- 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.
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.
.0213
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.
.0160
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.
.3137
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.
.25
What can you infer about the PSA test from the results of parts (a), (b), and (c)?
lower
2. 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.
0.3173
317
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.
0.0027
3
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?
substantial decrease
3. Consider a Poisson distribution with μ = 5. If needed, round your answer to four decimal digits.
a. Choose the appropriate Poisson probability mass function.
option i
0.0842
0.0337
0.9596
a. Develop a probability distribution for the job satisfaction score of a randomly selected senior executive.
0.05
0.09
0.3
0.42
0.14
1
b. Develop a probability distribution for the job satisfaction score of a randomly selected middle manager.
0.04
0.1
0.11
0.46
0.29
1
c. What is the probability that a randomly selected senior executive will report a job satisfaction score of 4 or 5?
0.56
d. What is the probability that a randomly selected middle manager is very satisfied?
0.29
e. Compare the overall job satisfaction of senior executives and middle managers.
Less satisfied
4. Consider the following exponential probability density function.
for x ≥ 0
If needed, round your answer to four decimal digits.
Choose the correct formula for P(x ≤ x0).
option iii
a. .3935
b. 0.4724
c. 0.7135
d. 0.320025863
5. 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.
a) What is the probability that the male patient has prostate cancer if the PSA test comes back positive? Round your answer to
.0213
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.
.0160
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.
.3137
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.
.25
What can you infer about the PSA test from the results of parts (a), (b), and (c)?
Lower
6. 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.
0.3173
317
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.
0.0124
12
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?
Substantial decrease
7. 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).
a. Develop a probability distribution for the job satisfaction score of a randomly selected senior executive.
0.05
0.09
0.4
0.42
0.04
1
b. Develop a probability distribution for the job satisfaction score of a randomly selected middle manager.
0.04
0.1
0.08
0.46
0.32
1
c. What is the probability that a randomly selected senior executive will report a job satisfaction score of 4 or 5?
0.46
d. What is the probability that a randomly selected middle manager is very satisfied?
0.32
e. Compare the overall job satisfaction of senior executives and middle managers.
Less satisfied
8. Consider the following exponential probability density function.
a. Choose the correct formula for P(x ≤ x0).
Option ii
0.4866
0.3679
0.8111
0.3200
9. 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 |
a. Develop a probability distribution for the job satisfaction score of a randomly selected senior executive.
0.05
.09
.34
.42
.10
1
b. Develop a probability distribution for the job satisfaction score of a randomly selected middle manager.
.04
.1
.16
.46
.24
1
c. What is the probability that a randomly selected senior executive will report a job satisfaction score of 4 or 5?
.52
d. What is the probability that a randomly selected middle manager is very satisfied?
.24
less satisfied
10. Consider a Poisson distribution with μ = 2. If needed, round your answer to four decimal digits.
a. Choose the appropriate Poisson probability mass function.
option iii
b. 0.2707
c. 0.2707
d. 0.5940
11. 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.
98 more
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.
.91
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.
.07
d. What is the probability that a randomly selected representative votes yes on El Salvador aid? Round your answer to three decimal places.
.52
12. 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.
3 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.
1.492
2.055
2.577
2.226
The largest
Is
13. 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.
3.879
3.725
1.906
Cluster 3
14. 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.
1.41
1.32
1.37
1.53
1.29
1.81
1.75
1.36
1.56
1.55
1.74
1.51
2.01
1.77
1.61
1.06
1.47
1.50
1.37
1.05
1.77
1.32
1.52
K = 3
15. 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.
Must
100
100
may
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.
is
more
c. Based on these top 10 rules, what general insight can Leggere gain on the purchase habits of these customers?
may
d. What will be the effect on the rules generated if Leggere decreases the minimum support and reruns the association rules analysis?
large
large
may
e. What will be the effect on the rules generated if Leggere decreases the minimum confidence and reruns the association rules analysis?
large
large
low
decrease
16. 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.
129
171
0.43
0.0286
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.
215
285
0.43
0.0221
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.
430
570
0.43
0.0157
d. In which of the preceding three cases, part (a) or part (b) or part (c), is the standard error of p smallest? Why?
part c
the same
part c
17. 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.
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
0.24
b. At 95% confidence, what is the margin of error? If required, round your answer to four decimal places.
0.0265
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.
0.2135 0.2665
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.
0.4991 0.5609
e. Which of the confidence intervals in parts (c) and (d) has the smaller margin of error? Why?
c
farther form
18. 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 x for itemized deductions for this population of taxpayers for each of the following sample sizes: 30, 50, 100, and 400?
16442
423.57
232.00
116.00
b. What is the advantage of a larger sample size when attempting to estimate the population mean?
Reduce
more
19. 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?
15
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.
0.0005
c. Using a 0.05 level of significance, would you conclude that the proportion of business owners providing gifts has decreased?
reject can
0
less than or equal to reject
yes
Reject can
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