CQE Study Questions,Answers and Solutions

This material is protected under United States copyright laws. It may not be reproduced, transmitted or distributed in any form without the express written consent of Carl Nocera. Copyright © 2004 by Carl D. Nocera. All rights reserved. CQEweb.com Produced in the United States of America.

PART I: QREVIEW STUDY QUESTIONS Basic Quality Concepts Basic Probability Statistics Statistical Inference Sampling Control Charts Reliability Regression and Correlation Cost of Quality Design of Experiments Metrology and Calibration

1 7 15 22 25 29 33 35 37 41 44

QReview Answers

46

PART II: QUESTIONS AND SOLUTIONS Basic Quality Concepts Basic Probability Statistics Statistical Inference Sampling Control Charts Reliability Regression and Correlation Cost of Quality Design of Experiments Metrology and Calibration

47 53 65 75 81 87 92 95 98 102 106

Basic Quality Concepts

1

PART 1 - QREVIEW STUDY QUESTIONS INTRODUCTION The following questions include key concepts which are representative of the CQE examination. Each question should be worked out as completely as possible before looking at the solution. Some questions have been taken from previously published ASQ CQE exams. BASIC QUALITY CONCEPTS 1. Certification of a company to the ISO 9002 standard does not include which of the following requirements? a) b) c) d)

Servicing Design control Contract review Internal quality audits

2. A quality control program is considered to be a) b) c) d)

a collection of quality control procedures and guidelines. a step by step list of all quality control check points. a summary of company quality control policies. a system of activities to provide quality of products and services.

3. The "quality function" of a company is best described as a) the degree to which the company product conforms to a design or specification. b) that collection of activities through which "fitness for use" is achieved. c) the degree to which a class or category of product possesses satisfaction for people generally. d) All of the above. 4. In preparing a product quality policy for your company, you should do all of the following except a) specify the means by which quality performance is measured. b) develop criteria for identifying risk situations and specify whose approval is required when there are known risks. c) include procedural matters and functional responsibilities. d) state quality goals. 5. What natural phenomenon created the necessity to control product and process quality?

2

QReview Study Questions

a) b) c) d)

Gravity Variation Human Error Management

6. The three basic elements of a quality system are a) Quality Management, Purchasing and Document Control b) Quality Management, Quality Control and Quality Assurance c) SPC, Inspection and Quality Assurance d) Quality Control, Quality Costs and Control Charts 7. What are the two basic categories of quality? a) b) c) d)

Design and Conformance Quality Good and Bad Quality Defective and Non-Defective Quality Attribute and Variable Quality

8. The Law of Large Numbers states that a) b) c) d)

individual occurrences are predictable and group occurrences are unpredictable. group data always follows a normal pattern. individual occurrences are unpredictable and group occurrences are predictable. the standard deviation of group data will always be greater than ten.

9. Statistical quality control is best described as a) keeping product characteristics within certain bounds. b) calculating the mean and standard deviation. c) the study of the characteristics of a product or process, with the help of numbers, to make them behave the way we want them to behave. d) the implementation of ISO 9000. 10. Which of the following is the most important element in Statistical Quality Control? a) The Feedback Loop b) Make Operation c) Inspection d) Quality of Incoming Material 11. When measurements are accurate and precise, a) the data are distributed randomly throughout the entire range. b) the data are clustered closely around the central value. c) minimum variation will exist.


d) the data are normally distributed. 12. All of the following are included in a quality system except a) b) c) d)

document control. corrective action. management responsibility. employee salaries.

13. Which of the following best describes a statistical distribution? a) b) c) d)

A model that shows how data are distributed over a range of measurements. An Analysis of Variance table. A sampling plan. A graph that contains data plotted on a normal curve.

14. Which of the following are two types of data used in statistical quality control? a) b) c) d)

Design and Conformance Data Precise and Accurate Data Variables and Attributes Data Mean and Variance Data

15. The primary reason for evaluating and maintaining surveillance over a supplier's quality program is to a) b) c) d)

perform product inspection at source. eliminate incoming inspection cost. motivate suppliers to improve quality. make sure the supplier's quality program is functioning effectively.

16. Which one of the following are ISO 9001 requirements? a) b) c) d)

Process Flow Chart Quality Manual Operations Manual TQM Program

17. Which of the following does not generate product-quality characteristics? a) b) c) d)

Designer Inspector Machinist Equipment engineer

3

4


18. Incoming material inspection is based most directly on a) b) c) d)

design requirements. purchase order requirements. manufacturing requirements. customer use of the end product.

19. The acronym ISO means a) b) c) d)

International Standards Organization. Internal Service Organization. equal. third party auditing organization.

20. Products should be subjected to tests which are designed to a) b) c) d)

demonstrate advertised performance. demonstrate basic function at minimum testing cost. approximate the conditions to be experienced in customer's application. assure that specifications are met under laboratory conditions.

21. The advantage of a written procedure is a) b) c) d)

it provides flexibility in dealing with problems. unusual conditions are handled better. it is a perpetual coordination device. coordination with other departments is not required.

22. In spite of the Quality Engineer's best efforts, situations may develop in which his decision is overruled. The most appropriate action would be to a) b) c) d)

resign from the position based upon convictions. report findings to an outside source such as a regulatory agency or the press. document findings, report them to superiors and move on the next assignment. discuss findings with co-workers in order to gain support, thereby forcing action.

23. If a test data does not support a Quality Engineer's expectations, the best thing to do is a) b) c) d)

adjust the data to support expectations if it is only slightly off. draw the expected conclusion omitting the data not supporting it. re-evaluate the expectations of the test based upon the data. report the data and expected conclusion with no reference to one another.


5

24. In case of conflict between contract specifications and shop practice, a) b) c) d)

arbitration is necessary. the customer is always right. good judgment should be exercised. contract specifications normally apply.

25. A quality audit program should begin with a) b) c) d)

a study of the quality documentation system. an evaluation of the work being performed. a report listing findings, the action taken and recommendations. a charter of policy, objectives and procedures.

26. Auditors should report to someone who is independent from a) b) c) d)

the company being audited. management. the function being audited. None of the above.

27. Analysis of data on all product returns is important because a) b) c) d)

failure rates change with length of product usage. changes in design and in customer use are often well reflected. immediate feedback and analysis of product performance becomes available. All of the above.

28. All of the following are considerations when a total quality management (TQM) program is implemented except a) b) c) d)

the use of statistical tools and techniques. a program of continuous quality improvement. the manager responsible for product quality. total involvement from management to production associates.

29. According to Juran, all of the following are widespread errors in perception that have led many managers astray except a) b) c) d)

the work force is mainly responsible for the company's quality problems. workers could do quality work but they lack the motivation to do so. quality will get top priority if upper management so decrees. return on investment is everything.

6


30. An essential technique in making training programs effective is to a) b) c) d)

set group goals. have training classes which teach skills and knowledge required. feed back to the employee meaningful measures of his performance. post results of performance before and after the training program.

31. An engineer has the job of providing a written plan of quality related tasks to his manager, including a detailed timeline, for the following year. Which of the following tools should be used? a) b) c) d)

Histogram Flow Chart Gantt Chart Frequency Distribution

Basic Probability

7

BASIC PROBABILITY 32. The time it takes to answer a technical support line has a continuous uniform distribution over an interval from 17 to 20 minutes. All of the following are true except a) b) c) d)

P(x = 18.5) = 1/2 P(x ≤ 20) = 1 P(17 ≤ x ≤ 18) = 1/3 P(x ≥ 17) = 1

33. For two events, A and B, which one of the following is a true probability statement? a) b) c) d)

P(A or B) = P(A) + P(B) if A and B are independent P(A or B) = P(A) + P(B) if A and B are mutually exclusive P(A and B) = P(A) x P(B) if A and B are mutually exclusive P(A or B) = P(A) x P(B) if A and B are independent

34. What is the probability of getting a head or a tail in 1 toss of a coin? a) b) c) d)

1/16 1/4 1/2 1

35. What is the probability of getting a head and a tail in 2 tosses of a fair coin? And, what is the probability of getting a head and a tail, in that order, in 2 tosses of a fair coin? a) b) c) d)

1/2, 1/2 1/4, 1/4 1/2, 1/4 1/4, 1/2

36. A coin is tossed 10 times. The first 9 tosses come up heads. What is the probability that the 10th toss will come up heads? a) b) c) d)

1/512 1/256 1/32 1/2

37. What is the probability of obtaining exactly 2 heads in 4 tosses of a fair coin? a) 1/4 b) 3/8 c) 1/2 d) 1/6 38. What is the probability of getting a 3 when rolling a single die? ( A die is one of a pair of

8


dice) a) b) c) d)

1/5 3/5 1/6 1/3

39. What is the probability of getting an odd number when rolling a pair of dice? (Spots on the two dice sum to odd number) a) b) c) d)

1/4 1/2 1/3 3/10

40. What is the probability of obtaining a sum of 7 when rolling a pair of dice? a) b) c) d)

1/5 3/5 1/6 1/3

Use the following information to answer questions 41, 42 and 43. The probability is 1/2 that Bob will pass the CQE exam, 1/3 that Amy will pass and 3/4 that Jon will pass. 41. What is the probability that Bob, Amy and Jon will all pass the exam? a) b) c) d)

1/8 4/9 4/11 1/3

42. What is the probability that neither Bob, Amy nor Jon will pass the exam? a) b) c) d)

1/9 7/8 1/12 2/3

43. What is the probability that only one of the three will pass the exam? a) 1/4 b) 1/3 c) 3/4 d) 3/8 44. Four people shoot at a target and the probability that each will hit the target is 1/2 (50%). What is the probability that the target will be hit?

