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Analytical Methods for Business (University of Arizona)

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ch07

*Student:*

1. A population consists of all items of interest in a statistical problem. True False

2. We calculate a parameter to make inferences about a statistic. True False

3. Bias refers to the tendency of a sample statistic to systematically over- or underestimate a population parameter.

True False

4. Selection bias occurs when the sample is mistakenly divided into strata, and random samples are drawn from each stratum.

True False

5. Nonresponse bias occurs when those responding to a survey or poll differ systematically from the non-respondents.

True False

6. A simple random sample is a sample of *n* observations which has the same probability of being selected from the population as any other sample of *n* observations.

True False

7. In stratified random sampling, the population is first divided up into mutually exclusive and collectively exhaustive groups, called strata. A stratified sample includes randomly selected observations from each stratum, which are proportional to the stratum’s size.

True False

8. A sample consists of all items of interest in a statistical problem, whereas a population is a subset of the population. We calculate a parameter to make inferences about the unknown sample statistic.

True False

9. If we had access to data that encompass the entire population, then the values of the parameters would be known and no statistical inference would be needed.

True False

10. A parameter is a random variable, whereas a sample statistic is a constant. True False

11. When a statistic is used to estimate a parameter, the statistic is referred to as an estimator. A particular value of the estimator is called an estimate.

True False

12. The standard deviation of equals the population standard deviation divided by the square root of the

sample size, or equivalently, .

True False

13. For any sample size *n*, the sampling distribution of is normal if the population from which the sample is drawn is uniformly distributed.

True False

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14. For any population with expected value µ and standard deviation **σ**, the sampling distribution of will be approximately normal if the sample size *n* is sufficiently small. As a general guideline, the

normal distribution approximation is justified when .

True False

15. For any population proportion *p*, the sampling distribution of will be approximately normal if the

sample size *n* is sufficiently large. As a general guideline, the normal distribution approximation is

justified when and . True False

16. Which of the following is an example of a sample statistic?

A.

B. *µ*

C. **σ**

D. **σ**2

17. Which of the following is *not* a population parameter?

A.

B. *µ*

C. **σ**

D. **σ**2

18. A census is an example of .

A. Sample data

B. Sample statistic

C. Population data

D. Population parameter

19. Bias can occur in sampling. Bias refers to .

A. The division of the population into overlapping groups

B. The creation of strata, which are proportional to the stratum’s size

C. The use of cluster sampling instead of stratified random sampling

D. The tendency of a sample statistic to systematically over- or underestimate a population parameter

20. Selection bias occurs when .

A. The population has been divided into strata

B. Portions of the population are excluded from the consideration for the sample

C. Cluster sampling is used instead of stratified random sampling

D. Those responding to a survey or poll differ systematically from the non-respondents

21. Nonresponse bias occurs when .

A. The population has been divided into strata

B. Portions of the population are excluded from the sample

C. Cluster sampling is used instead of stratified random sampling

D. Those responding to a survey or poll differ systematically from the non-respondents

22. Which of the following is not a form of bias?

A. Portions of the population are excluded from the sample.

B. Information from the sample is typical of information in the population.

C. Information from the sample overemphasizes a particular stratum of the population.

D. Those responding to a survey or poll differ systematically from the non-respondents.

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23. Which of the following meets the requirements of a simple random sample?

AA population contains 10 members under the age of 25 and 20 members over the age of 25. The sample

. will include six people who volunteer for the sample.

BA population contains 10 members under the age of 25 and 20 members over the age of 25. The sample

. will include six people chosen at random, without regard to age.

CA population contains 10 members under the age of 25 and 20 members over the age of 25. The sample

. will include six males chosen at random, without regard to age.

DA population contains 10 members under the age of 25 and 20 members over the age of 25. The sample

. will include two people chosen at random under the age of 25 and four people chosen at random over 25.

24. Which of the following meets the requirements of a stratified random sample?

AA population contains 10 members under the age of 25 and 20 members over the age of 25. The sample

. will include six people who volunteer for the sample.

BA population contains 10 members under the age of 25 and 20 members over the age of 25. The sample

. will include six people chosen at random, without regard to age.

CA population contains 10 members under the age of 25 and 20 members over the age of 25. The sample

. will include six males chosen at random, without regard to age.

DA population contains 10 members under the age of 25 and 20 members over the age of 25. The sample

. will include two people chosen at random under the age of 25 and four people chosen at random over 25.

