average Distance =
Cluster 3 to Cluster 1 Distance / Cluster 3 Average Distance =
Cluster 2 to Cluster 3 Distance / Cluster 2 Average Distance =
Cluster 3 to Cluster 2 Distance / Cluster 3 Average Distance =
Average =
k = 4
Cluster 1 to Cluster 2 Distance / Cluster 1 Average Distance =
Cluster 2 to Cluster 1 Distance / Cluster 2 Average Distance =
Cluster 1 to Cluster 3 Distance / Cluster 1 Average Distance =
Cluster 3 to Cluster 1 Distance / Cluster 3 Average Distance =
Cluster 1 to Cluster 4 Distance / Cluster 1 Average Distance =
Cluster 4 to Cluster 1 Distance / Cluster 4 Average Distance =
Cluster 2 to Cluster 3 Distance / Cluster 2 Average Distance =
Cluster 3 to Cluster 2 Distance / Cluster 3 Average Distance =
Cluster 2 to Cluster 4 Distance / Cluster 2 Average Distance =
Cluster 4 to Cluster 2 Distance / Cluster 4 Average Distance =
Cluster 3 to Cluster 4 Distance / Cluster 3 Average Distance =
Cluster 4 to Cluster 3 Distance / Cluster 4 Average Distance =
Average =
Based on the individual ratio values and the average ratio values for each value of k, it appears that is the best clustering.
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Leggere, an internet book retailer, is interested in better understanding the purchase decisions of its customers. For a set of 2,000 customer transactions, it has categorized the individual book purchases comprising those transactions into one or more of the following categories: Novels, Willa Bean series, Cooking Books, Bob Villa Do-It-Yourself, Youth Fantasy, Art Books, Biography, Cooking Books by Mossimo Bottura, Harry Potter series, Florence Art Books, and Titian Art Books. Leggere has conducted association rules analysis on this data set and would like to analyze the output. Based on a minimum support of 200 transactions and a minimum confidence of 50%, the table below shows the top 10 rules with respect to lift ratio.
Antecedent
Consequent
Support for A and C
Confidence
Lift Ratio
BotturaCooking
Cooking
0.227
1.00
1.16
Cooking, BobVilla
Art
0.205
0.54
1.12
Cooking, Art
Biography
0.204
0.61
1.10
Cooking, Biography
Art
0.204
0.53
1.10
Youth Fantasy
Novels, Cooking
0.245
0.55
1.075
Cooking, Art
BobVilla
0.205
0.61
1.055
Cooking, BobVilla
Biography
0.218
0.58
1.04
Biography
Novels, Cooking
0.293
0.53
1.035
Novels, Cooking
Biography
0.293
0.57
1.035
Art
Novels, Cooking
0.249
0.52
1.01
(a)
Explain why the top rule “If customer buys a Bottura cooking book, then they buy a cooking book,” is not helpful even though it has the largest lift ratio and 100% confidence.
A Bottura cooking book is a specific type of cooking book, so whenever a customer purchases a Bottura cooking book they also have bought a cooking book. Thus, the confidence of “If a customer purchases a Bottura cooking book, then they purchase a cooking book” is: %.
Thus, any rule with the antecedent being a subset of the consequent will have a confidence of % and provides no information. Furthermore, if we would interpret this rule as suggesting that if a customer is interested in a specific type of cooking book that they be interested in other types of cooking books, this still provides no profound insight.
(b)
Explain how the confidence of 53% and lift ratio of 1.10 was computed for the rule “If a customer buys a cooking book and a biography book, then they buy an art book.” Interpret these quantities.
A confidence of 53% means that for 53% of the transactions when a cooking book and biography are purchased, an art book purchased.
A lift ratio 1.10 means that a transaction in which a cooking book and biography is purchased is 10% likely to also have purchased an art book than a randomly-selected transaction.
(c)
Based on these top 10 rules, what general insight can Leggere gain on the purchase habits of these customers?
In the top 10 rules, there are two rules involving the item set {cooking, Bob Villa, art} and two rules involving the item set {biography, novels, cooking} and {cooking, art, biography}. Leggere be well served to investigate promotions involving these respective item sets.
(d)
What will be the effect on the rules generated if Leggere decreases the minimum support and reruns the association rules analysis?
Decreasing the minimum support will generate a pool of rules and therefore will likely result in association rules with larger lift ratios. A rule with a lift ratio suggests that it is effective at identifying transactions in which the consequent occurs. However, the transactions involving the item set of the association rule may not occur as frequently due to the lowered minimum support. Thus, the association rule be spurious, or the item set may apply to too few transactions to be a valuable business opportunity.
(e)
What will be the effect on the rules generated if Leggere decreases the minimum confidence and reruns the association rules analysis?
Decreasing the minimum confidence will generate a pool of rules and therefore likely result in association rules with larger lift ratios. A rule with a lift ratio suggests that it is effective at identifying transactions in which the consequent occurs. However, if this rule has a confidence then this means that there also is a low probability that the consequent will occur given the antecedent. That is, as the confidence of an association rule , the relationship between the antecedent and consequent will fail more often. Depending on the profit margin associated with the transaction, this may make the association rule not a viable business opportunity.
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Forty-three percent of primary care doctors think their patients receive unnecessary medical care. If required, round your answer to four decimal places.
(a)
Suppose a sample of 300 primary care doctors was taken. Calculate the mean and standad deviation of the sample proportion of doctors who think their patients receive unnecessary medical care.
np =
n(1-p) =
E(p) =
=
(b)
Suppose a sample of 500 primary care doctors was taken. Calculate the mean and standard deviation of the sample proportion of doctors who think their patients receive unnecessary medical care.
np =
n(1-p) =
E(p) =
=
(c)
Suppose a sample of 1,000 primary care doctors was taken. Calculate the mean and standard deviation of the sample proportion of doctors who think their patients receive unnecessary medical care.
np =
n(1-p) =
E(p) =
=
(d)
In which of the preceding three cases, part (a) or part (b) or part (c), is the standard error of psmallest? Why?
The standard error is the smallest in because pis in parts (a), (b), and (c) and the sample size is largest in .
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Forty-three percent of primary care doctors think their patients receive unnecessary medical care. If required, round your answer to four decimal places.
(a)
Suppose a sample of 300 primary care doctors was taken. Calculate the mean and standad deviation of the sample proportion of doctors who think their patients receive unnecessary medical care.
np =
n(1-p) =
E(p) =
=
(b)
Suppose a sample of 500 primary care doctors was taken. Calculate the mean and standard deviation of the sample proportion of doctors who think their patients receive unnecessary medical care.
np =
n(1-p) =
E(p) =
=
(c)
Suppose a sample of 1,000 primary care doctors was taken. Calculate the mean and standard deviation of the sample proportion of doctors who think their patients receive unnecessary medical care.
np =
n(1-p) =
E(p) =
=
(d)
In which of the preceding three cases, part (a) or part (b) or part (c), is the standard error of psmallest? Why?
The standard error is the smallest in because pis in parts (a), (b), and (c) and the sample size is largest in .
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Thirty-five percent of primary care doctors think their patients receive unnecessary medical care. If required, round your answer to four decimal places.
(a)
Suppose a sample of 300 primary care doctors was taken. Calculate the mean and standad deviation of the sample proportion of doctors who think their patients receive unnecessary medical care.
np =
n(1-p) =
E(p) =
=
(b)
Suppose a sample of 500 primary care doctors was taken. Calculate the mean and standard deviation of the sample proportion of doctors who think their patients receive unnecessary medical care.
np =
n(1-p) =
E(p) =
=
(c)
Suppose a sample of 1,000 primary care doctors was taken. Calculate the mean and standard deviation of the sample proportion of doctors who think their patients receive unnecessary medical care.
np =
n(1-p) =
E(p) =
=
(d)
In which of the preceding three cases, part (a) or part (b) or part (c), is the standard error of psmallest? Why?
The standard error is the smallest in because pis in parts (a), (b), and (c) and the sample size is largest in .
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Twenty-five percent of primary care doctors think their patients receive unnecessary medical care. If required, round your answer to four decimal places.
(a)
Suppose a sample of 300 primary care doctors was taken. Calculate the mean and standad deviation of the sample proportion of doctors who think their patients receive unnecessary medical care.
np =
n(1-p) =
E(p) =
=
(b)
Suppose a sample of 500 primary care doctors was taken. Calculate the mean and standard deviation of the sample proportion of doctors who think their patients receive unnecessary medical care.
np =
n(1-p) =
E(p) =
=
(c)
Suppose a sample of 1,000 primary care doctors was taken. Calculate the mean and standard deviation of the sample proportion of doctors who think their patients receive unnecessary medical care.
np =
n(1-p) =
E(p) =
=
(d)
In which of the preceding three cases, part (a) or part (b) or part (c), is the standard error of psmallest? Why?
The standard error is the smallest in because pis in parts (a), (b), and (c) and the sample size is largest in .
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One of the questions on a survey of 1,000 adults asked if today's children will be better off than their parents. Representative data are shown in the file named ChildOutlook. A response of Yes indicates that the adult surveyed did think today's children will be better off than their parents. A response of No indicates that the adult surveyed did not think today's children will be better off than their parents. A response of Not Sure was given by 23% of the adults surveyed.