Basic Probability

a) b) c) d)

9

1/16 15/16 1/2 1/4

45. A committee of 5 people is chosen at random from a room that contains 4 men and 6 women. What is the probability that the committee is composed of 2 men and 3 women? a) b) c) d)

1/2 10/21 5/21 1/3

46. A vendor is trying to sell you a box of 50 fuses that contains exactly 5 defective fuses. You select 2 fuses from the box for testing. If both are good you will buy the entire box. If one or both are defective, you will not buy the box. What is the probability that you will buy the box? a) b) c) d)

.7533 .8082 .9769 .8531

47. What is the probability of winning the Super Lotto? (Winning = getting all 6 numbers out of 47) a) b) c) d)

1/10,737,573 1/7,731,052,560 1/3,457,296 1/12,966,821

48. A box contains 12 connectors, 9 good and 3 defective. What is the probability of obtaining exactly 2 good and 1 defective connector in drawing 3 parts from the box without replacement? a) b) c) d)

.4219 .4909 .5022 .6915

10


49. A box contains 12 connectors, 9 good ones and 3 defective ones. What is the probability of obtaining exactly 2 good and 1 defective connector in drawing 3 parts from the box with replacement? a) b) c) d)

.4219 .4909 .5022 .6915

50. You have been asked to sample a lot of 300 units from a vendor whose past quality has been about 2% defective. A sample of 40 pieces is drawn from the lot and you have been told to reject the lot if you find two or more parts defective. What is the probability of rejecting the lot? a) b) c) d)

0.953 0.809 0.191 0.047

Use the following information to answer questions 51 and 52. A company produces capacitors by a process that normally yields 5% defective product. A sample of 4 capacitors is selected. 51. What is the probability that all 4 capacitors are good? a) b) c) d)

.9790 .9213 .8617 .8145

52. What is the probability that all 4 capacitors are defective? a) b) c) d)

.1383 .1855 .0000258 .00000625

Use the following information to answer questions 53, 54, 55 and 56. A company makes ball bearings that are found to be 10% defective in the long run. A sample of 10 bearings is selected. 53. What is the probability that 0 bearings will be defective? a) .3487 b) .3874 c) .4126 d) .1110 54. What is the probability of obtaining exactly 1 defective bearing?

Basic Probability

a) b) c) d)

11

.3487 .3874 .4126 .2574

55. What is the probability of obtaining exactly 3 defective bearings? a) b) c) d)

.0574 .4448 .7361 .1562

56. What is the probability of obtaining more than 1 defective bearing? a) b) c) d)

.3874 .4126 .2639 .2285

57. How many defective connectors would be expected in a sample of 200 parts if the process averages 2% defective? a) b) c) d)

1 2 4 7

58. What is the probability of obtaining exactly 2 defective connectors in a sample of 6 parts if the process averages 2% defective? a) b) c) d)

.0135 .0055 .0009 .0001

59. All of the following are probabilistic events except a) b) c) d)

the number rolled in a game of dice. the number of defects in a random sample. the acceleration of an apple when it drops from a tree. the number of games played in the world series.

Use the following information to answer problems 60, 61 and 62. A company produces integrated circuits (chips) by a process that normally yields 2000 ppm defective product for

12


electrical test requirements (ppm = defective parts per million). A sample of 5 chips is selected and tested. 60. What is the probability that all 5 chips are good? a) b) c) d)

.9900 .9603 .9213 .8563

61. What is the probability that 1 or more chips are defective? a) b) c) d)

.0051 .0009 .0269 .0100

62. What is the probability that more than 1 chip is defective? a) b) c) d)

.01931 .00510 .00008 .01000

Use the following information to answer problems 63, 64 and 65. A capability study was made to determine the defective rate of 28AZ transistors. The study showed the rate to be 5000 ppm. Ten of the transistors were shipped to a customer. 63. What is the probability that the shipment contains no defective transistors? a) b) c) d)

.9511 .9066 .8512 .9213

64. What is the probability that the shipment contains exactly 1 defective transistor? a) b) c) d)

.0001 .0478 .1048 .1165

65. What is the probability that the shipment contains 2 or more defective transistors? a) .0001

Basic Probability

13

b) .0478 c) .0237 d) .0011 Use the following information to answer problems 66, 67 and 68. A circuit board operation yields 2 defects per board on the average. A sample of 1 board is selected at random. 66. What is the probability of finding exactly 2 defects on the selected board? a) b) c) d)

.3522 .2706 .1550 .0295

67. What is the probability of finding less than 2 defects on the selected board? a) b) c) d)

.4060 .6352 .3522 .3849

68. What is the probability of finding more than 2 defects on the selected board? a) b) c) d)

.4060 .2706 .3522 .3235

Use the following information to answer problems 69, 70 and 71. In manufacturing material for automobile seats it was found that each 100-foot roll contained, on average, 2 defects (flaws). A sample of 1 roll is selected at random from the process. 69. What is the probability that the selected roll contains 0 defects? a) b) c) d)

.1353 .2707 .8647 .7293

70. What is the probability that the selected roll contains exactly 1 defect? a) .1353 b) .7293

14


c) .8647 d) .2706 71. What is the probability that the selected roll contains more than 1 defect? a) b) c) d)

.3233 .5941 .7293 .8647

Use the following information to answer problems 72 and 73. A firm that makes T-shirt decals has determined that their process yields, on average, 3 defects per day. Fifty decals are inspected each day. 72. What is the probability of finding exactly 2 defective decals in any given day? (Assume one defect per defective decal.) a) b) c) d)

.7361 .1494 .2240 .0746

73. What is the probability of buying a decal that contains more than 1 defect? a) b) c) d)

.0005 .0042 .0001 .0017

74. A parts dealer buys parts from a warehouse. Parts are made by either Company A or Company B but are not identified as to which company produces them. One company produces all parts in one shipment or lot. On the average, we know: Company A produces 2.5% defective parts. Company B produces 5.0% defective parts. The warehouse states that 70% of parts will come from Company A and 30% from Company B. If the dealer selects 4 parts at random from a lot and finds 1 defective part, what is the probability that the lot was produced by Company A? a) b) c) d)

.4422 .5580 .6915 .3085

Sampling

15

STATISTICS 75. What is the expected value of the random variable x for the following data? x 12 10 14 20 a) b) c) d)

f(x) 0.2 0.5 0.1 0.2

13.6 14.0 12.8 14.5

76. In the standard normal table, what value of z has 5% of the area in the tail beyond it? a) b) c) d)

1.960 1.645 2.576 1.282

77. Which distribution should be used to determine a confidence interval when σ is not known and the sample size is 10? a) b) c) d)

z t F χ2

78. Which of the following methods should be used to test 6 population means for statistical significance? a) b) c) d)

Chi Square Test Analysis of Variance F Test Duncan's Multiple Range Test

79. A sample size of 120 is taken from a process and is represented graphically on a histogram. What is the appropriate number of histogram cells to use? a) 1 - 8 b) 9 - 20 c) 21 - 35 d) 120 80. Which of the following conditions makes it possible for a process to produce a large

16


number of defective units while it is in statistical control? a) b) c) d)

When the specification limits are not set correctly. When the process capability is wider than the tolerance. When unknown external forces affect the process. When the sample size, from which the reject data is found, is too small.

81. For the normal probability distribution, which of the following is true about the relationship among the median, mean and mode? a) b) c) d)

They are all equal to the same value. The mean and mode have the same value but the median is different. Each has a value different from the other two. The mean and median are the same but the mode is different.

82. All of the following statistical techniques can be used to determine the effectiveness of a supplier improvement program except a) b) c) d)

Pareto analysis. x bar and R charts. a PERT chart. a flow chart.

83. A sample of n observations has a mean x and a standard deviation s > 0. If a single observation, which equals the value of the sample mean x , is removed from the sample, which of the following is true? a) b) c) d)

x x x x

and s both change and s remain the same remains the same but s increases remains the same but s decreases

84. The factory installed brake linings for a certain kind of car have a mean lifetime of 60,000 miles with a 6,000 mile standard deviation. A sample of 100 cars has been selected for testing. What is the standard error of x ? (Assume that the finite population correction may be ignored.) a) b) c) d)

60 miles 6000 miles 600 miles 6100 miles

Use the following information to answer problems 85 - 90. A sample of 7 rivets was taken from a shipment of 1000 rivets and the length was measured. The following data are

Sampling

17

obtained: Sample Number

Length (inches)

1 2 3 4 5 6 7

3.1 3.1 3.2 3.7 3.6 3.7 3.1

85. What is the mean length of the rivets? a) b) c) d)

3.20 inches 3.36 inches 4.00 inches 3.65 inches

86. What is the standard deviation of the length of the rivets (estimate of population standard deviation)? a) b) c) d)

0.27 inches 2.16 inches 0.29 inches 2.00 inches

Note: In the following 4 problems, the sample sizes are less than 30 and the t statistics should be used to solve the problems. Analyses of this type usually involve sample sizes of 30 or greater. Handle the problems just as if the sample sizes were greater than 30 and use the z statistics. 87. What percentage of rivets have lengths less than 2.80 inches? a) b) c) d)

2.69% 5.00% 1.22% 3.23%

88. What percentage of rivets have lengths greater than 3.65 inches? a) 17.1%

18


b) 14.2% c) 15.9% d) 7.10% 89. What percentage of rivets have lengths between 3.1 inches and 3.9 inches? a) b) c) d)

89.7% 78.4% 52.5% 99.7%

90. In the shipment of 1000 rivets, how many good parts will we find if a good part is defined as having a minimum of 3 inches and a maximum of 4 inches? a) b) c) d)

999 967 912 878

The following information is used to answer problems 91 - 95. Data are taken from a manufacturing process that produces optical glass. The sample size is 5 parts and the characteristic measured is the diameter of the plates. Sample Number

Diameter (mm)

1

30

2

31

3

29

4

33

5

34

91. What is the mean diameter of the optical glass? a) b) c) d)

31.4 mm 29.0 mm 31.0 mm 34.0 mm

92. What is the standard deviation of the population? a) 1.00 mm b) 2.07 mm

Sampling

19

c) 2.22 mm d) 1.22 mm 93. The specifications for the glass plates are 30.5 ± 2 mm. What percentage of parts made by this company will not meet specifications? a) b) c) d)

32.5% 5.00% 35.0% 37.9%

94. What percentage of parts will be less than 29.5 mm? a) b) c) d)

17.9% 7.21% 15.9% 24.3%

95. What percentage of parts will be greater than 33 mm? a) b) c) d)

78.5% 24.3% 15.9% 22.1%

96. The Zoglen Corporation markets a product, which is a blend of 3 ingredients (A, B, C). If the individual tolerances for the weight of the 3 ingredients are as shown, what should the tolerance be for the net weight of the product? A: 40.5 ± 2.236 grams, B: 30.4 ± 2.000 grams, C: 18.1 ± 1.732 grams a) b) c) d)

89.0 ± 2.443 grams 89.0 ± 3.464 grams 89.0 ± 5.968 grams 89.0 ± 4.732 grams

97. A random sample of size n is to be taken from a large population that has a standard deviation of 1 inch. The sample size is determined so that there will be a 95% chance that the sample average will be within ± 0.1 inch of the true mean. Which of the following values is nearest to the required sample size? a) 385

20


b) 200 c) 100 d) 40 98. All of the following conditions must be met for the process capability to be within the specification limits except a) b) c) d)

Cpk ≥ 1.0 Cp ≥ 1.0 Cp = Cpk a stable process.