25. Which of the following is true about statistics such as the sample mean or sample proportion?

A. A statistic is a constant.

B. A statistic is a parameter.

C. A statistic is always known.

D. A statistic is a random variable.

26. Statistics are used to estimate population parameters, particularly when it is impossible or too expensive

to poll an entire population. A particular value of a statistic is referred to as a(n) .

A. Mean

B. Stratum

C. Estimate

D. Finite correction factor

27. Which of the following is considered an estimator?

A.

B. *µ*

C. **σ**

D. **σ***2*

28. Which of the following is considered an estimate?

A.

B.

C.

D.

29. What is the relationship between the expected value of the sample mean and the expected value of the population?

A.

B.

C.

D.

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30. How does the variance of the sample mean compare to the variance of the population?

A. It is smaller and therefore suggests that averages have less variation than individual observations.

B. It is larger and therefore suggests that averages have less variation than individual observations.

C. It is smaller and therefore suggests that averages have more variation than individual observations.

D. It is larger and therefore suggests that averages have more variation than individual observations.

31. What is the relationship between the standard deviation of the sample mean and the population standard deviation?

A.

B.

C.

D.

32. A nursery sells trees of different types and heights. These trees average 60 inches in height with a standard deviation of 16 inches. Suppose that 75 pine trees are sold for planting at City Hall. What is the standard deviation for the sample mean?

A. 1.85

B. 3.41

C. 4

D. 16

33. If a population is known to be normally distributed, what can be said of the sampling distribution of the sample mean drawn from this population?

A. For any sample size *n*, the sampling distribution of the sample mean is normally distributed.

B. For a sample size , the sampling distribution of the sample mean is normally distributed.

C. For a sample size , the sampling distribution of the sample mean is normally distributed.

D. For a sample size , the sampling distribution of the sample mean is normally distributed.

34. Over the entire six years that students attend an Ohio elementary school, they are absent, on average,

28 days due to influenza. Assume that the standard deviation over this time period is days. Upon graduation from elementary school, a random sample of 36 students is taken and asked how many days of school they missed due to influenza.

Refer to Exhibit 7-1. What is the expected value for the sampling distribution of the number of school days missed due to influenza?

A. 6

B. 9

C. 28

D. 168

35. Over the entire six years that students attend an Ohio elementary school, they are absent, on average,

28 days due to influenza. Assume that the standard deviation over this time period is days. Upon graduation from elementary school, a random sample of 36 students is taken and asked how many days of school they missed due to influenza.

Refer to Exhibit 7-1. What is the standard deviation for the sampling distribution of the number of school days missed due to influenza?

A. 1.22

B. 1.50

C. 2.25

D. 9.00

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36. Over the entire six years that students attend an Ohio elementary school, they are absent, on average,

28 days due to influenza. Assume that the standard deviation over this time period is days. Upon graduation from elementary school, a random sample of 36 students is taken and asked how many days of school they missed due to influenza.

Refer to Exhibit 7-1. The probability that the sample mean is less than 30 school days is .

A. 0.0918

B. 0.4129

C. 0.5871

D. 0.9082

37. Over the entire six years that students attend an Ohio elementary school, they are absent, on average,

Refer to Exhibit 7-1. The probability that the sample mean is between 25 and 30 school days is

.

A. 0.0228

B. 0.0918

C. 0.8854

D. 0.9082

38. Suppose that, on average, electricians earn approximately dollars per year in the United

States. Assume that the distribution for electrician’s yearly earnings is normally distributed and that the

standard deviation is dollars.

Refer to Exhibit 7-2. Given a sample of four electricians, what is the standard deviation for the sampling distribution of the sample mean?

A. 6,000

B. 12,000

C. 36,000

D. 54,000

39. Suppose that, on average, electricians earn approximately dollars per year in the United

States. Assume that the distribution for electrician’s yearly earnings is normally distributed and that the

standard deviation is dollars.

Refer to Exhibit 7-2. What is the probability that the average salary of four randomly selected electricians exceeds $60,000?

A. 0.1587

B. 0.3085

C. 0.6915

D. 0.8413

40. Suppose that, on average, electricians earn approximately dollars per year in the United

States. Assume that the distribution for electrician’s yearly earnings is normally distributed and that the

standard deviation is dollars.

Refer to Exhibit 7-2. What is the probability that the average salary of four randomly selected electricians is less than $50,000?