Click on the datafile logo to reference the data.
(a)
What is the point estimate of the proportion of the population of adults who do think that today's children will be better off than their parents? If required, round your answer to two decimal places.
(b)
At 95% confidence, what is the margin of error? If required, round your answer to four decimal places.
(c)
What is the 95% confidence interval for the proportion of adults who do think that today's children will be better off than their parents? If required, round your answers to four decimal places. Do not round your intermediate calculations.
to
(d)
What is the 95% confidence interval for the proportion of adults who do not think that today's children will be better off than their parents? If required, round your answers to four decimal places. Do not round your intermediate calculations.
to
(e)
Which of the confidence intervals in parts (c) and (d) has the smaller margin of error? Why?
The confidence interval in part has the smaller margin of error. This is because p is 0.5.
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For the year 2010, 33% of taxpayers with adjusted gross incomes between $30,000 and $60,000 itemized deductions on their federal income tax return. The mean amount of deductions for this population of taxpayers was $16,642. Assume that the standard deviation is σ = $2,320. If required, round your answer to two decimal places.
(a)
What are the sampling errors of xfor itemized deductions for this population of taxpayers for each of the following sample sizes: 30, 50, 100, and 400?
E(x) = $
n
σx
30
$
50
$
100
$
400
$
(b)
What is the advantage of a larger sample size when attempting to estimate the population mean?
A larger sample the standard error and results in a(n) precise estimate of the population mean.
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Last year, 46% of business owners gave a holiday gift to their employees. A survey of business owners indicated that 25% plan to provide a holiday gift to their employees. Suppose the survey results are based on a sample of 60 business owners.
(a)
How many business owners in the survey plan to provide a holiday gift to their employees?
(b)
Suppose the business owners in the sample do as they plan. Compute the p value for a hypothesis test that can be used to determine if the proportion of business owners providing holiday gifts has decreased from last year. If required, round your answer to four decimal places. If your answer is zero, enter “0”. Do not round your intermediate calculations.
(c)
Using a 0.05 level of significance, would you conclude that the proportion of business owners providing gifts has decreased?
We the null hypothesis. We conclude that the proportion of business owners providing gifts has decreased.
What is the smallest level of significance for which you could draw such a conclusion? If required, round your answer to four decimal places. If your answer is zero, enter “0”. Do not round your intermediate calculations.
The smallest level of significance for which we could draw this conclusion is ; because p-value is the corresponding α, we the null hypothesis.
he Coca-Cola Company reported that the mean per capita annual sales of its beverages in the United States was 423 eight-ounce servings. Suppose you are curious whether the consumption of Coca-Cola beverages is higher in Atlanta, Georgia, the location of Coca-Cola's corporate headquarters. A sample of 36 individuals from the Atlanta area showed a sample mean annual consumption of 460.4 eight-ounce servings with a standard deviation of s = 101.9 ounces.
Using α = 0.05, do the sample results support the conclusion that mean annual consumption of Coca-Cola beverage products is higher in Atlanta?
What is your conclusion?
the null hypothesis. We conclude that Atlanta customers have a higher annual rate of consumption of Coca Cola beverages.
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Last year, 46% of business owners gave a holiday gift to their employees. A survey of business owners indicated that 45% plan to provide a holiday gift to their employees. Suppose the survey results are based on a sample of 60 business owners.
(a)
How many business owners in the survey plan to provide a holiday gift to their employees?
(b)
Suppose the business owners in the sample do as they plan. Compute the p value for a hypothesis test that can be used to determine if the proportion of business owners providing holiday gifts has decreased from last year. If required, round your answer to four decimal places. If your answer is zero, enter “0”. Do not round your intermediate calculations.
(c)
Using a 0.05 level of significance, would you conclude that the proportion of business owners providing gifts has decreased?
We the null hypothesis. We conclude that the proportion of business owners providing gifts has decreased.
What is the smallest level of significance for which you could draw such a conclusion? If required, round your answer to four decimal places. If your answer is zero, enter “0”. Do not round your intermediate calculations.
The smallest level of significance for which we could draw this conclusion is ; because p-value is the corresponding α, we the null hypothesis.
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If P(A ∩ B) = 0, _____.
Events A and B are mutually exclusive with P(A) = 0.40 and P(B) = 0.10. The probability of the complement of event B equals _____.
A sample point refers to a(n) _____.
Of five letters (A, B, C, D, and E), two letters are to be selected at random. How many possible selections are there?
In an experiment, events A and B are mutually exclusive. If P(A) = .6, then the probability of B _____.
There is a 30% chance of getting stuck in traffic when leaving the city. On two separate days, what is the probability that you get stuck in traffic both days?
If P(A) = 0.32, P(B) = 0.82, and P(A ∩ B) = 0.27; then P(A ∪ B) = _____.
If A and B are independent events with P(A) = .05 and P(B) = .65, then P(A | B) = _____.
An experiment consists of three steps. There are five possible results on the first step, four possible results on the second step, and two possible results on the third step. The total number of experimental outcomes is _____.
An experiment consists of three steps. There are three possible results on the first step, five possible results on the second step, and two possible results on the third step. The total number of experimental outcomes is _____.
A graphical device used for enumerating sample points in a multiple-step experiment is a _____.
Two events, A and B, are mutually exclusive and each has a nonzero probability. If event A is known to occur, the probability of the occurrence of event B is _____.
If A and B are mutually exclusive events with P(A) = .3 and P(B) = 0.5, then P(A ∩ B) =
If P(A) = .50, P(B) = .60, and P(A ∩ B) = .30, then events A and B are _____.
If A and B are independent events with P(A) = 0.1 and P(B) = .4, then _____.
If two events are independent, then _____.
The probability of the occurrence of event A in an experiment is 1/3. If the experiment is performed two times and event A did not occur, then on the third trial event A _____.
A couple has three children. Assuming each child has an equal chance of being a boy or a girl, what is the probability that they have at least one girl?
Bayes' theorem is used to compute _____.
he number of electrical outages in a city varies from day to day. Assume that the number of electrical outages ( x ) in the city has the following probability distribution.
x
f (x)
0
0.80
1
0.15
2
0.04
3
0.01
The mean and the standard deviation for the number of electrical outages (respectively) are _____.
Exhibit 5-4
A local bottling company has determined the number of machine breakdowns per month and their respective probabilities as shown below.
Number of
Breakdowns
Probability
0
.12
1
.38
2
.25
3
.18
4
.07
The expected number of machine breakdowns per month is _____.
A numerical description of the outcome of an experiment is called a _____.
In a binomial experiment consisting of five trials, the number of different values that x (the number of successes) can assume is _____.
Exhibit 5-5
AMR is a computer-consulting firm. The number of new clients that it has obtained each month has ranged from 0 to 6. The number of new clients has the probability distribution that is shown below.
Number of
New Clients
Probability
0
.05
1
.10
2
.15
3
.35
4
.20
5
.10
6
.05
The expected number of new clients per month is _____.
A large university consists of 45% of students who are 20 years of age or older. A random sample of 9 students is selected.
What is the probability that among the students in the sample at least 7 are younger than 20?
AMR is a computer-consulting firm. The number of new clients that it has obtained each month has ranged from 0 to 6. The number of new clients has the probability distribution that is shown below.
Number of
New Clients
Probability
0
0.05
1
0.05
2
0.10
3
0.30
4
0.15
5
0.25
6
0.10
The expected number of new clients per month is _____.
Excel's HYPGEOM.DIST function has how many inputs?
To compute the probability that in a random sample of n elements, selected without replacement, we will obtain x successes, we would use the _____.
Exhibit 5-11 The random variable x is the number of occurrences of an event over an interval of 10 minutes. It can be assumed the probability of an occurrence is the same in any two time periods of an equal length. It is known that the mean number of occurrences in 10 minutes is 5.3.
The probability there are less than 3 occurrences is _____.
Exhibit 5-4
A local bottling company has determined the number of machine breakdowns per month and their respective probabilities as shown below.
Number of
Breakdowns
Probability
0
.12
1
.38
2
.25
3
.18
4
.07
The probability of at least 3 breakdowns in a month is _____.
A description of how the probabilities are distributed over the values the random variable can assume is called a(n) _____.
In a binomial experiment, the probability of success is .06. What is the probability of two successes in seven trials?
The binomial probability distribution is used with _____.
Exhibit 5-8 The student body of a large university consists of 60% female students. A random sample of 8 students is selected.
What is the probability that among the students in the sample exactly two are female?
A large university consists of 45% of students who are 20 years of age or older. A random sample of 7 students is selected.
What is the probability that among the students in the sample at least 5 are younger than 20?
Exhibit 5-10 The probability Pete will catch fish when he goes fishing is .8. Pete is going fishing 3 days next week.
The probability that Pete will catch fish on 1 or fewer days is _____.
Suppose x is a normally distributed random variable with a mean of 8 and a standard deviation of 4. The probability that x is between 1.48 and 15.56 is _____.
The weight of items produced by a machine is normally distributed with a mean of 7 ounces and a standard deviation of 3 ounces. What percentage of items will weigh at least 12.55 ounces?