99. A value on the abscissa of the t distribution is 1.093. What is the area to the right of this value if the sample size is 11? a) b) c) d)

0.30 0.15 0.05 0.10

100. The spread of individual observations from a normal process capability distribution may be expressed numerically as a) b) c) d)

6 R /d2 2 x A2 R R /d2 D4 R

101. What percentage of data will normally fall within a process capability? a) b) c) d)

99.00% 99.73% 1.00% 0.27%

Use the following information to answer problems 102 - 105. A winding machine wraps wire around a metal core to make small transformers. The design engineers have determined that the nominal number of windings are to be 10,060 with a minimum of 10,025 and a maximum of 10,095. A sample of 300 transformers was selected in a three month period and the wire was unwrapped on each part to determine the number of windings. The results were:

x = 10,052 windings and s = 10 windings

Sampling

102. What is the process capability? a) b) c) d)

10020 - 10100 10052 - 10020 10022 - 10082 10020 - 10060

103. Compute the value of Cp? a) b) c) d)

30.0 10.0 0.67 1.17

104. Compute the value of Cpk? a) b) c) d)

0.90 0.72 3.22 2.67

105. What is the expected percent defective? a) b) c) d)

1.00% 0.35% 2.13% 0.49%

21

22


STATISTICAL INFERENCE

106. Which of the following cannot be a null hypothesis? a) b) c) d)

The population means are equal. p = 0.5 The sample means are equal. The difference in the population means is 3.85.

107. In a sampling distribution which of the following represents the critical region? a) b) c) d)

α β 1-β 1-α

108. In a hypothesis test which of the following represents the acceptance region? a) b) c) d)

α β 1-β 1-α

109. The Chi Square distribution is a) b) c) d)

a distribution of averages. a distribution of variances. a distribution of standard deviations. a distribution of frequencies.

110. Which of the following is a number derived from sample data that describes the data in some useful way? a) b) c) d)

constant statistic parameter critical value

111. A null hypothesis assumes that a process is producing no more than the maximum allowable rate of defective items. What does the type II error conclude about the process? a) b) c) d)

It is producing too many defectives when it actually isn't. It is not producing too many defectives when it actually is. It is not producing too many defectives when it is not. It is producing too many defectives when it is.

Sampling

23

112. A quality engineer wants to determine whether or not there is any difference between the means of the convolute paperboard cans supplied by two vendors, A and B. A random sample of 100 cans is selected from the output of each vendor. The sample from vendor A yielded a mean of 13.59 with a standard deviation of 5.94. The sample from vendor B yielded a mean of 14.43 with a standard deviation of 5.61. Which of the following would be a suitable null hypothesis? a) b) c) d)

µA = µB µA > µB µA < µB µA ≠ µB

113. A chi square test for independence has a 4 x 3 contingency table and a calculated chi square value of 11.5. At a .05 level of significance, which of the following are true about the decision regarding the null hypothesis? a) b) c) d)

7 df, accept Ho 6 df, reject Ho 6 df, accept Ho 11 df, accept Ho

114. A supplier states that the average diameter of optical glass plates is 37.50 mm. The specifications are 37.50 mm ± .40 mm. A shipment of 500 plates is received and a sample of 10 plates is selected and the diameters of each are measured. The following data are obtained: 37.42 mm 37.63

37.84 37.82

37.50 37.78

37.48 37.79

37.75 37.51

Does the entire population have an average diameter of 37.5 mm? Test at a level of significance of .05. a) b) c) d)

t = +2.97, Reject Ho t = +2.26, Accept Ho t = -2.26, Accept Ho t = +1.05, Accept Ho

115. Given: n = 40, p = .20 defects/unit Could this sample have possibly come from a process whose process average is .14 defects/unit? Test at a level of significance of .01. a) z = +1.75, Accept Ho b) z = -.075, Reject Ho c) z = +1.01, Accept Ho d) z = +2.47, Reject Ho 116. A study of 26 families across a certain state indicated that the average family income

24


during 1989 was $33,400 with a Standard Deviation of $3680. Test the hypothesis that the true average income in this state during 1989 was $32,000 against the alternative that it was not $32,000. Test at a level of significance of .05. a) b) c) d)

t = +.900, Accept Ho t = +1.94, Accept Ho t = -2.95, Reject Ho t = +3.16, Reject Ho

117. An inspection yields the following results: n = 20,

x = 10.5 lb.,

s = 1.2 lb.

Could this data have reasonably come from a process which usually has an average of 11.3 lb. or greater? Test at a level of significance of .01. a) b) c) d)

t = +3.10, Reject Ho t = -1.67, Accept Ho t = -2.98, Reject Ho t = +2.10, Accept Ho

118. In a hypothesis test what determines whether the test is one tailed or two tailed? a) b) c) d)

Critical Region Alternate Hypothesis Null Hypothesis α

119. Pooled variance is best described as a) b) c) d)

the variance of a sample that has been pulled from a population. the combination of variances of two or more samples. the combination of variances from multiple samples with dependant means. (n1 - 1) s12 + (n2 - 1)s22 + … + (nn - 1)snn

120. Goodness of Fit is best described as a) b) c) d)

how well two sample data standard deviations vary from each other. a test to determine if two or more samples follow the same distribution. a statistical tool for analyzing data using the χ2 distribution. the comparison of an observed sample distribution with a theoretical distribution.

Sampling

25

SAMPLING

121. The Dodge-Romig Tables are designed to minimize which parameter? a) b) c) d)

AOQL AQL ATI AOQ

122. What are the two unique quantities that determine a single sampling attributes plan? a) b) c) d)

AQL and LTPD Sample Size and Acceptance Number AQL and Producers Risk LTPD and Consumer's Risk

123. Characteristics for which 100% inspection may be practicable include all of the following except a) b) c) d)

dimensions subject to measurements or go/no-go gaging. performance characteristics subject to non-destructive testing. characteristics observable by visual inspection. ultimate physical properties (tensile strength, viscosity).

124. The term AQL as used in sampling inspection, means a) that level of lot quality for which there is a large risk of rejecting the lot. b) the Average Quality Limit. c) the maximum percent defective that can be considered satisfactory as a process average. d) the quality level. 125. Which of the following best describes what an operating characteristic curve shows? a) b) c) d)

The probability of accepting lots of various quality levels by sampling methods. The operating characteristics of a machine. How to operate a machine for best quality results. The probability that a lot contains a certain number of rejects.

126. In ANSI/ASQ Z1.4, the AQL is always determined at what Pa on the OC curve? a) 0.05 b) 0.10 c) 0.90 d) None of the above 127. Which of the following best describes what an average outgoing quality curve shows?

26


a) b) c) d)

The producer's risk. The outgoing quality level versus the incoming quality level. The probability of accepting lots of various quality levels. The consumer's risk.

Use the following information to answer problems 128 - 134. A sampling plan is needed which will satisfy the following requirements: • • • •

Accept acceptable quality product almost all (90%) of the time. Accept rejectable quality product almost none (12%) of the time. Acceptable quality is defined as a process average of 1% defective. Rejectable quality is defined as a process average of 7% defective.

128. What is the value of α? a) b) c) d)

90% 88% 12% 10%

129. What is the value of β? a) b) c) d)

88% 12% 95% 2.5%

130. What is the value of the AQL? a) b) c) d)

3% 99% 1% 97%

131. What is the value of the LTPD? a) b) c) d)

7% 10% 1% 5%

132. What is the sample size that will satisfy the requirements stated above? a) 35

Sampling

27

b) 40 c) 45 d) 50 133. What is the acceptance number that will satisfy the requirements stated above? a) b) c) d)

1 3 5 23

134. What is the approximate value of the AOQL? a) b) c) d)

.0165 .2000 .1375 .9999

Use the following data to answer problems 135 - 137. Given: AQL = .015 (1.5% defective), RQL = .050 (5.0% defective), α = .05, β = .10 135. What is the sample size of the sampling plan? a) b) c) d)

30 75 140 200

136. What is the acceptance number of the sampling plan? a) b) c) d)

1 3 6 9

137. What is the value of AOQL? a) .2000 b) .0160 c) .0107 d) .0029 138. A sampling plan specifies a sample size of 50 and an acceptance number of 3. What is the value of the AQL if α = .05? a) .0165 b) .0107

28


c) .1375 d) .0280 139. A sampling plan specifies a sample size of 50 and an acceptance number of 3. What is the approximate value of the AOQL? a) b) c) d)

.0280 .0107 .0370 .0490

140. All of the following are methods for checking product except a) b) c) d)

constant percentage sampling. random spot checking. no checking at all. safeguard checking.

141. All of the following are characteristics of lot by lot single sampling except a) a sample size is selected randomly from the lot. b) if the number of defects in a sample exceed the acceptance number, the entire lot is rejected. c) rejected lots are immediately scrapped. d) if the number of defects in a sample do not exceed the acceptance number, the entire lot is accepted. 142. All of the following are characteristics of continuous sampling except a) b) c) d)

it is only used where product flow is discrete. the clearing number must be specified. the amount of product checked may vary. the product is not able to be formed into lots.

143. The following parameters are required for sampling plan construction using the Binomial Nomograph except a) b) c) d)

AQL. RQL. α. Sample size.

Control Charts

CONTROL CHARTS

144. The primary use of a control chart is to a) b) c) d)

detect assignable causes of variation in the process. detect non-conforming product. measure the performance of all quality characteristics of a process. detect the presence of random variation in the process.

145. np charts are based on which distribution? a) b) c) d)

Poisson Binomial Shewhart Exponential

146. c and u charts are based on which distribution? a) b) c) d)

Poisson Binomial Shewhart Exponential

147. Why do x charts always follow a normal distribution? a) b) c) d)

Variance The Law of Large Numbers The Central Limit Theorem The Addition Theorem

148. When used together for variables data, which of the following pair of quantities is the most useful in preparing control charts? a) b) c) d)

AQL, p P, n x, R R, σ

149. A process is in control at x = 100, R = 7.3 with n = 4. If the process level shifts to 101.5, with the same R , what is the probability that the next x point will fall outside the old control limits? a) .016 b) .029 c) .122 d) .360 150. A process is checked by inspection at random samples of 4 shafts after a polishing

29

30


operation, x and R charts are maintained. A person making a spot check picks out 2 shafts, measures them accurately, and plots the value of each on the x chart. Both points fall just outside the control limits. He advises the department foreman to stop the process. This decision indicates that a) b) c) d)

the process level is out of control. both the level and dispersion are out of control. the process level is out of control but not the dispersion. the person is not using the chart correctly.