A. 0.2514

B. 0.3707

C. 0.6293

D. 0.7486

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41. Suppose that, on average, electricians earn approximately dollars per year in the United

standard deviation is dollars.

Refer to Exhibit 7-2. What is the probability that the average salary of four randomly selected electricians is more than $50,000 but less than $60,000?

A. 0.5899

B. 0.7486

C. 0.8413

D. 0.9048

42. Susan has been on a bowling team for 14 years. After examining all of her scores over that period of time, she finds that they follow a normal distribution. Her average score is 225, with a standard deviation of 13.

Refer to Exhibit 7-3. What is the probability that in a one-game playoff, her score is more than 227?

A. 0.2676

B. 0.4404

C. 0.5596

D. 0.7324

43. Susan has been on a bowling team for 14 years. After examining all of her scores over that period of time, she finds that they follow a normal distribution. Her average score is 225, with a standard deviation of 13.

Refer to Exhibit 7-3. If during a typical week Susan bowls 16 games, what is the probability that her average score is more than 230?

A. 0.0618

B. 0.3520

C. 0.6480

D. 0.9382

44. Susan has been on a bowling team for 14 years. After examining all of her scores over that period of time, she finds that they follow a normal distribution. Her average score is 225, with a standard deviation of 13.

Refer to Exhibit 7-3. If during a typical week Susan bowls 16 games, what is the probability that her average score for the week is between 220 and 228?

A. 0.0618

B. 0.2390

C. 0.7594

D. 0.8212

45. Susan has been on a bowling team for 14 years. After examining all of her scores over that period of time, she finds that they follow a normal distribution. Her average score is 225, with a standard deviation of 13.

Refer to Exhibit 7-3. If during a typical month Susan bowls 64 games, what is the probability that her average score in this month is above 227?

A. 0.1093

B. 0.4404

C. 0.5596

D. 0.8907

46. Professor Elderman has given the same multiple choice final exam in his Principles of Microeconomics class for many years. After examining his records from the past 10 years, he finds that the scores have a mean of 76 and a standard deviation of 12.

Refer to Exhibit 7-4. What is the probability that a class of 15 students will have a class average greater than 70 on Professor Elderman’s final exam?

A. 0.0262

B. 0.6915

C. 0.9738

D. Cannot be determined.

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47. Professor Elderman has given the same multiple choice final exam in his Principles of Microeconomics class for many years. After examining his records from the past 10 years, he finds that the scores have a mean of 76 and a standard deviation of 12.

Refer to Exhibit 7-4. What is the probability that a class of 36 students will have an average greater than

70 on Professor Elderman’s final exam?

A. 0.0014

B. 0.3085

C. 0.6915

D. 0.9986

48. Professor Elderman has given the same multiple choice final exam in his Principles of Microeconomics class for many years. After examining his records from the past 10 years, he finds that the scores have a mean of 76 and a standard deviation of 12.

Refer to Exhibit 7-4. Professor Elderman offers his class of 36 a pizza party if the class average is above

80. What is the probability that he will have to deliver on his promise?

A. 0.0228

B. 0.3707

C. 0.6293

D. 0.9772

49. Professor Elderman has given the same multiple choice final exam in his Principles of Microeconomics class for many years. After examining his records from the past 10 years, he finds that the scores have a mean of 76 and a standard deviation of 12.

Refer to Exhibit 7-4. What is the probability Professor Elderman’s class of 36 has a class average below 78?

A. 0.1587

B. 0.5675

C. 0.8413

D. Cannot be determined.

50. According to the central limit theorem, the distribution of the sample means is normal if

.

A. The underlying population is normal

B. The sample size

C. If the standard deviation of the population is known

D. Both A and B are correct

51. The central limit theorem states that, for any distribution, as *n* gets larger, the sampling distribution of the

sample mean .

A. Becomes larger

B. Becomes smaller

C. Is closer to a normal distribution

D. Is closer to the standard deviation

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52. A random sample of size 36 is taken from a population with mean and standard deviation

.

Refer to Exhibit 7-5. What is the expected value and the standard deviation for the sampling distribution of the sample mean?

A. Option A

B. Option B

C. Option C

D. Option D

53. A random sample of size 36 is taken from a population with mean and standard deviation

.

Refer to Exhibit 7-5. The probability that the sample mean is greater than 18 is .