A property of the exponential distribution is that the mean equals the _____.
Exhibit 6-7 f(x) = (1/10) e-x/10x ≥ 0
Refer to Exhibit 6-7. The probability that x is between 3 and 6 is _____.
The life expectancy of a particular brand of tire is normally distributed with a mean of 60,000 and a standard deviation of 4,000 miles. What percentage of tires will have a life of 53,000 to 67,000 miles?
What type of function defines the probability distribution of ANY continuous random variable?
Assume z is a standard normal random variable. Then P(1.05 ≤ z ≤ 2.13) equals _____.
The weight of football players is normally distributed with a mean of 220 pounds and a standard deviation of 20 pounds. What percent of players weigh between 215 and 225 pounds?
Excel's EXPON.DIST function has how many inputs?
The assembly time for a product is uniformly distributed between 6 and 11 minutes. The probability of assembling the product in 10 minutes or more is _____.
Corporate
triple A bond interest rates for 12 consecutive months are as follows:
9.6
9.4
9.5
9.7
9.9
9.5
9.8
10.6
10.1
9.9
9.5
9.8
(a)
Choose
the correct time series plot.
(i)
(ii)
(iii)
(iv)
What
type of pattern exists in the data?
(b)
Develop
three-month and four-month moving averages for this time series.
If
required, round your answers to two decimal places.
Month
Sales
3 Month
Moving Average
4 Month
Moving Average
1
9.6
2
9.4
3
9.5
4
9.7
5
9.9
6
9.5
7
9.8
8
10.6
9
10.1
10
9.9
11
9.5
12
9.8
Enter
the Mean Square Errors for the three-month and the four-month moving
average forecasts. If required, round your answers to three decimal digits.
3-month
moving average
4-month
moving average
MSE
Does
the three-month or the four-month moving average provide the better
forecasts based on MSE? Explain.
(c)
What
is the moving average forecast for the next month? If required, round your
answer to two decimal places.
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Consider
the following time series data:
Month
1
2
3
4
5
6
7
Value
24
13
21
14
20
23
15
(a)
Choose the correct time series plot.
(i)
(ii)
(iii)
(iv)
What type of pattern exists in
the data?
(b)
Develop a three-month moving
average for this time series. Compute MSE and a forecast for month 8.
If required, round your answers
to two decimal places. Do not round intermediate calculation.
MSE:
The forecast for month 8:
(c)
Use α = 0.2 to
compute the exponential smoothing values for the time series. Compute MSE and
a forecast for month 8.
If required, round your answers
to two decimal places. Do not round intermediate calculation.
MSE:
The forecast for month 8:
(d)
Compare the three-month moving
average forecast with the exponential smoothing forecast using α =
0.2. Which appears to provide the better forecast based on MSE?
(e)
Use trial and error to find a
value of the exponential smoothing coefficient α that
results in the smallest MSE.
Do not round intermediate
calculations. Use a two-decimal digit precision for the exponential smoothing
coefficient.
α =
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Consider
the following time series data.
Week
1
2
3
4
5
6
Value
18
13
16
11
17
14
Using
the naïve method (most recent value) as the forecast for the next week,
compute the following measures of forecast accuracy.
(a)
Mean absolute error
If required, round your answer to one decimal place.
(b)
Mean squared error
If required, round your answer to one decimal place.
(c)
Mean absolute percentage error
If required, round your intermediate calculations and final answer
to two decimal places.
(d)
What is the forecast for week 7?
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Consider
the following time series data.
Quarter
Year 1
Year 2
Year 3
1
3
6
8
2
2
4
8
3
4
7
9
4
6
9
11
(a)
Choose the correct time series plot.
(i)
(ii)
(iii)
(iv)
What type of pattern exists in the data?
(b)
Use a multiple regression model with dummy variables as follows to
develop an equation to account for seasonal effects in the data: Qtr1 = 1
if Quarter 1, 0 otherwise; Qtr2 = 1 if Quarter 2, 0 otherwise; Qtr3 = 1 if
Quarter 3, 0 otherwise.
If required, round your answers to three decimal places. For
subtractive or negative numbers use a minus sign even if there is a + sign
before the blank (Example: -300). If the constant is "1" it must
be entered in the box. Do not round intermediate calculation.
ŷ = + Qtr1
+ Qtr2
+ Qtr3
(c)
Compute the quarterly forecasts for next year based on the model you
developed in part (b).
If required, round your answers to three decimal places. Do not
round intermediate calculation.
Year
Quarter
Ft
4
1
4
2
4
3
4
4
(d)
Use a multiple regression model to develop an equation to account
for trend and seasonal effects in the data. Use the dummy variables you
developed in part (b) to capture seasonal effects and create a
variable t such that t = 1 for Quarter 1
in Year 1, t = 2 for Quarter 2 in Year 1,… t =
12 for Quarter 4 in Year 3.
If required, round your answers to three decimal places. For
subtractive or negative numbers use a minus sign even if there is a + sign
before the blank (Example: -300).
ŷ = + Qtr1
+ Qtr2
+ Qtr3
+ t
(e)
Compute the quarterly forecasts for next year based on the model you
developed in part (d).
Do not round your interim computations and round your final answer
to three decimal places.
Year
Quarter
Period
Ft
4
1
13
4
2
14
4
3
15
4
4
16
(f)
Calculate the MSE for the regression models developed in parts (b)
and (d).
If required, round your intermediate calculations and final answer
to three decimal places.
Model developed in part (b)
Model developed in part (d)
MSE
Is the model you developed in part (b) or the model you developed in
part (d) more effective?
The model developed in is
more effective because it has the MSE.
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The
president of a small manufacturing firm is concerned about the continual
increase in manufacturing costs over the past several years. The table below
provides a time series of the cost per unit for the firm's leading product
over the past eight years.
Click
on the datafile logo to reference the data.
Year
Cost/Unit ($)
Year
Cost/Unit ($)
1
20.00
5
26.60
2
24.50
6
30.00
3
28.20
7
31.00
4
27.50
8
36.00
(a)
Choose the correct time series plot.
(i)
(ii)
(iii)
(iv)
What type of pattern exists in the data?
(b)
Use simple linear regression analysis to find the parameters for the
line that minimizes MSE for this time series.
If required, round your answers for the parameters to four decimal
places and your answer for the MSE to two decimal places. Do not round your
intermediate calculations.
y-intercept, b0 =
Slope, b1 =
MSE =
(c)
What is the average cost increase that the firm has been realizing
per year?
If required, round your answer to two decimal places.
$
(d)
Compute an estimate of the cost/unit for next year.
If required, round your answer to two decimal places. Do not round
your intermediate calculations.
$
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Problem 07-08 Algo (Assessing the Fit of the Simple Linear Regression
Model)
Click on the datafile logo to
reference the data.
The
Toyota Camry is one of the best-selling cars in North America. The cost of a
previously owned Camry depends on many factors, including the model year,
mileage, and condition. To investigate the relationship between the car’s
mileage and the sales price for Camrys, the following data show the mileage
and sale price for 19 sales (PriceHub web site, February 24, 2012).
Miles
(1,000s)
Price
($1,000s)
22
16.2
29
16.0
36
13.8
47
11.5
63
12.5
77
12.9
73
11.2
87
13.0
92
11.8
101
10.8
110
8.3
28
12.5
59
11.1
68
15.0
68
12.2
91
13.0
42
15.6
65
12.7
110
8.3
(a)
Choose a scatter chart below
with ‘Miles (1000s)’ as the independent variable.
(i)
(ii)
(iii)
(iv)
What does the scatter chart
indicate about the relationship between price and miles?
The scatter chart indicates
there may be a linear
relationship between miles and price. Since a Camry with higher miles will
generally sell for a lower price, a relationship is
expected between these two variables. This scatter chart consistent with
what is expected.
(b)
Develop an estimated
regression equation showing how price is related to miles. What is the
estimated regression model?
Let x represent
the miles.
If required, round your
answers to four decimal places. For subtractive or negative numbers use a
minus sign even if there is a + sign before the blank. (Example: -300)
= + x
(c)
Test whether each of the regression
parameters β0 and β1 is
equal to zero at a 0.01 level of significance. What are the correct
interpretations of the estimated regression parameters? Are these
interpretations reasonable?
(i)
We can conclude that
both β0 and β1 are
equal to zero, where β0 is the estimated
change in price in thousands of dollars for a increase of 1,000 miles
and β1 is the estimated price in thousands of
dollars when the number of miles is zero. The interpretation of β0 is
not reasonable but the interpretation of β1 is
reasonable.
(ii)
We cannot conclude that
neither β0 nor β1 are
equal to zero, where β0 is the estimated
price in thousands of dollars when the number of miles is zero and β1 is
the estimated change in price in thousands of dollars for a increase of
1,000 miles. The interpretation of β0 is
reasonable but the interpretation of β1 is
not reasonable.
(iii)
We can conclude that
neither β0 nor β1 are
equal to zero, where β0 is the estimated
price in thousands of dollars when the number of miles is zero and β1 is
the estimated change in price in thousands of dollars for a increase of
1,000 miles. The interpretation of β0 is not
reasonable but the interpretation of β1 is
reasonable.