151. The hardness of rivets is normally distributed with µ = 60.0 and standard deviation σ = 1.2. What are the 3 sigma control limits for the x chart using samples of size 5? a) b) c) d)

57.95, 58.0, 58.38, 56.50,

62.15 62.0 61.62 63.5

152. A possible cause of a cycle pattern in a control chart include all of the following except a) b) c) d)

seasonal effects. differences between suppliers. different inspectors. new supplier.

Use the following information to answer problems 153 - 158. The following data are obtained from measuring the length of a metal bracket. Sample #1 x = 1.51" R = .03" n=5

Sample #2 x = 1.50" R = .02" n=5

Sample #3 x = 1.52" R = .04" n=5

153. What are the values of x , UCL x and LCL x for an x chart? a) b) c) d)

1.500", 1.515", 1.515", 1.525",

1.517", 1.532", 1.491", 1.532",

1.443" 1.498" 1.336" 1.498"

154. What are the values of R , UCLR, LCLR for an R chart? a) .020", .9970", .012" b) .025", .0634", -.012" c) .030", .0634", 0 d) .030", 0, -.062 155. What are the values of the sample size and number of samples?

Sample #4 x = 1.53" R = .03" n=5

Control Charts

a) b) c) d)

31

5, 5 4, 5 5, 4 4, 4

156. What is the standard error? a) b) c) d)

.0012" .0058" .0632" .0729"

157. What is the standard deviation of the individual data points? a) b) c) d)

.013" .030" .010" 1.52"

158. What is the process capability for the individual data points? a) b) c) d)

1.498" to 1.532" 1.500" to 1.530" 0.020" to 0.040" 1.476" to 1.554"

Use the following information to answer questions (problems) 159 - 162. The following data are the result of inspecting aircraft seat belt buckles. Number of buckles in the sample 50 60 40 50 55 55 50 60

Number of defective buckles 2 3 1 2 0 1 0 1

32


159. What is the average percent defective ( p )? a) b) c) d)

.833 .024 .009 .997

160. What are the upper and lower control limits? a) b) c) d)

.087, .024, 1.50, .087,

-.024 0 .013 0

161. What is the appropriate type of control chart for these data? a) b) c) d)

u chart p chart c chart np chart

162. An additional sample of 75 buckles contains 4 defectives. Is this sample within the control limits? a) b) c) d)

Yes, p = .053 Yes, p = .024 No, p = .053 No, p = .024

Reliability

33

RELIABILITY

163. A reliability data system usually implies collecting data on a) b) c) d)

process machine downtime. product failures and operating time. maintenance costs. repair times.

164. Which of the following best describes the failure rate in the wear out phase of the bathtub curve? a) b) c) d)

Constant Random Improves Gets Worse

165. The greatest contribution of a reliability effort is made in the a) b) c) d)

design area. manufacturing area. shipping area. field service area.

166. Reliability prediction is a) the process of estimating performance. b) the process of estimating the probability that a product will perform its intended function for a stated time. c) the process of telling "how you can get there from here." d) All of the above. 167. Maintainability is a) the probability of a system being restored to functional operation within a given period of time. b) performing adequate maintenance on a system. c) probability of survival of a system for a given period of time. d) maintaining a machine in satisfactory working condition. 168. A set of components has a MTBF of 1000 hours. What percentage will fail if the components are tested for 500 hours? a) 25% b) 39% c) 61% d) 50% 169. What is the reliability of a system at 850 hours, if the average usage on the system was

34


400 hours for 1650 items and the total number of failures was 145? Assume an exponential distribution. a) b) c) d)

0% 36% 18% 83%

Use the following information to answer problems 170, 171 and 172. λ A = .001,

λ B = .002,

λ C = .003,

λ D = .0025,

A

C

B

D

t = 100 hours

170. What is the reliability of the system? a) b) c) d)

.9302 .4000 .8805 .4403

171. What is the MTBF for components A, B, C and D respectively? a) b) c) d)

500, 400, 333, 999 1000, 750, 333, 500 700, 500, 333, 500 1000, 500, 333, 400

172. What is the probability that component C will fail before 100 hours. a) b) c) d)

.5184 .2592 .3460 .1296

173. The MTBF of a complex piece of repairable radar equipment is determined to be 950 hours. The equipment has been in continuous operation for 150 hours. What is the probability that the equipment will continue to operate without failure for at least another 375 hours? a) b) c) d)

0.5754 0.6376 0.6739 0.8521

Regression and Correlation

35

REGRESSION AND CORRELATION

174. All of the following about multiple regression are true except a) extrapolation beyond the region of observations can lead to erroneous predictions. b) that it always involves at least 3 variables. c) that it involves one independent and two or more dependant variables. d) that it may be linear in the coefficients. 175. All of the following statements are true about a correlation coefficient except a) b) c) d)

it is a mathematical measure of the degree of correlation. negative correlation will result in a negative correlation coefficient. a correlation coefficient of 0 indicates no correlation. a correlation coefficient of ±1 indicates a cause and effect relationship.

176. A study was conducted on the relationship between the speed of different cars and their gasoline mileage. The correlation coefficient was found to be 0.35 from the study. Later, it was discovered that there was a defect in the speedometers and they had all been set 5 miles per hour too fast. The correlation coefficient was computed using the correct scores. What is the new correlation coefficient? a) b) c) d)

0.30 0.35 0.40 0.45

Use the following information to answer problems 177 - 182. Quality Training (Costs per employee per year)

Quality Cost Savings per employee per year

$500

$2700

800

3500

1000

4600

1200

7000

1400

9500

2000

11500

177. What is the intercept of the regression line?

36


a) b) c) d)

42.3 -100.7 -1027.4 567.52

178. What is the slope of the regression line? a) b) c) d)

6.52 3.21 24.4 79.3

179. What is the correlation coefficient? a) b) c) d)

.52 .23 .48 .97

180. What is the formula for the regression line? a) b) c) d)

Y = 6.52 -1027.42x Y = 79.3 + 42.3x Y = -100.7 + 24.4x Y = -1027.42 + 6.52x

181. What will the projected quality costs savings be when $1700 per employee is invested in quality training? a) b) c) d)

$97,000.00 $10,056.58 $152,231.50 $212,520.63

182. If the cost savings were $8000 per employee, what are the probable training costs per employee? a) b) c) d)

$1,000.00 $1,523.33 $1,384.57 $682.80

Cost of Quality

37

COST OF QUALITY

183. The basic objective of a quality cost program is to a) b) c) d)

identify the source of quality failures. interface with the accounting department. improve the profit of your company. identify quality control department costs.

184. Analysis of quality costs consists of a) b) c) d)

reviewing manpower utilization against standards. evaluating seasonal productivity. establishing management tools to determine net worth. examining each cost element in relation to other elements and the total.

185. In selecting a base for measuring quality costs, which of the following should be considered? a) b) c) d)

Sensitivity to increases and decreases in production schedule. Affects by seasonal product sales. Sensitivity to material price fluctuations. All of the above.

186. Which of the following quality cost indices is likely to have the greatest appeal to top management as an indicator of relative cost? a) b) c) d)

Quality cost per unit of product Quality cost per hour of direct production labor Quality cost per unit of processing cost Quality cost per unit of sales

187. If prevention costs are increased to pay for engineering work in quality control, and this results in a reduction in the number of product defects, this yields a reduction in a) b) c) d)

appraisal costs. quality costs. failure costs. manufacturing costs.

38


188. A process that sorts good product from defective product falls into which of the following quality cost categories? a) b) c) d)

Prevention Appraisal Internal failure External failure

189. Cost of calibrating test and inspection equipment would be included in a) b) c) d)

prevention costs. appraisal costs. failure costs. material-procurement cost.

190. The cost of writing instructions and operating procedures for inspection and testing should be charged to a) b) c) d)

prevention costs. appraisal costs. internal failure costs. external failure costs.

191. Failure costs include costs due to a) b) c) d)

quality control engineering. inspection set-up for tests. certification of special process suppliers. supplier analysis of non-conforming hardware.

192. Which of the following is least likely to be reported as a failure related cost? a) Sorting lots rejected by a sampling procedure. b) Downtime caused by late delivery of a purchased part rejected by the supplier's final inspection. c) Repair of field failures. d) Re-testing of a repaired product. 193. Which of the following activities is not normally charged as a preventive cost? a) b) c) d)

Quality Training Design and Development of Quality Measurement Equipment Quality Planning Laboratory Acceptance Testing

Cost of Quality

39

194. In deciding whether sampling inspection of parts would be more economical than 100% inspection, you need to determine all of the following except a) b) c) d)

cost of inspecting the parts. cost of correcting defective parts. cost of not finding defective parts. cost of improving the production process.

195. Quality cost trend analysis is facilitated by comparing quality costs to a) b) c) d)

manufacturing costs over the same time period. cash flow reports. appropriate measurement bases. QC department budget.

196. For a typical month, the 3D Manufacturing Company identified and reported the following quality costs: Inspection wages…………………………………………….. Quality planning………………………………………………. Source inspection…………………………………………….. In-plant scrap and rework………………………………….… Final product test…………………………….…..…………… Retest and troubleshooting………………………………….. Field warranty cost…………………………………………… Evaluation and processing of deviation requests………… What is the total failure cost from this data? a) b) c) d)

$244,000 $261,000 $205,000 $332,000

197. Which of the following is a typical external failure cost? a) b) c) d)

Material Repair Material Scrap Customer Appraisal Training

$2,000 $4,000 $2,000 $88,000 $110,000 $39,000 $205,000 $6,000

40


Use the following information to answer problems 198 - 201. A manufacturer produces an electronic memory device. The following costs are incurred on a yearly basis. Activity Research and Development Manufacturing Engineering Quality Engineering Plant Facilities Training Direct Labor Process Control (Labor) Supervision Materials Inspection Scrap Costs Rework Costs

198. What are the annual Prevention costs? a) b) c) d)

$10,000 $6,000 $4,000 $18,000

199. What are the annual Appraisal costs? a) b) c) d)

$18,000 $22,000 $12,000 $33,000

200. What are the annual Failure costs? a) b) c) d)

$18,000 $66,000 $33,000 $27,000

201. What is the percentage of quality costs to total costs a) b) c) d)

47% 8% 10% 14%

Cost $50,000 40,000 5,000 60,000 1,000 80,000 5,000 30,000 90,000 13,000 21,000 12,000

Metrology and Calibration

41

DESIGN OF EXPERIMENTS

202. In performing an Analysis of Variance for a single factor experiment, a fundamental assumption which is made is that the factor a) b) c) d)

means are equal. means are unequal. variances are equal. variances are unequal.