A. 0.1587

B. 0.4325

C. 0.5675

D. 0.8413

54. A random sample of size 36 is taken from a population with mean and standard deviation

.

Refer to Exhibit 7-5. The probability that the sample mean is less than 15 is .

A. 0.0228

B. 0.3707

C. 0.6293

D. 0.9772

55. A random sample of size 36 is taken from a population with mean and standard deviation

.

Refer to Exhibit 7-5. The probability that the sample mean is between 15 and 18 is .

A. 0.0228

B. 0.8185

C. 0.8413

D. 0.8641

56. Using the central limit theorem, applied to the sampling distribution of the sample proportion, what conditions must be met?

A.

B.

C.

D.

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57. A random sample of size 100 is taken from a population described by the proportion .

Refer to Exhibit 7-6. What is the expected value and the standard deviation for the sampling distribution of the sample proportion?

A. Option A

B. Option B

C. Option C

D. Option D

58. A random sample of size 100 is taken from a population described by the proportion .

Refer to Exhibit 7-6. The probability that the sample proportion is greater than 0.62 is .

A. 0.3409

B. 0.4082

C. 0.6591

D.

59. A random sample of size 100 is taken from a population described by the proportion .

Refer to Exhibit 7-6. The probability that the sample proportion is less than 0.55 is .

A.

B. 0.1539

C. 0.3669

D. 0.8461

60. A random sample of size 100 is taken from a population described by the proportion .

Refer to Exhibit 7-6. The probability that the sample proportion is between 0.55 and 0.62 is .

A. 0.1539

B. 0.5052

C. 0.6591

D. 0.8130

61. A university administrator expects that 25% of students in a core course will receive an A. He looks at the grades assigned to 60 students.

Refer to Exhibit 7-7. What are the expected value and the standard deviation for the proportion of students that receive an A?

A. Option A

B. Option B

C. Option C

D. Option D

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62. A university administrator expects that 25% of students in a core course will receive an A. He looks at the grades assigned to 60 students.

Refer to Exhibit 7-7. The probability that the proportion of students that receive an A is 0.20 or less is

.

A. 0.1867

B. 0.6266

C. 0.8133

D. 0.8900

63. A university administrator expects that 25% of students in a core course will receive an A. He looks at the grades assigned to 60 students.

Refer to Exhibit 7-7. The probability that the proportion of students who receive an A is between 0.20

and 0.35 is .

A. 0.1867

B. 0.7766

C. 0.8133

D. 0.9633

64. A university administrator expects that 25% of students in a core course will receive an A. He looks at the grades assigned to 60 students.

Refer to Exhibit 7-7. The probability that the proportion of students who receive an A is NOT between

0.20 and 0.30 is .

A. 0.1867

B. 0.3734

C. 0.6266

D. 0.8133

65. The labor force participation rate is the number of people in the labor force divided by the number of people in the country that are of working age and not institutionalized. The BLS reported in February of 2012 that the labor force participation rate in the United States was 63.7% (Calculatedrisk.com). A marketing company asks 120 working-age people if they either have a job or are looking for a job, or, in other words, whether they are in the labor force.

Refer to Exhibit 7-8. What is the expected value and the standard deviation for a labor participation rate in the company’s sample?

A. Option A

B. Option B

C. Option C

D. Option D

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66. The labor force participation rate is the number of people in the labor force divided by the number of people in the country that are of working age and not institutionalized. The BLS reported in February of 2012 that the labor force participation rate in the United States was 63.7% (Calculatedrisk.com). A marketing company asks 120 working-age people if they either have a job or are looking for a job, or, in other words, whether they are in the labor force.

Refer to Exhibit 7-8. For the company’s sample, the probability that the proportion of people who are in

the labor force is greater than 0.65 is .

A. 0.1179

B. 0.3000

C. 0.3821

D. 0.6179

67. The labor force participation rate is the number of people in the labor force divided by the number of people in the country that are of working age and not institutionalized. The BLS reported in February of 2012 that the labor force participation rate in the United States was 63.7% (Calculatedrisk.com). A marketing company asks 120 working-age people if they either have a job or are looking for a job, or, in other words, whether they are in the labor force.

Refer to Exhibit 7-8. What is the probability that less than 60% of those surveyed are members of the labor force?