(iv)
We can conclude that β0 =
0 but β1 ≠ 0, where β0 is
the estimated change in price in thousands of dollars for a increase of
1,000 miles and β1 is the estimated price in
thousands of dollars when the number of miles is zero. Both
interpretations are reasonable.
(d)
How much of the variation in
the sample values of price does the model estimated in part (b) explain?
If required, round your
answer to two decimal places.
%
(e)
For the model estimated in
part (b), calculate the predicted price and residual for each automobile in
the data. Identify the two automobiles that were the biggest bargains.
If required, round your
answer to the nearest whole number.
The best
bargain is the Camry # in
the data set, which has miles,
and sells for $ less
than its predicted price.
The second best
bargain is the Camry # in
the data set, which has miles,
and sells for $ less
than its predicted price.
(f)
Suppose that you are
considering purchasing a previously owned Camry that has been driven 60,000
miles. Use the estimated regression equation developed in part (b) to
predict the price for this car.
If required, round your
answer to one decimal place. Do not round intermediate calculations. Enter
your answer in dollars. For example, 12 thousand should be entered as 12,000.
Predicted price: $
Is this the price you would
offer the seller? Explain.
(i)
Regardless of other factors
not considered in the model (various options, the physical condition of
the body and interior, etc.), this is not a reasonable price to expect to
pay for a Camry that has been driven 60,000 miles miles.
(ii)
Depending on other factors
not considered in the model (various options, the physical condition of
the body and interior, etc.), this is a reasonable price to expect to pay
for a Camry that has been driven 60,000 miles miles.
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Problem 07-10
(Inference and Regression)
Resorts
& Spas, a magazine devoted to upscale
vacations and accommodations, published its Reader's Choice List of the top
20 independent beachfront boutique hotels in the world. The data shown are
the scores received by these hotels based on the results from Resorts
& Spas' annual Readers' Choice Survey. Each score represents the
percentage of respondents who rated a hotel as excellent or very good on one
of three criteria (comfort, amenities, and in-house dining). An overall score
was also reported and used to rank the hotels. The highest ranked hotel, the
Muri Beach Odyssey, has an overall score of 94.3, the highest component of
which is 97.7 for in-house dining. If required, round your answers to three
decimal places.
Click
on the datafile logo to reference the data.
Hotel
Overall
Comfort
Amenities
In-House Dining
Muri
Beach Odyssey
94.3
94.5
90.8
97.7
Pattaya
Resort
92.9
96.6
84.1
96.6
Sojourner's
Respite
92.8
99.9
100.0
88.4
Spa
Carribe
91.2
88.5
94.7
97.0
Penang
Resort and Spa
90.4
95.0
87.8
91.1
Mokihana
Hōkele
90.2
92.4
82.0
98.7
Theo's
of Cape Town
90.1
95.9
86.2
91.9
Cap
d'Agde Resort
89.8
92.5
92.5
88.8
Spirit
of Mykonos
89.3
94.6
85.8
90.7
Turismo
del Mar
89.1
90.5
83.2
90.4
Hotel
Iguana
89.1
90.8
81.9
88.5
Sidi
Abdel Rahman Palace
89.0
93.0
93.0
89.6
Sainte-Maxime
Quarters
88.6
92.5
78.2
91.2
Rotorua
Inn
87.1
93.0
91.6
73.5
Club
Lapu-Lapu
87.1
90.9
74.9
89.6
Terracina
Retreat
86.5
94.3
78.0
91.5
Hacienda
Punta Barco
86.1
95.4
77.3
90.8
Rendezvous
Kolocep
86.0
94.8
76.4
91.4
Cabo
de Gata Vista
86.0
92.0
72.2
89.2
Sanya
Deluxe
85.1
93.4
77.3
91.8
(a)
Determine
the estimated multiple linear regression equation that can be used to
predict the overall score given the scores for comfort, amenities, and
in-house dining.
Let x1 represent
Comfort.
Let x2 represent
Amenities.
Let x3 represent
In-House Dining.
= + x1 + x2 + x3
(b)
Use
the t test to determine the significance of each
independent variable. What is the conclusion for each test at the 0.01
level of significance? If your answer is zero, enter "0".
The p-value
associated with the estimated regression parameter b1 is .
Because this p-value is than
the level of significance, we the
hypothesis that β1 = 0. We conclude that
there a
relationship between the score on comfort and the overall score at the 0.01
level of significance when controlling for .
The p-value
associated with the estimated regression parameter b2 is .
Because this p-value is than
the level of significance, we the
hypothesis that β2 = 0. We conclude that there a
relationship between the score on amenities and the overall score at the
0.01 level of significance when controlling for .
The p-value
associated with the estimated regression parameter b3 is .
Because this p-value is than
the level of significance, we the
hypothesis that β3 = 0. We conclude that
there a
relationship between the score on in-house dining and the overall score at
the 0.01 level of significance when controlling for .
(c)
Remove
all independent variables that are not significant at the 0.01 level of
significance from the estimated regression equation. What is your
recommended estimated regression equation? Enter a coefficient of zero for
any independent variable you chose to remove.
= + x1 + x2 + x3
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Problem
07-18 (Modeling Nonlinear Relationships)
In
2011, home prices and mortgage rates fell so far that in a number of cities
the monthly cost of owning a home was less expensive than renting. The
following data show the average asking rent for 10 markets and the monthly
mortgage on the median priced home (including taxes and insurance) for 10
cities where the average monthly mortgage payment was less than the average
asking rent (The Wall Street Journal, November 26–27, 2011).
Click
on the datafile logo to reference the data.
City
Rent
($)
Mortgage
($)
Atlanta
840
539
Chicago
1,062
1,002
Detroit
823
626
Jacksonville
779
711
Las Vegas
796
655
Miami
1,071
977
Minneapolis
953
776
Orlando
851
695
Phoenix
762
651
St. Louis
723
654
(a)
Develop a scatter chart for
these data, treating the average asking rent as the independent variable.
Choose the correct scatter chart below.
(i)
(ii)
(iii)
(iv)
Does a simple linear
regression model appear to be appropriate?
The scatter chart suggests
that rent is related to
mortgage. It is that the
relationship is linear, and so a simple linear regression model be appropriate.
(b)
Use a simple linear
regression model to develop an estimated regression equation to predict the
monthly mortgage on the median priced home given the average asking rent.
If required, round your answers to three decimal places. For subtractive or
negative numbers use a minus sign even if there is a + sign before the
blank. (Example: -300)
ŷ = + x
Construct a plot of the
residuals against the independent variable rent. Based on this residual
plot, does a simple linear regression model appear to be appropriate?
(c)
Using a quadratic regression
model, develop an estimated regression equation to predict the monthly
mortgage on the median-priced home, given the average asking rent. If
required, round your answers to three decimal places.
Let x represent
Rent ($).
Let x2 represent
Rent Squared.
ŷ = + x + x2
(d)
Do you prefer the estimated
regression equation developed in part (a) or part (c)?
Create a plot of the linear
and quadratic regression lines overlaid on the scatter chart of the monthly
mortgage on the median-priced home and the average asking rent to help you
assess the two regression equations. Choose the correct scatter chart
below.
(i)
(ii)
(iii)
(iv)
Explain your conclusions.
The graph shows that there
is difference
between the fit of the linear and quadratic regression models to the data.
The coefficient of determination for the linear regression model is than that of the
quadratic regression model, implying the model is
superior.
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Problem
07-06 Algo (Assessing the Fit of the Simple Linear Regression Model)
A
marketing professor at Givens College is interested in the relationship
between hours spent studying and total points earned in a course. Data
collected on 156 students who took the course last semester are provided in
the file MktHrsPts.
Click
on the datafile logo to reference the data.
(a)
Choose
a scatter chart below with Hours Spent Studying as the independent
variable.
(i)
(ii)
(iii)
(iv)
What
does the scatter chart indicate about the relationship between hours spent
studying and total points earned?
The
scatter chart indicates there may be a linear
relationship between hours spent studying and total points earned. Students
who spend more time studying generally earn more points, and this scatter
chart is consistent with what is expected.
(b)
Develop
an estimated regression equation showing how total points earned is related
to hours spent studying. What is the estimated regression model?
Let x represent
the hours spent studying.
If
required, round your answers to three decimal places. For subtractive or
negative numbers use a minus sign even if there is a + sign before the
blank. (Example: -300)
= + x
(c)
Test
whether each of the regression parameters β0 and β1 is
equal to zero at a 0.01 level of significance. What are the correct
interpretations of the estimated regression parameters? Are these
interpretations reasonable?
(i)
We
can conclude that β0 = 0 but β1 ≠
0, where β0 is the estimated change in total
points earned for a one hour increase in time spent studying and β1 is
the estimated total points earned when the hours spent studying is zero.
Both interpretations are reasonable.
(ii)
We
can conclude that both β0 and β1 are
equal to zero, where β0 is the estimated
change in total points earned for a one hour increase in time spent
studying and β1 is the estimated total points
earned when the hours spent studying is zero. The interpretation of β0 is
not reasonable but the interpretation of β1 is
reasonable.