203. To state that a model in an experimental design is fixed indicates that a) b) c) d)

the levels used for each factor are the only ones of interest. the levels were chosen from a fixed population. the equipment from which the data are collected must not be moved. the factors under consideration are qualitative.

204. An experiment with two factors, in which all levels of one variable are run at each level of the second variable, is called a a) b) c) d)

One-way experiment. Latin square experiment. factorial experiment. fractional factorial experiment.

205. A two-way Analysis of Variance has r levels for the first variable and c levels for the second variable with 2 observations per cell. The degrees of freedom for interaction is a) b) c) d)

2(r x c) (r - 1)(c - 1) rc - 1 2(r - 1)(c - 1)

206. An analysis of variance results in a calculated F value of F(10, 12) = 2.75. What is the level of significance? a) b) c) d)

p < 0.01 p = .01 p = .05 p > .05

207. A 32 experiment means that we are considering

42


a) b) c) d)

two levels of three factors. two dependent variables and three independent variables. two go/no-go variables and three continuous variables. three levels of two factors.

208. Which of the following distributions is needed to evaluate the results of analysis of variance (ANOVA)? a) b) c) d)

binomial distribution Chi Square distribution F distribution Z distribution

209. The primary advantage of the Latin Square design compared to the factorial design, is that a) b) c) d)

it requires less data. it eliminates the need for interaction analysis. it allows higher significance levels. it does not require homogeneity of variance.

210. Consider the SS and MS columns of an Analysis of Variance table for a single factor design. The appropriate ratio for testing the null hypothesis of no treatment effect is a) b) c) d)

SS treatments divided by SS residual. MS treatments divided by MS residual. SS treatments divided by MS residual. MS treatments divided by SS residual.

211. A completely randomized design is best described as a) a design in which the linear equation coefficients are sorted randomly. b) a design in which each factor is selected randomly. c) a design in which all treatments are assigned to the experimental units in a completely random manner. d) a design in which certain factors are randomly selected for analysis.

Use the following information to answer problems 212 - 216. The following factorial experiment was conducted to determine the effect of study time and study material used on test scores. The numbers in the box represent the test scores. You will need to construct the ANOVA table to complete all of the questions.


Study Material Used

1 2

Hours of Study 4 6 60, 85 77, 92 55, 65 60, 80

212. How many factors are used in this experiment? a) b) c) d)

2 4 6 8

213. How many levels are being examined for each factor? a) b) c) d)

2 4 6 8

214. What is the sum of squares for the residual? a) b) c) d)

675.0 242.0 364.5 1283.5

215. Which factor has the greatest impact on the outcome? a) b) c) d)

Study Hours Study Material Used Interaction Residual

216. Which of the following factors are significant? a) b) c) d)

Study Hours and Interaction Study Material Used Interaction None of the above

43

44


METROLOGY AND CALIBRATION

217. The key to designing an effective calibration program is to balance a) b) c) d)

reliability and performance metrics. cost and quality considerations. test and result considerations. cost and variance metrics.

218. Reliability metrics can be used to measure equipment a) b) c) d)

availability. waste. performance. redundancies.

219. Which of the following terms is defined as the probability that measurement equipment will be found to be in tolerance after a specified period of time? a) b) c) d)

Availability Activism Reproducibility Reliability

220. Suppose a test procedure is performed in order to determine the value of a product measurement. What is the order of equipment type, from lowest to highest accuracy, if NIST traceability is to be achieved? a) b) c) d)

Working standards, ISO standards, transfer standards, reliability standards Primary standards, ISO standards, transfer standards Primary standards, transfer standards, working standards Working standards, transfer standards, primary standards

221. Which procedures are performed on product and test equipment in order to determine if predefined specifications have been met? a) b) c) d)

Inspection Identification Labeling Traceability

222. Which of the following is used to qualify measurement instruments in reference to national or internationally recognized standards?


a) b) c) d)

Reliability Traceability Variability Dependability

223. A measurement standard with a stated uncertainty of 0.1 V, 95% confidence level, is used to test a device that has a nominal value of 100 V, and a tolerance of ±10 V. What is the TUR for this measurement? a) b) c) d)

10 100 9.5 95

224. A requirement of the ISO 9000 series standards is that all M&TE used for product inspection a) b) c) d)

is calibrated at least annually. meets Mil-Std 45662A requirements. is identified and labeled appropriately. has passed the null test.

225. What ISO 9001 element addresses equipment calibration? a) b) c) d)

4.1 4.5 4.11 4.17

45

46


QREVIEW ANSWERS 1. b 2. d 3. b 4. c 5. b 6. b 7. a 8. c 9. c 10. a 11. b 12. d 13. a 14. c 15. d 16. b 17. b 18. b 19. c 20. c 21. c 22. c 23. c 24. d 25. d 26. c 27. d 28. c 29. d 30. c 31. c 32. a 33. b 34. d 35. c 36. d 37. b 38. c 39. b 40. c 41. a 42. c 43. d 44. b 45. b

46. b 47. a 48. b 49. a 50. c 51. d 52. d 53. a 54. b 55. a 56. c 57. c 58. b 59. c 60. a 61. d 62. c 63. a 64. b 65. d 66. b 67. a 68. d 69. a 70. d 71. b 72. c 73. d 74. b 75. c 76. b 77. b 78. b 79. b 80. b 81. a 82. d 83. c 84. c 85. b 86. c 87. a 88. c 89. b 90. d

91. a 92. b 93. d 94. a 95. d 96. b 97. a 98. c 99. b 100. a 101. b 102. c 103. d 104. a 105. b 106. c 107. a 108. d 109. b 110. b 111. b 112. a 113. c 114. a 115. c 116. b 117. c 118. b 119. b 120. d 121. c 122. b 123. d 124. c 125. a 126. d 127. b 128. d 129. b 130. c 131. a 132. d 133. a 134. a 135. d

136. c 137. b 138. d 139. c 140. d 141. c 142. a 143. d 144. a 145. b 146. a 147. c 148. c 149. a 150. d 151. c 152. d 153. b 154. c 155. c 156. b 157. a 158. d 159. b 160. d 161. b 162. a 163. b 164. d 165. a 166. b 167. a 168. b 169. d 170. c 171. d 172. b 173. a 174. c 175. d 176. b 177. c 178. a 179. d 180. d

181. b 182. c 183. c 184. d 185. d 186. d 187. c 188. c 189. a 190. a 191. d 192. b 193. d 194. d 195. c 196. d 197. a 198. b 199. a 200. c 201. d 202. c 203. a 204. c 205. b 206. c 207. d 208. c 209. a 210. b 211. c 212. a 213. a 214. a 215. b 216. d 217. b 218. c 219. d 220. d 221. a 222. b 223. b 224. c 225. c


47

PART II - QREVIEW SOLUTIONS BASIC QUALITY CONCEPTS

1. Certification of a company to the ISO 9002 standard does not include which of the following requirements? b) Design control Servicing, design control, contract review and internal quality audits are all requirements of ISO 9001. The only change in ISO 9002 is the elimination of design control. ISO 9002 is intended for organizations that do not design products. 2. A quality control program is considered to be d) a system of activities to provide quality of products and services. A quality control program is more than a set of procedures, step by step lists of check points or summaries of policies and data. It includes all activities performed to ensure that products and services meet quality requirements. 3. The "quality function" of a company is best described as b) that collection of activities through which "fitness for use" is achieved. The quality function of a company is a collection or system of activities to ensure quality. “Fitness for use” is an outcome of all these activities. 4. In preparing a product quality policy for your company, you should do all of the following except c) include procedural matters and functional responsibilities. Procedures are separate documents, but the quality policy may state that procedures need to be included in the system. Functional responsibilities are also usually a separate document but the quality policy may have a provision that they must exist. 5. What natural phenomenon created the necessity to control product and process quality? b) Variation It is because of variation that the science of quality control exists. All measurements exhibit variation and this is why it is necessary to control product and process quality. 6. The three basic elements of a quality system are

48


b) Quality Management, Quality Control and Quality Assurance Quality Management is the means of implementing and carrying out quality policy. Quality Control monitors and improves the conformance of products, processes and services. Quality Assurance takes action to assure that products or services will satisfy the specified requirements. 7. What are the two basic categories of quality? a) Design and Conformance Quality Design quality defines the level of characteristics that go into the product. Conformance quality measures how well the design intent is met. 8. The Law of Large Numbers states that c) individual occurrences are unpredictable and group occurrences are predictable. The number of marriages, births and deaths in the United States next year can be predicted with some degree of accuracy, but exactly who will get married, who will be born or who will die cannot be predicted. This concept can be applied to a manufacturing process. A statistical study can determine that products from a certain process are on average two percent defective, but in any sample, the specific parts that will be defective cannot be predicted. 9. Statistical quality control is best described as c) the study of the characteristics of a product or process, with the help of numbers, to make them behave the way we want them to behave. The word statistical means having to do with numbers, or more specifically, with drawing conclusions from numbers. The word quality means much more than the goodness or defectiveness of the product. It refers to the qualities or characteristics of the product or process being studied. The word control means to keep something within boundaries or to regulate it so that its outcome may be predicted with some degree of accuracy.


49

10. Which of the following is the most important element in Statistical Quality Control? a) The Feedback Loop In statistical process control, the feedback loop is between the process control function and the device that regulates the process or the person responsible for adjustments. Continuous feedback and the appropriate corrective action are what make statistical quality and process control work to achieve the desired results. 11. When measurements are accurate and precise, b) the data are clustered closely around the central value. Accurate measurements fall within an acceptable range but are distributed randomly within that range. Precise measurements have minimum variation, which means that they are clustered close together. Accurate and precise measurements would then imply that the data is clustered closely around the central value. 12. All of the following are included in a quality system except d) employee salaries. Document control, corrective action and management responsibility are all essential parts of a quality system according to the ISO 9001/Q9001 quality systems standards. Companies usually develop a specific policy for employee salaries. 13. Which of the following best describes a statistical distribution? a) A model that shows how data are distributed over a range of measurements. Because of variation between measurements of individual parts, data when plotted will form a distribution. A distribution model describes how the data are dispersed. A plot of the distribution will show a center value and the range of measurements. 14. Which of the following are two types of data used in statistical quality control? c) Variables and Attributes Data Data classified as good/bad, pass/fail, go/no-go, etc., are called attribute or discrete data. When actual measurements are taken and recorded, the data are called variables or continuous data.