A. 0.2005

B. 0.7995

C. 0.8400

D. 0.9706

68. The labor force participation rate is the number of people in the labor force divided by the number of people in the country that are of working age and not institutionalized. The BLS reported in February of 2012 that the labor force participation rate in the United States was 63.7% (Calculatedrisk.com). A marketing company asks 120 working-age people if they either have a job or are looking for a job, or, in other words, whether they are in the labor force.

Refer to Exhibit 7-8. What is the probability that between 60% and 62.5% of those surveyed are members of the labor force?

A. 0.0243

B. 0.1931

C. 0.2005

D. 0.3936

69. Super Bowl XLVI was played between the New York Giants and the New England Patriots in Indianapolis. Due to a decade-long rivalry between the Patriots and the city’s own team, the Colts, most Indianapolis residents were rooting heartily for the Giants. Suppose that 90% of Indianapolis residents wanted the Giants to beat the Patriots.

Refer to Exhibit 7-9. What is the probability that, of a sample of 100 Indianapolis residents, at least 15% were rooting for the Patriots in Super Bowl XLVI?

A. 0.0300

B. 0.0475

C. 0.4763

D. 0.9525

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70. Super Bowl XLVI was played between the New York Giants and the New England Patriots in Indianapolis. Due to a decade-long rivalry between the Patriots and the city’s own team, the Colts, most Indianapolis residents were rooting heartily for the Giants. Suppose that 90% of Indianapolis residents wanted the Giants to beat the Patriots.

Refer to Exhibit 7-9. What is the probability that from a sample of 100 Indianapolis residents, fewer than 95% were rooting for the Giants in Super Bowl XLVI?

A. 0.0300

B. 0.0475

C. 0.4763

D. 0.9525

71. Super Bowl XLVI was played between the New York Giants and the New England Patriots in Indianapolis. Due to a decade-long rivalry between the Patriots and the city’s own team, the Colts, most Indianapolis residents were rooting heartily for the Giants. Suppose that 90% of Indianapolis residents wanted the Giants to beat the Patriots.

Refer to Exhibit 7-9. What is the probability that from a sample of 40 Indianapolis residents, fewer than 95% were rooting for the Giants in Super Bowl XLIV?

A. 0.0474

B. 0.1469

C. 0.8531

D. Cannot be determined

72. Super Bowl XLVI was played between the New York Giants and the New England Patriots in Indianapolis. Due to a decade-long rivalry between the Patriots and the city’s own team, the Colts, most Indianapolis residents were rooting heartily for the Giants. Suppose that 90% of Indianapolis residents wanted the Giants to beat the Patriots.

Refer to Exhibit 7-9. What is the probability that from a sample of 200 Indianapolis residents, fewer than 170 were rooting for the Giants in Super Bowl XLIV?

A. 0.0091

B. 0.0212

C. 0.4954

D. 0.9908

73. According to the 2011 Gallup daily tracking polls (www.gallup.com, February 3, 2012), Mississippi is the most conservative U.S. state, with 53.4 percent of its residents identifying themselves as conservative.

Refer to Exhibit 7-10. What is the probability that at least 60% of a random sample of 200 Mississippi residents identify themselves as conservative?

A. 0.0307

B. 0.3530

C. 0.4847

D. 0.9693

74. According to the 2011 Gallup daily tracking polls (www.gallup.com, February 3, 2012), Mississippi is the most conservative U.S. state, with 53.4 percent of its residents identifying themselves as conservative.

Refer to Exhibit 7-10. What is the probability that at least 100 but fewer than 115 respondents of a random sample of 200 Mississippi residents identify as conservative?

A. 0.1685

B. 0.3370

C. 0.7085

D. 0.8770

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75. According to the 2011 Gallup daily tracking polls (www.gallup.com, February 3, 2012), Mississippi is the most conservative U.S. state, with 53.4 percent of its residents identifying themselves as conservative.

Refer to Exhibit 7-10. What is the probability that at least 50 respondents of a random sample of 100 Mississippi residents do NOT identify themselves as conservative?

A. 0.0499

B. 0.2483

C. 0.4966

D. 0.7517

76. According to the 2011 Gallup daily tracking polls (www.gallup.com, February 3, 2012), Mississippi is the most conservative U.S. state, with 53.4 percent of its residents identifying themselves as conservative.

Refer to Exhibit 7-10. What is the probability that fewer than 45 respondents of a random sample of 100 Mississippi residents do NOT identify themselves as conservative?

A. 0.0499

B. 0.1873

C. 0.3745

D. 0.6255

77. Under what condition is the finite population correction factor used for computing the standard deviations

of and ?