(iii)
We
cannot conclude that neither β0 nor β1 are
equal to zero, where β0 is the estimated
total points earned when the hours spent studying is zero and β1 is
the estimated change in total points earned for a one hour increase in
time spent studying. The interpretation of β0 is
reasonable but the interpretation of β1 is
not reasonable.
(iv)
We
can conclude that neither β0 nor β1 are
equal to zero, where β0 is the estimated
total points earned when the hours spent studying is zero and β1 is
the estimated change in total points earned for a one hour increase in
time spent studying. The interpretation of β0 is
not reasonable but the interpretation of β1 is
reasonable.
(d)
How
much of the variation in the sample values of total point earned does the
model you estimated in part (b) explain?
If
required, round your answer to two decimal places.
%
(e)
Mark
Sweeney spent 95 hours studying. Use the regression model you estimated in
part (b) to predict the total points Mark earned.
If
required, round your answer to the nearest whole number. Do not round
intermediate calculations.
$
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Problem 07-15 (Categorical
Independent Variables)
The
U.S. Department of Energy's Fuel Economy Guide provides fuel efficiency data
for cars and trucks (U.S. Department of Energy website, February 22, 2008). A
portion of the data for 311 compact, midsize, and large cars follows. The
column labeled Class identifies the size of the car: Compact, Midsize, or
Large. The column labeled Displacement shows the engine's displacement in
liters. The column labeled Fuel Type shows whether the car uses premium (P)
or regular (R) fuel, and the column labeled HwyMPG shows the fuel efficiency
rating for highway driving in terms of miles per gallon. The complete data
set is contained in the DATAfile named FuelData.
Click
on the datafile logo to reference the data.
(a)
Develop
an estimated regression equation that can be used to predict the fuel
efficiency for highway driving, given the engine's displacement.
Let x represent
the engine's displacement.
If
required, round your answers to three decimal places. For subtractive or
negative numbers use a minus sign even if there is a + sign before the
blank. (Example: -300)
= + x
Is
there a relationship between HwyMPG and displacement at the 0.05 level of
significance?
How
much of the variation in the sample values of HwyMPG is is explained by
this estimated regression equation?
If
required, round your answer to one decimal places.
%
(b)
Create
a scatter chart in Excel with HwyMPG on the y-axis and
displacement on the x-axis for which the points representing
compact, midsize, and large automobiles are shown in different shapes and
or colors. Choose the correct chart below.
(i)
(ii)
(iii)
(iv)
What
does this chart suggest about the relationship between the class of
automobile (compact, midsize, and large) and HwyMPG?
This
chart suggests that for each class of automobile (compact, midsize, and
large) HwyMPG as
displacement increases. The chart also suggests that automobiles
generally have the highest HwyMPG while automobiles
generally have the lowest HwyMPG.
(c)
Consider
the addition of the dummy variables ClassMidsize and ClassLarge to the
simple linear regression model in part (a). The value of ClassMidsize is 1
if the car is a midsize car and 0 otherwise; the value of ClassLarge is 1
if the car is a large car and 0 otherwise. Thus, for a compact car, the
value of ClassMidsize and the value of ClassLarge are both 0. Develop the
estimated regression equation that can be used to predict the fuel
efficiency for highway driving, given the engine's displacement and the
dummy variables ClassMidsize and ClassLarge.
Let x1 represent
engine's displacement.
Let x2 represent
variable ClassMidsize.
Let x3 represent
variable ClassLarge.
If
required, round your answers to three decimal places. For subtractive or
negative numbers use a minus sign even if there is a + sign before the
blank. (Example: -300)
= + x1 + x2 + x3
How
much of the variation in the sample values of HwyMPG does this estimated
regression equation explain?
If
required, round your answer to one decimal places.
%
(d)
Use
significance level of 0.05 to determine whether the dummy variables added
to the model in part (c) are significant. Explain.
We conclude
that β2 ≠ 0 which implies there a
difference in HwyMPG between midsized automobiles and compact automobiles.
We conclude
that β3 ≠ 0 which implies there a
difference in HwyMPG between large automobiles and compact automobiles.
(e)
Consider
the addition of the dummy variable FuelPremium, where the value of
FuelPremium is 1 if the car uses premium fuel and 0 if the car uses regular
fuel. Develop the estimated regression equation that can be used to predict
the fuel efficiency for highway driving, given the engine's displacement,
the dummy variables ClassMidsize and ClassLarge, and the dummy variable
FuelPremium.
Let x1 represents
engine's displacement.
Let x2 represents
variable ClassMidsize.
Let x3 represents
variable ClassLarge.
Let x4 represents
variable FuelPremium.
If
required, round your answers to three decimal places. For subtractive or
negative numbers use a minus sign even if there is a + sign before the
blank. (Example: -300)
= + x1 + x2 + x3 + x4
How
much of the variation in the sample values of HwyMPG does this estimated
regression equation explain?
If
required, round your answer to one decimal places.
%
(f)
For
the estimated regression equation developed in part (e), test for the significance
of the relationship between each of the independent variables and the
dependent variable using the 0.05 level of significance for each test.
Explain
(i)
We
can reject the hypothesis β1 = 0, β2 =
0, β3 = 0, and β4 =
0. There is a relationship between HwyMPG and displacement. There is a
difference in HwyMPG between midsized automobiles and compact
automobiles. There is a difference in HwyMPG between large automobiles
and compact automobiles. There is a difference in HwyMPG between premium
fuel and regular fuel.
(ii)
We
cannot reject the hypotheses β1 = 0, β2 =
0, β3 = 0, and β4 =
0. There is not a relationship between HwyMPG and displacement. There is
not a difference in HwyMPG between midsized automobiles and compact
automobiles. There is not a difference in HwyMPG between large
automobiles and compact automobiles. There is not a difference in HwyMPG
between premium fuel and regular fuel.
(iii)
We
can reject the hypotheses β1 = 0 and β2 =
0, but neither the hypotheses β3 = 0
nor β4 = 0. There is a relationship between
HwyMPG and displacement. There is a difference in HwyMPG between midsized
automobiles and compact automobiles. There is not a difference in HwyMPG
between large automobiles and compact automobiles. There is not a
difference in HwyMPG between premium fuel and regular fuel.
(iv)
We
can only reject the hypothesis β1 = 0, but
fail to reject the hypotheses β2 = 0, β3 =
0, β4 = 0. There is a relationship between
HwyMPG and displacement. There is not a difference in HwyMPG between
midsized automobiles and compact automobiles. There is not a difference
in HwyMPG between large automobiles and compact automobiles. There is not
a difference in HwyMPG between premium fuel and regular fuel.
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Casey
Deesel is a sports agent negotiating a contract for Titus Johnston, an
athlete in the National Football League (NFL). An important aspect of any NFL
contract is the amount of guaranteed money over the life of the contract.
Casey has gathered data on 506 NFL athletes who have recently signed new
contracts. Each observation (NFL athlete) includes values for percentage of
his team's plays that the athlete is on the field (SnapPercent), the number
of awards an athlete has received recognizing on-field performance (Awards),
the number of games the athlete has missed due to injury (GamesMissed), and
millions of dollars of guaranteed money in the athlete's most recent contract
(Money, dependent variable).
Casey
has trained a full regression tree on 304 observations and then used the
validation set to prune the tree to obtain a best-pruned tree. The
best-pruned tree (as applied to the 202 observations in the validation set)
is:
(a)
Titus Johnston's variable values are: SnapPercent = 96, Awards = 7,
and GamesMissed = 3. How much guaranteed money does the regression tree
predict that a player with Titus Johnson's profile should earn in his
contract?
If required, round your answers to two decimal places.
The predicted result is $ million
of guaranteed money.
(b)
Casey feels that Titus was denied an additional award in the past
season due to some questionable voting by some sports media. If Titus had
won this additional award, how much additional guaranteed money would the
regression tree predict for Titus versus the prediction in part (a)?
I.An additional award would increase
the amount of guaranteed money by $8.91 million.
II.An additional award would increase
the amount of guaranteed money by $13.99 million.
III.An additional award would increase
the amount of guaranteed money by $17.79 million.
IV.An additional award would increase
the amount of guaranteed money by $26.02 million.
V.An additional award would not
change the amount of guaranteed money.
(c)
As Casey reviews the best-pruned tree, he is confused by the leaf
node corresponding to the sequence of decision rules of "SnapPercent ≥
90.28, SnapPercent < 95.37, Awards < 6.75, GamesMissed <
1.5." This sequence of decision rules results in an estimate of $50
million of guaranteed money, but the tree states that zero observations
occur in the corresponding partition. If zero observations occur in this
partition, how can the regression tree provide an estimate of $50 million?
Explain this part of the regression tree to Casey by referring to how the
best-pruned tree is obtained.
The predicted guaranteed money of $50 million for observations
satisfying "SnapPercent ≥ 90.28, SnapPercent < 95.37, Awards <
6.75, GamesMissed < 1.5" is based on the average guaranteed money
of the observations in the set
that satisfy this sequence of decision rules. The best-pruned tree is
obtained by the
initial regression tree to obtain the tree with the leaf
nodes while achieving the minimum classification error rate on the set.