15. The primary reason for evaluating and maintaining surveillance over a supplier's quality

50


program is to d) make sure the supplier's quality program is functioning effectively. Purchased parts have a significant impact on the quality of a final product. It is important that supplier's have an effective quality program in place. An effective quality program and specific product quality are closely related. 16. Which one of the following are ISO 9001 requirements? b) Quality Manual A quality manual will usually address the elements of the ISO standard. Specific quality tools and instructions are not included as part of the ISO requirements. 17. Which of the following does not generate product-quality characteristics? b) Inspector A designer, machinist and equipment engineer all play a role in the manufacture of a product and therefore have the ability to generate product-quality characteristics. An inspector only examines a product and cannot alter it in any way. 18. Incoming material inspection is based most directly on b) purchase order requirements. In since incoming inspection is concerned with the quality of purchased products, the requirements should be listed on the purchase order. 19. The acronym ISO means c) equal. The International Organization for Standardization is also known as ISO. ISO is not an acronym for this organization but it is a Latin word which translates to equal. 20. Products should be subjected to tests which are designed to c) approximate the conditions to be experienced in customer's application. Products need to be tested in the environment where they are used. Various other types of tests can be performed for prove in, but every condition varies to some degree and the best way to prove a product is under actual working conditions. 21. The advantage of a written procedure is


51

c) it is a perpetual coordination device. A written procedure allows everyone concerned to know exactly what is required. It may always be referred to when coordinating activities. Unwritten procedures usually cause confusion because of communication failure. 22. In spite of the Quality Engineer's best efforts, situations may develop in which his decision is overruled. The most appropriate action would be to c) document findings, report them to superiors and move on the next assignment. Human conflict can be minimized but never eliminated. There may be times when engineering decisions are overruled or employee suggestions not adopted. In these cases, the findings should be documented for possible review at a later date, then the engineer or employee can move on to the next assignment. In any conflict, whether it is between management and employees, between management and unions, or between employees, good judgement must be used. 23. If a test data does not support a Quality Engineer's expectations, the best thing to do is c) re-evaluate the expectations of the test based upon the data. Objective data that has been collected should never be altered or the analysis of the data should never be modified to fit expectations. The expectations were just probably not realizable for the application under test. 24. In case of conflict between contract specifications and shop practice, d) contract specifications normally apply. Contract specifications take precedence over all other practices. Any deviation from the specifications must be negotiated with the customer. 25. A quality audit program should begin with d) a charter of policy, objectives and procedures. All quality programs must have clear policies, objectives and documented procedures. Without these, the effort will be aimless. 26. Auditors should report to someone who is independent from c) the function being audited. It is good policy for auditors to not have a vested interest in the activity being audited. For example, the fox guarding the chicken coop will usually result in fewer chickens. 27. Analysis of data on all product returns is important because d) All of the above.

52


Analysis of data on all product returns is important because it may provide feedback as to how a customer's specific application of the product may affect the product quality. It will also help to determine failure rates based on the longevity of the product. Immediate feedback and analysis may be obtained for the specific products returned as opposed to collecting data over some specified time and generalizing product problems. 28. All of the following are considerations when a total quality management (TQM) program is implemented except c) the manager responsible for product quality. No one person is responsible for product quality in a TQM program. TQM involves the use of statistical techniques, continuous improvement and total involvement by all. 29. According to Juran, all of the following are widespread errors in perception that have led many managers astray except d) return on investment is everything. Juran has listed many widespread errors in perception in his books (Juran on Leadership for Quality, 1989), but does not include "return on investment is everything." This statement may not necessarily lead managers astray. 30. An essential technique in making training programs effective is to c) feedback to the employee meaningful measures of his performance. Feedback lets an employee know whether the concepts learned from training have been grasped. This helps the employee know where their strengths and weaknesses are. It also can help the employee gain confidence in their abilities. 31. An engineer has the job of providing a written plan of quality related tasks to his manager, including a detailed timeline, for the following year. Which of the following tools should be used? c) Gantt Chart Gantt charts are typically horizontal bar charts, where each bar represents a task that spans from the beginning time to the ending time. Histograms and frequency distributions are analytical tools and a flow chart is typically used for relationship modeling.

Basic Probability

53

BASIC PROBABILITY

32. The time it takes to answer a technical support line has a continuous uniform distribution over an interval from 17 to 20 minutes. All of the following are true except a) P(x = 18.5) = 1/2 In a continuous distribution, the probability of exact point values cannot be calculated. Probability statements must specify a range less than or greater than a point value.

33. For two events, A and B, which one of the following is a true probability statement? b) P(A or B) = P(A) + P(B) if A and B are mutually exclusive For any two events, P(A or B) = P(A) + P(B) - P(A & B). For independent events P(A & B) = P(A) x P(B) and for mutually exclusive events P(A & B) = 0 as illustrated below. Mutually Exclusive Independent

A

A

A&B B

B

34. What is the probability of getting a head or a tail in 1 toss of a coin? d) 1 P(Head) or P(Tail) = P(H) + P(T) = 1/2 + 1/2 = 1 35. What is the probability of getting a head and a tail in 2 tosses of a fair coin? And, What is the probability of getting a head and a tail, in that order, in 2 tosses of a fair coin? c) 1/2, 1/4 P(Head & Tail in any order) = P(Head) & P(Tail) or P(Tail) & P(Head) = P(Head) x P(Tail) + P(Tail) x P(Head) = 1/2 x 1/2 + 1/2 x 1/2 = 1/4 + 1/4 = 1/2 P(Head & Tail in that order) = P(Head) & P(Tail) = P(Head) x P(Tail) = 1/2 x 1/2 = 1/4 36. A coin is tossed 10 times. The first 9 tosses come up heads. What is the probability that

54


the 10th toss will come up heads? d) 1/2 All tosses are independent events. The probability of getting a head on the 10th toss is the same as the probability of getting a head on any toss, which is 50%. 37. What is the probability of obtaining exactly 2 heads in 4 tosses of a fair coin? b) 3/8 Possible events:

Toss 1 H H H H T T T T H T H T H T H T

Toss 2 H H H T T T T H T H T H H T T H

Toss 3 H H T T T T H H H T T H T H H T

Toss 4 H T T T T H H H T H H T H T H T

The total number of events is 16 and the number of favorable events is 6, therefore P(2 Heads in 4 tosses) = number of favorable events / total number of events = 6/16 = 3/8 38. What is the probability of getting a 3 when rolling a single die? ( A die is one of a pair of dice) c) 1/6 A single die has 6 sides numbered 1 - 6, therefore P(rolling a 3) = number of favorable events / total number of events = 1/6

Basic Probability

55

39. What is the probability of getting an odd number when rolling a pair of dice? (Spots on the two dice sum to odd number) b) 1/2 Possible events:

1,1 2,1 3,1 4,1 5,1 6,1

1,2 2,2 3,2 4,2 5,2 6,2

1,3 2,3 3,3 4,3 5,3 6,3

1,4 2,4 3,4 4,4 5,4 6,4

1,5 2,5 3,5 4,5 5,5 6,5

1,6 2,6 3,6 4,6 5,6 6,6

The total number of events is 36 and the number of favorable events is 18, therefore P(odd number) = number of favorable events / total number of events = 18/36 = ½ 40. What is the probability of obtaining a sum of 7 when rolling a pair of dice? c) 1/6 Possible events:

1,1 2,1 3,1 4,1 5,1 6,1

1,2 2,2 3,2 4,2 5,2 6,2

1,3 2,3 3,3 4,3 5,3 6,3

1,4 2,4 3,4 4,4 5,4 6,4

1,5 2,5 3,5 4,5 5,5 6,5

1,6 2,6 3,6 4,6 5,6 6,6

The total number of events is 36 and the number of favorable events is 6, therefore P(rolling a 7) = number of favorable events / total number of events = 6/36 = 1/6 Use the following information to answer questions 41, 42 and 43. The probability is 1/2 that Bob will pass the CQE exam, 1/3 that Amy will pass and 3/4 that Jon will pass. 41. What is the probability that Bob, Amy and Jon will all pass the exam? a) 1/8 P(all will pass) = P(Bob will pass) x P(Amy will pass) x P(Jon will pass) = 1/2 x 1/3 x 3/4 = 1/8

56


42. What is the probability that neither Bob, Amy nor Jon will pass the exam? c) 1/12 P(none will pass) = [1 - P(Bob will pass)] x [1 - P(Amy will pass)] x [1 - P(Jon will pass)] = (1 - 1/2) x (1 - 1/3) x (1 - 3/4) = 1/2 x 2/3 x 1/4 = 1/12 43. What is the probability that only one of the three will pass the exam? d) 3/8 P(only one will pass) = [P(Bob will pass) & P(Amy will fail) & P(Jon will fail)] or [P(Bob will fail) & P(Amy will pass) & P(Jon will fail)] or [P(Bob will fail) & P(Amy will fail) & P(Jon will pass)] = [1/2 x (1 - 1/3) x (1 - 3/4)] + [(1 - 1/2) x 1/3 x (1 - 3/4)] + [(1 - 1/2) x (1 - 1/3) x 3/4] = 2/24 + 1/24 + 6/24 = 3/8 44. Four people shoot at a target and the probability that each will hit the target is 1/2 (50%). What is the probability that the target will be hit? b) 15/16 The target can be hit 0 times, 1 time, 2 times, 3 times or 4 times, therefore P(0) + P(1) + P(2) + P(3) + P(4) = 1 P(0) = 1/2 x 1/2 x 1/2 x 1/2 = 1/16 P(hitting target) = P(1) + P(2) + P(3) + P(4) = 1 - P(0) = 1 - 1/16 = 15/16 45. A committee of 5 people is chosen at random from a room that contains 4 men and 6 women. What is the probability that the committee is composed of 2 men and 3 women? b) 10/21 Use the hypergeometric probability distribution:  4  6      2  3 (6)(20) 120 10 = = = =.4762 P(2 men and 3 women) = 252 252 21  10    5 46. A vendor is trying to sell you a box of 50 fuses that contains exactly 5 defective fuses. You select 2 fuses from the box for testing. If both are good you will buy the entire box. If one or both are defective, you will not buy the box. What is the probability that you will

Basic Probability

57

buy the box? b) .8082 Use the hypergeometric probability distribution: P(buying box) = P(selecting 2 good fuses) =

 45  5     2   0  50    2

=

(990)(1) =.8082 1225

47. What is the probability of winning the Super Lotto? (Winning = getting all 6 numbers out of 47) a) 1/10,737,573 Use the hypergeometric probability distribution:

 6  41 P(winning) =     6  0 

(1)(1)  47  = .00000009313  =  6  10,737,573

48. A box contains 12 connectors, 9 good ones and 3 defective ones. What is the probability of obtaining exactly 2 good and 1 defective connector in drawing 3 parts from the box without replacement? b) .4909 Use the hypergeometric probability distribution:

P(2 good and 1 defective) =

 9  3      2  1  12    3

=

(36)(3) = 108 = 27 =.4909 220

220

55

49. A box contains 12 connectors, 9 good ones and 3 defective ones. What is the probability of obtaining exactly 2 good and 1 defective connector in drawing 3 parts from the box with replacement? a) .4219 Use the Binomial probability distribution: 2

1

 3  9   3  P(2 good and 1 defective) =       = 3(.5625)(.25) =.4219  2  12   12  50. You have been asked to sample a lot of 300 units from a vendor whose past quality has been about 2% defective. A sample of 40 pieces is drawn from the lot and you have been told to reject the lot if you find two or more parts defective. What is the probability of rejecting the lot?