A.

B.

C.

D.

78. The finite correction factor is always

.

A.Less than one, and therefore increases the standard deviations of

and computed under the

assumption of infinite population

B.Less than one, and therefore decreases the standard deviations of and computed under the assumption of infinite population

C.Greater than one, and therefore increases the standard deviations of and computed under the assumption of infinite population

D Greater than one, and therefore decreases the standard deviations ofand computed under the

. assumption of infinite population

79. A local company makes snack size bags of potato chips. Each day, the company produces batches of 400 snack size bags using a process designed to fill each bag with an average of 2 ounces of potato chips. However, due to imperfect technology, the actual amount placed in a given bag varies. Assume the amount placed in each of the 400 bags is normally distributed and has a standard deviation of 0.1 ounces. What is the probability that a sample of 40 bags has an average weight of at least 2.02 ounces?

A. 0.0150 B. 0.0918 C. 0.1038 D. 0.4207

80. Suppose 35% of homes in a Miami, Florida, neighborhood are under water (in other words, the amount due on the mortgage is larger than the value of the home). There are 160 homes in the neighborhood and 30 of those homes are owned by your friends. What is the probability that less than 30% of your friend’s homes are under water?

A. 0.2611 B. 0.2843 C. 0.6400 D. 0.7389

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81. Successful firms must focus on the quality of the products and services that they offer. Which factor(s) contributes to the quest for quality?

A. Global competition

B. Consumer expectations

C. Technological advances

D. All of the above

82. Acceptance sampling is a(n) .

A. Division of the population into strata.

B. Plot of calculated statistics of the production process over time.

C. Inspection of a portion of the products at the completion of the production process.

D. Determination of a point at which the production process does not conform to specifications.

83. The detection approach to statistical quality control .

A. Divides the population into strata

B. Inspects a portion of the products at the completion of the production process

C. Determines at which point the production process does not conform to specifications

D. Uses the finite correction factor when the sample size is not much smaller than the population size

84. In any production process, variation in the quality of the end product is inevitable. Chance variation, or

common variation, refers to .

A. The variation caused by stratified random sampling

B. The variation caused by the use of the finite correction factor

C. Specific events or factors that can usually be identified and eliminated

D. A number of randomly occurring events that are part of the production process.

85. In any production process, variations in the quality of the end product are inevitable. Assignable variation

refers to .

A. The variation caused by stratified random sampling

B. The variation caused by the use of the finite correction factor

C. Specific events or factors that can usually be identified and eliminated

D. A number of randomly occurring events that are part of the production process

86. A local company makes snack size bags of potato chips. The company produces batches of 400 snack size bags using a process designed to fill each bag with an average of 2 ounces of potato chips. However, due to imperfect technology, the actual amount placed in a given bag varies. Assume the population of filling weights is normally distributed with a standard deviation of 0.1 ounces. The company periodically weighs samples of 10 bags to ensure the proper filling process. The last five sample means, in ounces, were 1.99, 2.02, 2.07, 1.96, and 2.01. Is the production process under control?

A. No, since the sample means show a downward trend

B. Yes, since the sample means show a downward trend

C. No, since the sample means fall within the upper and lower control limits

D. Yes, since the sample means fall within the upper and lower control limits

87. A manufacturing process produces computer chips in batches of 100. The firm believes that the percent of defective computer chips is 2%. If in five batches the percent defective were 3%, 8%, 1%, 2%, and 7%, how many of these fell outside of the upper or lower control limits for the proportion of defective computer chips in a batch?

A. 0

B. 1

C. 2

D. 3

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88. A random sample of 49 cast aluminum pots is taken from a production line once every day. The number of defective pots is counted. The proportion of defective pots has been closely examined in the past and is believed to be 0.05.

Refer to Exhibit 7-11. What are the upper and lower control limits for the chart?

A. Option A

B. Option B

C. Option C

D. Option D

89. A random sample of 49 cast aluminum pots is taken from a production line once every day. The number of defective pots is counted. The proportion of defective pots has been closely examined in the past and is believed to be 0.05.

Refer to Exhibit 7-11. The sample proportions for the week are shown in the accompanying table.

Is the production process in control?