In this case, the set
has zero observations that satisfy "SnapPercent ≥ 90.28, SnapPercent
< 95.37, Awards < 6.75, GamesMissed < 1.5" which just means
that this leaf node to
the classification error rate of this tree.
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A
university is applying classification methods in order to identify alumni who
may be interested in donating money. The university has a database of 58,205
alumni profiles containing numerous variables. Of these 58,205 alumni, only
576 have donated in the past. The university has oversampled the data and
trained a random forest of 100 classification trees. For a cutoff value of
0.5, the following confusion matrix summarizes the performance of the random
forest on a validation set:
Predicted
Actual
Donation
No Donation
Donation
268
20
No Donation
5,375
23,439
The
following table lists some information on individual observations from the
validation set:
Probability of
Predicted
Observation ID
Actual Class
Donation
Class
A
Donation
0.8
Donation
B
No Donation
0.1
No Donation
C
No Donation
0.6
Donation
(a)
Choose the correct explanation for how the probability of Donation
was computed for the three observations.
(i)
The probability of Donation for each observation is the
proportion of the 100 individual classification trees that classified
the observation as "Donation."
(ii)
The probability of Donation for each observation is the
proportion of the 100 individual classification trees that classified
the observation as "No Donation."
(iii)
The probability of Donation for each observation is the ratio of
the individual classification trees that classified the observation as
"Donation" and those that classified it as "No
Donation."
(iv)
The probability of Donation for each observation is the ratio of
the individual classification trees that classified the observation as
"No Donation" and those that classified it as
"Donation."
Why were Observations A and C classified as Donation and Observation
B was classified as No Donation?
If required, round your answers to one decimal place.
The probability of Donation for Observation A
is .
It is than
0.5, so Observation A is classified as Donation by the random forest.
The probability of Donation for Observation B
is .
It is than
0.5, so Observation B is classified as No Donation by the random forest.
The probability of Donation for Observation C
is .
It is than
0.5, so Observation C is classified as Donation by the random forest.
(b)
Compute the values of accuracy, sensitivity, specificity, and
precision. Explain why accuracy is a misleading measure to consider in this
case. Evaluate the performance of the random forest, particularly
commenting on the precision measure.
If required, round your answer to three decimal places.
Accuracy =
If required, round your answers to the nearest whole percentage.
Accuracy is not the best measure to use for unbalanced data sets
because less than %
of the alumni in the data have donated.
If required, round your answers for Sensitivity and Specificity to
three decimal places and round your answer for Precision to four decimal
places.
Sensitivity =
Specificity =
Precision =
The value of precision seems disturbingly .
The precision measure represents the percentage of alumni classified by the
random forest as that
are donors. Comparing the value of precision with the proportion of
observations corresponding to donations, there a
tremendous improvement in the ability to target alumni who may be more
likely to donate.
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The dating web site Oollama.com requires its users to create profiles based on a survey in which they rate their interest (on a scale from 0 to 3) in five categories: physical fitness, music, spirituality, education, and alcohol consumption. A new Oollama customer, Erin O'Shaughnessy, has reviewed the profiles of 40 prospective dates and classified whether she is interested in learning more about them.
Based on Erin's classification of these 40 profiles, Oollama has applied a logistic regression to predict Erin's interest in other profiles that she has not yet viewed. The resulting logistic regression model is as follows:
For the 40 profiles (observations) on which Erin classified her interest, this logistic regression model generates that following probability of Interested.
Probability of
Probability of
Observation
Interested
Interested
Observation
Interested
Interested
35
1
1.000
13
0
0.412
21
1
0.999
2
0
0.285
29
1
0.999
3
0
0.219
25
1
0.999
7
0
0.168
39
1
0.999
9
0
0.168
26
1
0.990
12
0
0.168
23
1
0.981
18
0
0.168
33
1
0.974
22
1
0.168
1
0
0.882
31
1
0.168
24
1
0.882
6
0
0.128
28
1
0.882
20
0
0.128
36
1
0.882
15
0
0.029
16
0
0.791
5
0
0.020
27
1
0.791
14
0
0.015
30
1
0.791
19
0
0.011
32
1
0.791
8
0
0.008
34
1
0.791
10
0
0.001
37
1
0.791
17
0
0.001
40
1
0.791
4
0
0.001
38
1
0.732
11
0
0.000
(a)
Using a cutoff value of 0.5 to classify a profile observation as Interested or not, construct the confusion matrix for this 40-observation training set.
Predicted
Actual
0
1
0
1
Compute sensitivity, specificity, and precision measures and interpret them within the context of Erin's dating prospects.
If required, round your answers to two decimal places. Do not round intermediate calculations.
The sensitivity of the model is . This suggests that the model is reasonably at identifying the profiles that Erin is interested in.
The specificity of the model is . This suggests that the model is reasonably at avoiding recommending profiles to Erin that she will not be interested in.
The precision of the model is . This suggests that the model is reasonably at suggesting profiles of interest to Erin.
(b)
Oollama understands that its clients have a limited amount of time for dating and therefore use decile-wise lift charts to evaluate their classification models. For the training data, what is the first decile lift resulting from the logistic regression model? Interpret this value.
The first decile lift of this classification is . It means that the first decile of the logistic regression model the number of profiles that Erin is interested in versus random selection.
(c)
A recently posted profile has values of Fitness = 3, Music = 1, Education = 3, and Alcohol = 1. Use the estimated logistic regression equation to compute the probability of Erin's interest in this profile.
If required, round your answers to three decimal places. Do not round intermediate calculations.
Log odds =
Probability of Interest =
(d)
Now that Oollama has trained a logistic regression model based on Erin's initial evaluations of 40 profiles, what should its next steps be in the modeling process?
Oollama should use their model to suggest profiles to Erin in order to compute classification accuracy measures on a validation set.
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(All answers were generated using 1,000 trials and native Excel functionality.)
In preparing for the upcoming holiday season, Fresh Toy Company (FTC) designed a new doll called The Dougie that teaches children how to dance. The fixed cost to produce the doll is $100,000. The variable cost, which includes material, labor, and shipping costs, is $34 per doll. During the holiday selling season, FTC will sell the dolls for $42 each. If FTC overproduces the dolls, the excess dolls will be sold in January through a distributor who has agreed to pay FTC $10 per doll. Demand for new toys during the holiday selling season is uncertain. The normal probability distribution with an average of 60,000 dolls and a standard deviation of 15,000 is assumed to be a good description of the demand. FTC has tentatively decided to produce 60,000 units (the same as average demand), but it wants to conduct an analysis regarding this production quantity before finalizing the decision.
(a)
Create a what-if spreadsheet model using formulas that relate the values of production quantity, demand, sales, revenue from sales, amount of surplus, revenue from sales of surplus, total cost, and net profit. What is the profit when demand is equal to its average (60,000 units)?
$
(b)
Modeling demand as a normal random variable with a mean of 60,000 and a standard deviation of 15,000, simulate the sales of The Dougie doll using a production quantity of 60,000 units. What is the estimate of the average profit associated with the production quantity of 60,000 dolls? Round your answer to the nearest dollar.
$
How does this compare to the profit corresponding to the average demand (as computed in part (a))?
The average profit from the simulation is the profit computed in part (a).
(c)
Before making a final decision on the production quantity, management wants an analysis of a more aggressive 70,000-unit production quantity and a more conservative 50,000-unit production quantity. Run your simulation with these two production quantities. What is the average profit associated with each? Round your answers to the nearest dollar.
When ordering 50,000 units, the average profit is approximately $. When ordering 70,000 units, the average profit is approximately $.
(d)
Besides average profit, what other factors should FTC consider in determining a production quantity? Compare the four production quantities (40,000; 50,000; 60,000; and 70,000) using all these factors.
If required, round Probability of a Loss to three decimal places and Probability of a Shortage to two decimal places. Round the other answers to the nearest dollar.
Production Quantity
Average Net Profit
Profit Standard Deviation
Maximum Net Profit
Probability of a Loss
Probability of a Shortage
40,000
$
$
$
50,000
$
$
$
60,000
$
$
$
70,000
$
$
$
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All answers were generated using 1,000 trials and native Excel functionality.)
At a local university, the Student Commission on Programming and Entertainment (SCOPE) is preparing to host its first music concert of the school year. To successfully produce this music concert, SCOPE has to complete several activities. The following table lists information regarding each activity. An activity’s immediate predecessors are the activities that must be completed before the considered activity can begin. The table also lists duration estimates (in days) for each activity.
Activity
Immediate Predecessors
Minimum Time
Likely Time
Maximum Time
A: Negotiate contract with selected musicians
—
5
6
9
B: Reserve site
—
8
12
15
C: Logistical arrangements for music group
A
5
6
7
D: Screen and hire security personnel
B
3
3
3
E: Advertising and ticketing
B, C
1
5
9
F: Hire parking staff
D
4
7
10
G: Arrange concession sales
E
3
8
10
The following network illustrates the precedence relationships in the SCOPE project. The project begins with activities A and B, which can start immediately (time 0) because they have no predecessors. On the other hand, activity E cannot be started until activities B and C are both completed. The project is not complete until all activities are completed.