58


c) 0.191 Use the Binomial probability distribution: P(rejecting lot) = P(2) + P(3) + … + P(40) = 1 – [P(0) + P(1)]  40  P(0) =  (.02) 0 (.98) 40 = .4457 0

 40  P(1) =  (.02)1 (.98) 39 = .3638  1 P(rejecting lot) = 1 – [.4457 + .3638] = 1 - .8095 = 0.191 Use the following information to answer questions 51 and 52. A company produces capacitors by a process that normally yields 5% defective product. A sample of 4 capacitors is selected. 51. What is the probability that all 4 capacitors are good? d) .8145 Use the Binomial probability distribution:  4 4 0 P(all 4 are good) =   (.95) (.05) = 1(.8145)(1) =.8145  4

52. What is the probability that all 4 capacitors are defective? d) .00000625 Use the Binomial probability distribution:  4 4 0 P(all 4 are defective) =   (.05) (.95) = 1(.00000625)(1) =.00000625  4

Basic Probability

59

Use the following information to answer questions 53, 54, 55 and 56. A company makes ball bearings that are found to be 10% defective in the long run. A sample of 10 bearings is selected. 53. What is the probability that 0 bearings will be defective? a) .3487 Use the Binomial probability distribution:  10 0 10 P(0) =   (.10) (.90) = 1(1)(.3487) =.3487  0

54. What is the probability of obtaining exactly 1 defective bearing? b) .3874 Use the Binomial probability distribution:  10 1 9 P(1) =   (.10) (.90) = 10(.10)(.3874) =.3874  1 55. What is the probability of obtaining exactly 3 defective bearings? a) .0574 Use the Binomial probability distribution:  10 3 7 P(3) =   (.10) (.90) = 120(.001)(.4783 ) =.0574  3 56. What is the probability of obtaining more than 1 defective bearing? c) .2639 Use the Binomial probability distribution: P(more than 1 defective) = P(2) + P(3) + … + P(10) = 1 - [P(0) + P(1)] = 1 - [.3487 + .3874] = .2639 57. How many defective connectors would be expected in a sample of 200 parts if the process averages 2% defective? c) 4 The average number of defective parts is equal to the sample size multiplied by the average percent defective. The average number of defective parts = 200 x .02 = 4. 58. What is the probability of obtaining exactly 2 defective connectors in a sample of 6 parts

60


if the process averages 2% defective? b) .0055 Use the Binomial probability distribution and the average number of defective parts from the previous problem:  6 2 4 P(2) =   ( 02) (.98) = 15(.0004)(.9224) =.0055  2

59. All of the following are probabilistic events except c) the acceleration of an apple when it drops from a tree. The number rolled in a game of dice, the number of defects in a random sample, and the number of games played in the world series are all probabilistic events because the exact outcome is not predictable 100% of the time. The acceleration of an apple when it drops from a tree is a deterministic event because it will always accelerate at the same rate (9.81 m/s2). Use the following information to answer problems 60, 61 and 62. A company produces integrated circuits (chips) by a process that normally yields 2000 ppm defective product for electrical test requirements (ppm = defective parts per million). A sample of 5 chips is selected and tested. 60. What is the probability that all 5 chips are good? a) .9900 Use the Binomial probability distribution: defective rate = 2000 ppm = 2000/1,000,000 = .002 P(all 5 are good) = P(0 defective) =

(50 )(.002)0 (.998)5 = 1 (1)(.99) = .99

61. What is the probability that 1 or more chips are defective? d) .0100 P(1 or more defective) = P(1) + P(2) + … + P(5) = 1 - P(0) = 1 - .99 = .0100

Basic Probability

62. What is the probability that more than 1 chip is defective? c) .00008 P(more than 1 defective) = P(2) + P(3) + P(4) + P(5) = 1 - [P(0) + P(1)] Use the following information to answer problems 63, 64 and 65. A capability study was made to determine the defective rate of 28AZ transistors. The study showed the rate to be 5000 ppm. Ten of the transistors were shipped to a customer. 63. What is the probability that the shipment contains no defective transistors? a) .9511 Use the Binomial probability distribution: defective rate = 5000 ppm = 5000/1,000,000 = .005 P(0 defective) =

( )(.005) (.995) 10 0

0

10

= 1(1)(.9511) = .9511

64. What is the probability that the shipment contains exactly 1 defective transistor? b) .0478 Use the Binomial probability distribution: defective rate = 5000 ppm = 5000/1,000,000 = .005  10 1 9 P(1 defective) =   (.005) (.995) = 10(.005 )(.9559) =.0478  1

65. What is the probability that the shipment contains 2 or more defective transistors? d) .0011 P(2 or more defective) = P(2) + P(3) + … + P(10) = 1 - [P(0) + P(1)] = 1 - [.9511 + .0478] (from the previous two problems) = .0011

61

62


Use the following information to answer problems 66, 67 and 68. A circuit board operation yields 2 defects per board on the average. A sample of 1 board is selected at random. 66. What is the probability of finding exactly 2 defects on the selected board? b) .2706 Use the Poisson probability distribution: p = 2 defects/board, n = 1, np = 2 e −2 2 2 (.1353)(4) P(x = 2) = = =.2706 2! 2

67. What is the probability of finding less than 2 defects on the selected board? a) .4060 Use the Poisson probability distribution: p = 2 defects/board, n = 1, np = 2 P(x < 2) = P(0) + P(1) P(x < 2) =

e −2 2 0 e −2 21 (.1353)(1) (.1353)(2) + = + =.1353 +.2706 =.4059 0! 1! 1 1

68. What is the probability of finding more than 2 defects on the selected board? d) .3235 Use the Poisson probability distribution: p = 2 defects/board, n = 1, np = 2 P(x > 2) = 1 - P(x < 3) = 1 - [P(0) + P(1) + P(2)]  e −2 2 0 e − 2 2 1 e − 2 2 2  P(x > 2) = 1 −  + +  1! 2!   0! = 1 - (.1353)(1)/1 - (.1353)(2)/1 - (.1353)(4)/2 = 1 - .1353 - .2706 - .2706 = .3235

Basic Probability

63

Use the following information to answer problems 69, 70 and 71. In manufacturing material for automobile seats it was found that each 100-foot roll contained 2 defects (flaws) on the average. A sample of 1 roll is selected at random from the process. 69. What is the probability that the selected roll contains 0 defects? a) .1353 Use the Poisson probability distribution: p = 2 defects/board, n = 1, np = 2 e −2 2 0 (.1353)(1) = =.1353 P(x = 0) = 0! 1 70. What is the probability that the selected roll contains exactly 1 defect? d) .2706 Use the Poisson probability distribution: p = 2 defects/board, n = 1, np = 2 e −2 21 (.1353)(2) = =.2706 P(x = 1) = 1! 1 71. What is the probability that the selected roll contains more than 1 defect? b) .5941 Use the Poisson probability distribution: p = 2 defects/board, n = 1, np = 2 P(x > 1) = P(2) + P(1) + … + P(∞) = 1 - [P(0) + P(1)] = 1 - [.1353 + .2706] = 1 - .4059 = .5941 Use the following information to answer problems 72 and 73. A firm that makes T-shirt decals has determined that their process yields, on average, 3 defects per day. Fifty decals are inspected each day. 72. What is the probability of finding exactly 2 defective decals in any given day? (Assume one defect per defective decal.) c) .2240 Use the Poisson probability distribution: p = 3 defects/day, n = 1, np = 3 e −3 3 2 (.0498)(9) = =.2240 P(x = 2) = 2! 2 73. What is the probability of buying a decal that contains more than 1 defect?

64


d) .0017 Use the Poisson probability distribution: p = 3/50 = .06, n = 1, np = .06 P(x > 1) = P(2) + P(1) + … + P(∞) = 1 - [P(0) + P(1)]  e −.06 .06 0 e −.06 .06 1  = 1−  +  ! 0 1!   = 1 - [(.9418)(1)/1 + (.9418)(.06)/1] = 1 - .9983 = .0017 74. A parts dealer buys parts from a warehouse. Parts are made by either Company A or Company B but are not identified as to which company produces them. One company produces all parts in one shipment or lot. On the average, we know: Company A produces 2.5% defective parts. Company B produces 5.0% defective parts. The warehouse states that 70% of parts will come from Company A and 30% from Company B. If the dealer selects 4 parts at random from a lot and finds 1 defective part, what is the probability that the lot was produced by Company A? b) .5580 The question to be answered is “What is the probability that the lot came from company A given that the shipment contains one defective part or P(A/D)?” Use Bayes formulas for conditional probability: P(A) = .70, P(B) = .30, D = defective part  4  4 P(D/A) =   (.025)1 (.975) 3 = .0927, P(D/B) =   (.05)1 (.95) 3 = .1715  1  1 P(D) = P(D/A)P(A) + P(D/B)P(B) = (.0927)(.70) +(.1715)(.30) = .0649 + .0514 = .1163 P(A & D) = P(D/A)P(A) = (.0927)(.07) = .0649 P(A/D) = P(A & D)/P(D) = .0649/.1163 = .5580