A. No, since the sample proportions show a downward trend

B. No, since the sample proportions fall within the upper and lower control limits

C. Yes, since the sample proportions show a downward trend

D. Yes, since the sample proportions fall within the upper and lower control limits

90. The California Department of Education wants to gauge the difficulty of a new exam by having a sample of students at a particular school take the exam. The quality of the students at the chosen school varies widely and the school administrators are allowed to choose who gets to take the exam. The administrators have a strong incentive for the school to do well on the exam. Do you think the results will represent the true ability of the students at school? What kind of bias if any do you think will be present? Explain.

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91. The campaign manager for a candidate for governor in Arizona wants to conduct a poll to better understand his candidate’s chances for the upcoming election.

a. What is the population of interest?

b. Why may the poll be biased if a simple random sample of voters in the last gubernatorial election (four years prior) is taken?

92. It is known that college students at a local community college study 12 hours per week with a standard deviation of 5 hours. What is the expected value and variance for a sample of nine students?

93. A fast food restaurant uses an average of 110 grams of meat per burger patty. Suppose the amount of meat in a burger patty is normally distributed with a standard deviation of 20 grams. What is the probability that the average amount of meat in four randomly selected burgers is less than 105 grams?

94. Suppose residents in a well-to-do neighborhood pay an average overall tax rate of 25% with a standard deviation of 8%. Assume tax rates are normally distributed. What is the probability that the mean tax rate of 16 randomly selected residents is between 20% and 30%.

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95. Suppose the average casino patron in Las Vegas loses $110 dollars per day, with a standard deviation of $700. Assume winnings/losses are normally distributed.

a. What is the probability that a random group of nine people average more than $500 in winnings on their one day trip to Las Vegas?

b. What is the probability that a random group of nine people average more than $500 in losses on their one day trip to Las Vegas?

96. A ski resort gets an average of 2,000 customers per weekday with a standard deviation of 800 customers. Assume the underlying distribution is normal. What is the probability a ski resort averages between 1,500 customers and 3,000 customers per weekday over the course of four weekdays?

97. A mining company made some changes to their mining process in an attempt to save fuel. Before the changes were made, it took an average of 20 gallons of diesel fuel to mine 1,000 pounds of copper. Suppose the standard deviation of fuel used per 1,000 pounds of copper mined is 6 gallons. After the changes were made, the company only used an average of 18 gallons of diesel for the next 30,000 pounds of copper mined.

a. How unusual would it be to get a sample average of 18 gallons or less for 30,000 pounds of copper mined if the changes to the mining process had no effect?

b. Do you think the changes in the mining process actually lowered the fuel used? Explain.

98. A gym knows that each member, on average, spends 70 minutes at the gym per week, with a standard deviation of 20 minutes. Assume the amount of time each customer spends at the gym is normally distributed.

a. What is the probability that a randomly selected customer spends less than 65 minutes at the gym?

b. Suppose the gym surveys a random sample of 49 members about the amount of time they spend at the gym each week. What is the expected value and standard deviation of the sample mean of the time spent at the gym?

c. If 49 members are randomly selected, what is the probability that the average time spent at the gym exceeds 75 minutes?

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99. A book publisher knows that it takes an average of nine business days from when the material for the book is finalized until the first edition is printed and ready to sell. Suppose the exact amount of time has a standard deviation of four days.

a. Suppose the publisher examines the printing time for a sample of 36 books. What is the probability that the sample mean time is shorter than eight days?

b. Suppose the publisher examines the printing time for a sample of 36 books. What is the probability that the sample mean time is between 7 and 10 days?

c. Suppose the publisher signs a contract for the printer to print 100 books. If the average printing time for the 100 books is longer than 9.3 days, the printer must pay a penalty. What is the probability the penalty clause will be activated?

d. Suppose the publisher signs a contract for the printer to print 10 books. If the average printing time for the 10 books is longer than 9.7 days, the printer must pay a penalty. What is the probability the penalty clause will be activated?

100.In a large metropolitan area, the top providers for television and Internet services are a phone company, a satellite company, and a cable company. The satellite company serves 43% of the homes in the area. What is the probability that in a survey of 1,000 homes, more than 447 of them are served by the satellite company?

101.In early 2012, the United States Congress approval rating was approximately 10% (Reuters.com). In a poll of 400 Americans, what is the probability that their approval rating is between 8% and 12%?

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102.A tutoring company claims that 75% of the high school students who hire one of their tutor’s will improve their grades.

a. In a sample of 100 high school students, what is the probability that 80% or more improved their grades?

b. In a sample of 200 high school students, what is the probability that 80% or more improved their grades?

c. Comment on the reason for the difference between the computed probabilities in parts a and b.