(a)
Using the triangular distribution to represent the duration of each activity, construct a simulation model to estimate the average amount of time to complete the concert preparations.
Round your answers to one decimal place.
Project Duration
Average
days
Standard Deviation
days
(b)
What is the likelihood that the project will be complete in 23 days or less?
Round your answer to the nearest whole number.
%
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(All answers were generated using 1,000 trials and native Excel functionality.)
The management of Brinkley Corporation is interested in using simulation to estimate the profit per unit for a new product. The selling price for the product will be $45 per unit. Probability distributions for the purchase cost, the labor cost, and the transportation cost are estimated as follows:
Procurement Cost ($)
Probability
Labor Cost ($)
Probability
Transportation Cost ($)
Probability
10
0.25
20
0.10
3
0.75
11
0.45
22
0.25
5
0.25
12
0.30
24
0.35
25
0.30
(a)
Construct a simulation model to estimate the average profit per unit. What is a 95% confidence interval around this average?
Round your answers to two decimal places.
Lower Bound: $
Upper Bound: $
(b)
Management believes that the project may not be sustainable if the profit per unit is less than $5. Use simulation to estimate the probability that the profit per unit will be less than $5. What is a 95% confidence interval around this proportion?
Round your answers to one decimal of a percentage.
Lower Bound: %
Upper Bound: %
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(All answers were generated using 1,000 trials and native Excel functionality.)
The Iowa Wolves is scheduled to play against the Maine Red Claws in an upcoming game in the National Basketball Association (NBA) G League. Because a player in the NBA G League is still developing his skills, the number of points he scores in a game can vary substantially. Assume that each player's point production can be represented as an integer uniform random variable with the ranges provided in the following table:
Player
lowa Wolves
Maine Red Claws
1
[5,20]
[7,12]
2
[7,20]
[15,20]
3
[5,10]
[10,20]
4
[10,40]
[15,30]
5
[6,20]
[5,10]
6
[3,10]
[1,20]
7
[2,5]
[1,4]
8
[2,4]
[2,4]
(a)
Develop a spreadsheet model that simulates the points scored by each team and the difference in their point totals. What are the average and standard deviation of points scored by the Iowa Wolves?
Round your answers to one decimal place.
Average:
Standard Deviation:
What is the shape of the distribution of points scored by the Iowa Wolves?
(b)
What are the average and standard deviation of points scored by the Maine Red Claws?
Round your answers to one decimal place.
Average:
Standard Deviation:
What is the shape of the distribution of points scored by the Maine Red Claws?
(c)
Let Point Differential = Iowa Wolves points – Maine Red Claws points. What is the average Point Differential between the Iowa Wolves and Maine Red Claws? What is the standard deviation of the Point Differential? If your answer is negative, enter a minus sign in the input box.
Round your answers to one decimal place.
Average:
Standard Deviation:
What is the shape of the point differential distribution?
(d)
What is the probability that the Iowa Wolves scores more points than the Maine Red Claws?
Round your answer to the nearest whole number.
%
(e)
The coach of the Iowa Wolves feels that they are the underdog and is considering a riskier game strategy. The effect of this strategy is that the range of each Wolves player's point production increases symmetrically so that the new range is [0, original upper bound + original lower bound]. For example, Wolves player 1's range with the risky strategy is [0, 25]. How does the new strategy affect the average and standard deviation of the Wolves point total? How does that affect the probability of the Iowa Wolves scoring more points than the Maine Red Claws?
Round first two numerical answers to one decimal place and the last answer to a whole percentage.
The average Iowa Wolves point total will be points and the standard deviation of the Iowa Wolves point total will be points. The probability of the Iowa Wolves scoring more points than the Maine Red Claws will become %.
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(All answers were generated using 1,000 trials and native Excel functionality.)
Orange Tech (OT) is a software company that provides a suite of programs that are essential to everyday business computing. OT has just enhanced its software and released a new version of its programs. For financial planning purposes, OT needs to forecast its revenue over the next few years. To begin this analysis, OT is considering one of its largest customers. Over the planning horizon, assume that this customer will upgrade at most once to the newest software version, but the number of years that pass before the customer purchases an upgrade varies. Up to the year that the customer actually upgrades, assume there is a 0.50 probability that the customer upgrades in any particular year. In other words, the upgrade year of the customer is a random variable. For guidance on an appropriate way to model upgrade year, refer to Appendix 11.1. Furthermore, the revenue that OT earns from the customer's upgrade also varies (depending on the number of programs the customer decides to upgrade). Assume that the revenue from an upgrade obeys a normal distribution with a mean of $100,000 and a standard deviation of $25,000. Using the template in the file OrangeTech, complete a simulation model that analyzes the net present value of the revenue from the customer upgrade. Use an annual discount rate of 10%.
Click on the datafile logo to reference the data.
(a)
What is the average net present value that OT earns from this customer?
Round your answer to the nearest whole number. Do not round your intermediate calculation.
$
(b)
What is the standard deviation of net present value?
Round your answer to the nearest whole number. Do not round your intermediate calculation.
$
How does this compare to the standard deviation of the revenue? Explain.
The input in the box below will not be graded, but may be reviewed and considered by your instructor.
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Eastman Publishing Company is considering publishing an electronic textbook about spreadsheet applications for business. The fixed cost of manuscript preparation, textbook design, and web-site construction is estimated to be $150,000. Variable processing costs are estimated to be $7 per book. The publisher plans to sell single-user access to the book for $49.
(a)
Build a spreadsheet model in Excel to calculate the profit/loss for a given demand. What profit can be anticipated with a demand of 3,400 copies? For subtractive or negative numbers use a minus sign.
$
(b)
Use a data table to vary demand from 1,000 to 6,000 in increments of 200 to test the sensitivity of profit to demand. Breakeven occurs where profit goes from a negative to a positive value, that is, breakeven is where total revenue = total cost yielding a profit of zero. In which interval of demand does breakeven occur?
(i) Breakeven appears in the interval of 3,000 to 3,200 copies.
(ii) Breakeven appears in the interval of 3,400 to 3,600 copies.
(iii) Breakeven appears in the interval of 3,600 to 3,800 copies.
(iv) Breakeven appears in the interval of 3,800 to 4,000 copies.
(c)
Use Goal Seek to determine the access price per copy that the publisher must charge to break even with a demand of 3,400 copies. If required, round your answer to two decimal places.
$
(d)
Consider the following scenarios:
Scenario 1
Scenario 2
Scenario 3
Scenario 4
Scenario 5
Variable Cost/Book
$7
$10
$11
$8
$13
Access Price
$49
$40
$45
$50
$55
Demand
3,000
2,000
4,500
6,000
1,000
For each of these scenarios, the fixed cost remains $150,000. Use Scenario Manager to generate a summary report that gives the profit for each of these scenarios. Which scenario yields the highest profit? Which scenario yields the lowest profit? For subtractive or negative numbers use a minus sign.
yields the highest profit of $.
yields the lowest profit of $.
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Let S represent the amount of steel produced (in tons). Steel production is related to the amount of labor used (L) and the amount of capital used (C) by the following function:
S = 20 L0.30C0.70
In this formula L represents the units of labor input and C the units of capital input. Each unit of labor costs $50, and each unit of capital costs $100.
(a)
Formulate an optimization problem that will determine how much labor and capital are needed in order to produce 50,000 tons of steel at minimum cost. If your answer is zero, enter “0”.
Min
L
+
C
s.t.
L0.30C0.70
L, C
(b)
Solve the optimization problem you formulated in part a. Hint: When using Excel Solver, start with an initial L > 0 and C > 0.
Do not round intermediate calculations. Round your answers to the nearest whole number.
L =
C =
Cost = $
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A second version of the Markowitz portfolio model maximizes expected return subject to a constraint that the variance of the portfolio must be less than or equal to some specified amount. Consider the Hauck Financial Service data.
Click on the datafile logo to reference the data.
Annual Return (%)
Mutual Fund
Year 1
Year 2
Year 3
Year 4
Year 5
Foreign Stock
10.06
13.12
13.47
45.42
-21.93
Intermediate-Term Bond
17.64
3.25
7.51
-1.33
7.36
Large-Cap Growth
32.41
18.71
33.28
41.46
-23.26
Large-Cap Value
32.36
20.61
12.93
7.06
-5.37
Small-Cap Growth
33.44
19.40
3.85
58.68
-9.02
Small-Cap Value
24.56
25.32
-6.70
5.43
17.31
(a)
Construct this version of the Markowitz model for a maximum variance of 30.
Let:
FS = proportion of portfolio invested in the foreign stock mutual fund
IB = proportion of portfolio invested in the intermediate-term bond fund
LG = proportion of portfolio invested in the large-cap growth fund
LV = proportion of portfolio invested in the large-cap value fund
SG = proportion of portfolio invested in the small-cap growth fund
SV = proportion of portfolio invested in the small-cap value fund
= the expected return of the portfolio
Rs= the return of the portfolio in years
If required, round your answers to two decimal places. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank (Example: -300). If the constant is "1" it must be entered in the box. If your answer is zero enter “0”.
Max
s.t.