Answer: a. 0.1251 b. 0.0516 c. The standard deviation of is lower with a larger sample size.

Feedback: a. Transform into ;

b. Transform into ;

c. The larger sample size in part b makes for a smaller standard deviation of and a smaller probability

of event .

103.A school is required by the government to give some randomly chosen students a standardized test. From previous experience, the school knows about 68% of their students will receive passing scores in math and English. To improve funding, the school needs to score at least 70% on the standardized test. This year the school can decide if it wants to test 100 or 200 students. Should the school test 100 or 200 students? Explain.

104.The Office of Career Services at a major university knows that 74% of its graduates find full-time positions in the field of their choosing within six months of graduation. Suppose the Office of Career Services surveys 25 alumni six months after graduation.

a. What is the probability that at least 80% of the alumni have a job in the field of their choosing? b. What is the probability that between 60% and 76% of the alumni have a job in the field of their choosing?

c. What is the probability that fewer than 60% of the alumni have a job in the field of their choosing?

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105.Administrative assistants in a local university have been asked to prove their proficiency in the use of spreadsheet software by taking a proficiency test. Historically, the mean test score has been 74 with a standard deviation of 4. A random sample of size 40 is taken from the 100 administrative assistants and asked to complete the proficiency test.

a. Calculate the expected value and the standard deviation of the sample mean.

b. What is the probability that the sample mean score is more than 75, the predetermined passing score?

106.In a small town, there are 3,000 registered voters. An editor of a local newspaper would like to predict the outcome of the next election; in particular he is interested in the likelihood that Eli Brady will be elected. The editor believes that Eli, a local hero, will garner 54% of the vote. A poll of 500 registered voters is taken. Assuming that the editor’s belief is true, calculate:

a. The expected value and the standard deviation of the sample proportion.

b. The probability that the sample proportion score is more than 0.58.

107.A random sample of nine cast aluminum pots is taken from a production line once every hour. The interior diameter of the pots is measured and the sample mean is calculated. The target for the diameter is 12′ and the standard deviation for the pot diameter is 0.05′. Assume the pot diameter is normally distributed.

a. Construct the centerline and the upper and lower control limits for the chart.

b. The means of the samples for a given eight-hour day are 12.01, 12.06, 11.97, 12.08, 11.92, 11.95,

11.97, and 12.04. Plot these values on the chart.

c. Does it appear that the process is under control? Explain.

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108.In a recent investigation, the National Highway Traffic Safety Administration (NHTSA) found that the Chevrolet Volt and other electric vehicles do not pose a greater risk of fire than gasoline-powered vehicles (*The Boston Globe,* January 25, 2012). Specifically, it was determined that “no discernible defect trend exists.” Suppose a consumer advocacy group wants to verify some of these claims by constructing

a chart. The group expects 2% of electric cars to catch fire each month. For each of the last six months, 500 electric car owners are asked if their cars have caught fire. The following sample proportions are obtained:

0.010 0.020 0.015 0.030 0.025 0.015

a. Assuming that the group expectation is correct, construct the centerline and the upper and lower control

limits for the chart.

b. Do the consumer group’s findings support those of the NHTSA? Explain.

109.A bottled water plant utilizes a production process designed to fill bottles with 20 ounces of water.

The population of filling volumes is normally distributed with a standard deviation of 1.3 ounces.

Periodically, process engineers take 20-bottle samples and compute the sample mean.

a. What are the upper and lower control limits?

b. Suppose the last five sample means were 19.4, 20.2, 20.5, 20.7, and 21.1 ounces. Is the process under control?

110.A manufacturing process produces tubeless mountain bike tires in batches of 200. Past records show that 6% of the tires will not hold air. An engineer tests five batches, each one week apart, and shows the proportion of tires that will not hold air below.

Proportion of tires that will not hold air:

a. Construct the centerline and the upper control limit for chart.

b. Should the engineer be worried? Comment on any trend in the proportion of tubeless tires that will not hold air.

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111.A large accounting firm gives out 1,000 job offers every year to new college graduates. Suppose that 85% of those that received offers accept the position. The following shows the number of graduates that have accepted jobs in the last four years.

Number of job offers accepted:

a. Construct the centerline and the upper and lower control limits for the chart.

b. Does the company need to worry about its ability to attract college graduates to the firm?

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