FS
+ IB
+ LG
+ LV
+ SG
+ SV
R1
FS
+ IB
+ LG
+ LV
+ SG
+ SV
R2
FS
+ IB
+ LG
+ LV
+ SG
+ SV
R3
FS
+ IB
+ LG
+ LV
+ SG
+ SV
R4
FS
+ IB
+ LG
+ LV
+ SG
+ SV
R5
FS
+ IB
+ LG
+ LV
+ SG
+ SV
FS, IB, LG, LV, SG, SV
(b)
Solve the model developed in part (a).
If required, round your answers to two decimal places. If your answer is zero, enter “0”.
FS
%
IB
%
LG
%
LV
%
SG
%
SV
%
Portfolio Expected Return = %
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Jim's Camera shop sells two high-end cameras, the Sky Eagle and Horizon. The demand for these two cameras are as follows (DS = demand for the Sky Eagle, Ps is the selling price of the Sky Eagle, DH is the demand for the Horizon and PH is the selling price of the Horizon):
Ds = 230 - 0.5 Ps + 0.38 PH
DH = 260 + 0.1 Ps - 0.62 PH
The store wishes to determine the selling price that maximizes revenue for these two products. Select the revenue function for these two models. Choose the correct answer below.
Do not round intermediate calculations. If required, round your answers to two decimal places.
Optimal Solution:
Selling price of the Sky Eagle (Ps): $
Selling price of the Horizon (PH): $
Total Revenue: $
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Amy Lloyd is interested in leasing a new Honda and has contacted three automobile dealers for pricing information. Each dealer offered Amy a closed-end 36-month lease with no down payment due at the time of signing. Each lease includes a monthly charge and a mileage allowance. Additional miles receive a surcharge on a per-mile basis. The monthly lease cost, the mileage allowance, and the cost for additional miles are as follows:
Dealer
Monthly Cost
Mileage Allowance
Cost per Additional Mile
Hepburn Honda
$299
36,000
$0.15
Midtown Motors
$310
45,000
$0.20
Hopkins Automotive
$325
54,000
$0.15
Amy decided to choose the lease option that will minimize her total 36-month cost. The difficulty is that Amy is not sure how many miles she will drive over the next three years. For purposes of this decision, she believes it is reasonable to assume that she will drive 12,000 miles per year, 15,000 miles per year, or 18,000 miles per year. With this assumption Amy estimated her total costs for the three lease options. For example, she figures that the Hepburn Honda lease will cost her 36($299) + $0.15*max(0,36,000 - 36,000) = $10,764 if she drives 12,000 miles per year, 36($299) + $0.15*max(0,45,000 - 36,000) = $12,114 if she drives 15,000 miles per year, or 36($299) + $0.15*max(0,54,000 - 36,000) = $13,464 if she drives 18,000 miles per year.
(a)
What is the decision, and what is the chance event? Choose the correct answer below.
(i)
The decision is to select the monthly cost and the chance event is the three alternatives (Hepburn Honda, Midtown Motors, and Hopkins Automotive).
(ii)
The decision is to select the number of miles Amy will drive and the chance event is the three alternatives (Hepburn Honda, Midtown Motors, and Hopkins Automotive).
(iii)
The decision is to select the best lease option from three alternatives (Hepburn Honda, Midtown Motors, and Hopkins Automotive) and the chance event is the number of miles Amy will drive.
(iv)
The decision is to select the best lease option from three alternatives (Hepburn Honda, Midtown Motors, and Hopkins Automotive) and the chance event is the monthly cost that Amy will incur.
(b)
Construct a payoff table for Amy's problem.
Actual Miles Driven Annually
Dealer
12,000
15,000
18,000
Hepburn Honda
$
$
$
Midtown Motors
$
$
$
Hopkins Automotive
$
$
$
(c)
If Amy has no idea which of the three mileage assumptions is most appropriate, what is the recommended decision (leasing option) using the optimistic, conservative, and minimax regret approaches?
Optimistic approach
Conservative approach
Minimax approach
(d)
Suppose that the probabilities that Amy drives 12,000, 15,000, and 18,000 miles per year are 0.5, 0.4, and 0.1, respectively. What option should Amy choose using the expected value approach?
(e)
Develop a risk profile for the decision selected in part (d). What is the most likely cost?
$
What is its probability?
If required, round your answer to one decimal place.
(f)
Suppose that after further consideration, Amy concludes that the probabilities that she will drive 12,000, 15,000, and 18,000 miles per year are 0.3, 0.4, and 0.3, respectively. What decision should Amy make using the expected value approach?
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Seneca Hill Winery recently purchased land for the purpose of establishing a new vineyard. Management is considering two varieties of white grapes for the new vineyard: Chardonnay and Riesling. The Chardonnay grapes would be used to produce a dry Chardonnay wine, and the Riesling grapes would be used to produce a semidry Riesling wine. It takes approximately four years from the time of planting before new grapes can be harvested. This length of time creates a great deal of uncertainty concerning future demand and makes the decision about the type of grapes to plant difficult. Three possibilities are being considered: Chardonnay grapes only; Riesling grapes only; and both Chardonnay and Riesling grapes. Seneca management decided that for planning purposes it would be adequate to consider only two demand possibilities for each type of wine: strong or weak. With two possibilities for each type of wine, it was necessary to assess four probabilities. With the help of some forecasts in industry publications, management made the following probability assessments:
Riesling Demand
Chardonnay Demand
Weak
Strong
Weak
0.05
0.50
Strong
0.25
0.20
Revenue projections show an annual contribution to profit of $20,000 if Seneca Hill plants only Chardonnay grapes and demand is weak for Chardonnay wine, and $70,000 if Seneca plants only Chardonnay grapes and demand is strong for Chardonnay wine. If Seneca plants only Riesling grapes, the annual profit projection is $25,000 if demand is weak for Riesling grapes and $45,000 if demand is strong for Riesling grapes. If Seneca plants both types of grapes, the annual profit projections are shown in the following table:
Riesling Demand
Chardonnay Demand
Weak
Strong
Weak
$22,000
$40,000
Strong
$26,000
$60,000
(a)
What is the decision to be made, what is the chance event, and what is the consequence? Identify the alternatives for the decisions and the possible outcomes for the chance events.
The decision to be made is . The chance event is . The consequence is . The alternatives for the decisions are . The possible outcomes for the chance events are .
(b)
Choose the correct decision tree.
(i)
(ii)
(iii)
(iv)
(c)
Use the expected value approach to recommend which alternative Seneca Hill Winery should follow in order to maximize expected annual profit.
(d)
Suppose management is concerned about the probability assessments when demand for Chardonnay wine is strong. Some believe it is likely for Riesling demand to also be strong in this case. Suppose that the probability of strong demand for Chardonnay and weak demand for Riesling is 0.05 and that the probability of strong demand for Chardonnay and strong demand for Riesling is 0.40. How does this change the recommended decision? Assume that the probabilities when Chardonnay demand is weak are still 0.05 and 0.50. Choose the correct alternative.
(e)
Other members of the management team expect the Chardonnay market to become saturated at some point in the future, causing a fall in prices. Suppose that the annual profit projections fall to $50,000 when demand for Chardonnay is strong and only Chardonnay grapes are planted. Using the original probability assessments, determine how this change would affect the optimal decision. Choose the correct alternative.
Carlos has just created a new subnet for the finance department. He needs to be able to allow the employees in finance to retrieve files from the sales server, which is located in another subnet. Which of the following OSI model layers would it be necessary to categorize the device into? a. Layer 4 b. Layer 6 c. Layer 2 d. Layer 3 All of the cubicles in a company's office have cables that run up to the ceiling and across to an IDF. Inside the IDF, they connect directly to the latest generation switch available from the networking equipment vendor that they have chosen. Which of the following describes the physical topology most likely in use? a. mesh
Some of the advantages of using time boxes and cycles in project coordination efforts include creating urgency, measuring progress, and allowing for predictable measurements. A) True 2. Even though most project managers are not contract specialists, they need to understand the process well enough to coordinate with the team. For the current assignment, you are looking at a short-term and small effort with a contractor of just a few hours without significant clarity. Which of the following would be the most applicable contract to use in this situation? A) Time and materials 3. The project you are working on has had modifications to the plan from the start and even how the project is run. Project governance covers all of the 3 following except: A) Naming The project manager 4. Of the following, which is most likely a trigger condition defined early in the project? A) Alerting Governance board if 10 percent over schedule 5. Of the following options, which stand
A process or product that is insensitive to normal variation is referred to as being Select one: a. in specification b. capable c. robust d. out of control Feedback Your answer is correct. A completed failure mode and effects analysis (FMEA) results in the following assessment rating. Occurrence = 4 Severity = 8 Detection = 10 What is the risk priority number (RPN) for this FMEA? Select one: a. 42 b. 22 c. 320 d. 120 Feedback Your answer is correct. In a visual inspection situation, one of the best ways to minimize deterioration of the quality level is to: Select one: a. have a program of frequent eye exams. b. retrain the inspector frequently. c. add variety to the task. d. have a standard to compare against as an element of the operation. Feedback Your answer is correct. Which of the following elements is least necessary to a good corrective action feedback report? Select one: a. What caused the failure b. Who caused the failure c. What correction has been made d. Wh
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