Statistics

65

STATISTICS

75. What is the expected value of the random variable x for the following data? x 12 10 14 20

f(x) 0.2 0.5 0.1 0.2

a) 12.8 The mean or expected value of x is µ = E(X) =

∑ xf ( x ) . x

E(x) = 12(0.2) + 10(0.5) + 14(0.1) + 20(0.2) = 12.8 76. In the standard normal table, what value of z has 5% of the area in the tail beyond it? b) 1.645 The value that has 5% of the area to the right must have 95% of the area to the left. Most normal tables provide the total area to the left of the z value, but in some cases the table may show the area from the left of the z value to the midpoint of the normal curve for positive z values. In the initial case, the z value that corresponds to an area of .9500 is the desired value. In the latter case, the z value that corresponds to an area of .4500 is the desired value. 77. Which distribution should be used to determine a confidence interval when σ is not known and the sample size is 10? b) t When the population variance is estimated by S2, and the sample size is small, then S2 will fluctuate considerably from sample to sample and the distribution of the random variable will deviate from the normal distribution. The t distribution is derived from the random variable containing the sample standard deviation and is used for small sample sizes. The t distribution should be used for sample sizes of 30 or less. 78. Which of the following methods should be used to test 6 population means for statistical significance? b) Analysis of Variance The Analysis of Variance or ANOVA is a widely used technique for dealing with more than two populations. Analysis of Variance analyzes the total variation by dividing it into meaningful components. 79. A sample size of 120 is taken from a process and is represented graphically on a

66


histogram. What is the appropriate number of histogram cells to use? b) 9 - 20 Grouping data into cells condenses large amounts of data that can be hard to manage. Typically, between 5 and 20 cells should be used choosing the number approximately equal to the square root of the sample size. Also, the cells should be of uniform width and the lower limit of the first cell should be just below the smallest data value. 80. Which of the following conditions makes it possible for a process to produce a large number of defective units while it is in statistical control? b) When the process capability is wider than the tolerance. If the process capability is wider than the tolerance, a percentage of parts will be outside of the specification limits. Any parts outside of the specification limits are considered to be defective. 81. For the normal probability distribution, which of the following is true about the relationship among the median, mean and mode? a) They are all equal to the same value. The normal probability distribution is symmetrical. If a distribution is symmetrical the median, mean and mode are located at the same location. 82. All of the following statistical techniques can be used to determine the effectiveness of a supplier improvement program except d) a flow chart. A flow chart is used to illustrate the order in which a sequence of events will occur and therefore cannot supply any meaningful data for evaluating a supplier improvement program. Pareto analysis and x bar and R charts are effective tools for evaluating a program based on numerical data collected from the entity being evaluated. The program evaluation and review technique (PERT) is also an effective tool for evaluating a particular program, particularly in evaluating scheduling requirements. 83. A sample of n observations has a mean x and a standard deviation s > 0. If a single observation, which equals the value of the sample mean x , is removed from the sample, which of the following is true? c)

x remains the same but s increases

Removing an observation that is equal to the average does not change the average. The standard deviation increases because the removed point helped to keep the variation of the data from the mean to a minimum. 84. The factory installed brake linings for a certain kind of car has a mean lifetime of 60,000 miles with a 6,000 mile standard deviation. A sample of 100 cars has been selected for

Statistics

67

testing. What is the standard error of x ? (Assume that the finite population correction may be ignored.) c) 600 miles The standard error ( σ x ) is the standard deviation for a distribution of averages. σx = s = 6000 = 600 miles n 100 Use the following information to answer problems 85 - 90. A sample of 7 rivets was taken from a shipment of 1000 rivets and the length was measured. The following data are obtained: Sample Number

Length (inches)

1

3.1

2

3.1

3

3.2

4

3.7

5

3.6

6

3.7

7

3.1

85. What is the mean length of the rivets? b) 3.36 inches x=

∑x

i

n

23.5 = 3.36" 7

=

86. What is the standard deviation of the length of the rivets (estimate of population standard deviation)? c) 0.29 inches

∑ (x − x ) s=

2

i

n −1

=

1.76 = 0.29" 6

Note: In the following 4 problems, the sample sizes are less than 30 and the t statistics should be used to solve the problems. Analyses of this type usually involve sample sizes of 30 or greater. Handle the problems just as if the sample sizes were greater than 30 and use

68


the z statistics. 87. What percentage of rivets have lengths less than 2.80 inches? a) 2.69% The percentage of rivets that are less than 2.80" is represented by area B. The area of B can be found by obtaining the area that corresponds to the following z value from the standard normal table.

z=

x i − x 2.80 − 3.36 −.56 = = = −193 . s .29 .29

A B 2.80

3.36

-1.93

0

x z

Area B = .0269, therefore 2.69% of the rivets have lengths less than 2.80". 88. What percentage of rivets have lengths greater than 3.65 inches? c) 15.9% The percentage of rivets that are greater than 3.65" is represented by area D. The area of D can be found by subtracting the area to the left of D from the total area under the curve. The area to the left D can be found by obtaining the area that corresponds to the following z value from the standard normal table. z=

x i − x 3.65 − 3.36 .29 . = = = 10 .29 .29 s

C

D

3.36 3.65

x

z 0 1.0 The area to the left of D = .8413 and the area of D = 1 - .8413 = .1587, therefore 15.9% of the rivets have lengths greater than 3.65". 89. What percentage of rivets have lengths between 3.1 inches and 3.9 inches? b) 78.4%

Statistics

69

Area A plus area C represent the percentage of rivets that are between 3.1" and 3.9". The area of A plus the area of C can be found by obtaining the areas that correspond to the following two z values from the standard normal table and subtracting the first from the second. z1 =

x i − x 3.1 − 3.36 = = −.90 .29 s

A

z2 =

x i − x 3.9 − 3.36 . = = 186 .29 s

C D

B 3.1

3.36

3.9

0

+1.86

-.90

x z

Area B = .1841, area BAC = .9685 and area BAC - B = .9685 - .1841 = .7844, therefore 78.4% of the rivets have lengths between 3.1" and 3.9". 90. In the shipment of 1000 rivets, how many good parts will we find if a good part is defined as having a minimum of 3 inches and a maximum of 4 inches? d) 878 Area A plus area C represent the percentage of rivets that are between 3.0" and 4.0". The area of A plus the area of C can be found by obtaining the areas that correspond to the following two z values from the standard normal table and subtracting the first from the second. z1 =

x i − x 3.0 − 3.36 . = = −124 .29 s

z2 =

A

x i − x 4.0 − 3.36 = = +2.20 .29 s

C D

B 3.0

3.36

4.0

-1.24

0

+2.20

x z

Area B = .1076, area BAC = .9861 and area BAC - B = .9861 - .1076 = .8785, therefore 87.85% of the rivets in the shipment have lengths between 3.0" and 4.0" which equates to 878 good parts. Remember, do not round up because you cannot have a partially good part. The following information is used to answer problems 91 - 95. Data are taken from a manufacturing process that produces optical glass. The sample size is 5 parts and the characteristic measured is the diameter of the plates.

70


Sample Number

Diameter (mm)

1 2 3 4 5

30 31 29 33 34

91. What is the mean diameter of the optical glass? a) 31.4 mm x=

∑x

i

n

=

157 . . mm = 314 5

92. What is the standard deviation of the population? b) 2.07 mm

∑ (x s=

i

− x)

2

=

n−1

8.29 = 2.07mm 4

93. The specifications for the glass plates are 30.5 ± 2 mm. What percentage of parts made by this company will not meet specifications? d) 37.9% Area B plus area D represent the percentage of parts that will not meet specification. To find the area of B plus D, it is first necessary to obtain the areas that correspond to the following two z values from the standard normal table. z1 =

x i − x 28.5 − 314 . . = = −14 2.07 s

z2 =

A

x i − x 32.5 − 314 . = = +.53 2.07 s

C D

B 28.5

31.4

32.5

-1.4

0

+.53

x z

Area B = .0808, area BAC = .7019, area D = 1 - BAC = 1 - .7019 = .2981 and area B + D = .0808 + .2981 = .3789, therefore 37.9% will not meet specifications. 94. What percentage of parts will be less than 29.5 mm? a) 17.9%

Statistics

71

The percentage of rivets that are less than 29.5 mm is represented by area B. The area of B can be found by obtaining the area that corresponds to the following z value from the standard normal table. z=

x i − x 29.5 − 314 . = = −.92 2.07 s

A B 29.5

31.4

-.92

0

x z

Area B = .1788, therefore 17.9% of the parts will be less than 29.5 mm. 95. What percentage of parts will be greater than 33 mm? d) 22.1% The percentage of parts that are greater than 33 mm is represented by area D. The area of D can be found by subtracting the area to the left of D from the total area under the curve. The area to the left D can be found by obtaining the area that corresponds to the following z value from the standard normal table. z=

x i − x 33.0 − 316 . = = +.77 2.07 s

C

D

31.4 33.0 0

.77

x z

The area to the left of D = .7793 and the area of D = 1 - .7793 = .2207, therefore 22.1% of the parts will be greater than 33 mm.

96. The Zoglen Corporation markets a product, which is a blend of 3 ingredients (A, B, C). If the individual tolerances for the weight of the 3 ingredients are as shown, what should the tolerance be for the net weight of the product?

72


A: 40.5 ± 2.236 grams, B: 30.4 ± 2.000 grams, C: 18.1 ± 1.732 grams b) 89.0 ± 3.464 grams nominal value = nominal A + nominal B + nominal C = 40.5 + 30.4 + 18.1 = 89.0 2 . = 3.464 tolerance = TA2 + TB2 + TC2 = 2.236 2 + 2.0 2 + 1732 This is known as the statistical method of relating tolerances. The net weight of most of the products will fall within 89.0 ± 3.464 grams.

97. A random sample of size n is to be taken from a large population that has a standard deviation of 1 inch. The sample size is determined so that there will be a 95% chance that the sample average will be within ± 0.1 inch of the true mean. Which of the following values is nearest to the required sample size? a) 385 The sample size is determined by using the formula for variables data. The z value corresponding to 95% of the area under the normal curve is 1.96. 2

2

. )(1)   Zs   (196 n=  =  = 385  E  0.1  98. All of the following conditions must be met for the process capability to be within the specification limits except c) Cp = Cpk The value of Cp is based on a ratio of the specification limits to the measured process variability (6σ). A Cp less than 1.0 (large variance) will yield a distribution that falls outside of the specification limits. A value of Cpk less than 1.0 will represent a shift of the mean value of the data from the target value, which will cause some of the data to possibly fall outside of a specification limit. A stable process implies that all the data will fall within the specification limits. The process capability index and the process performance index may equal each other and fall outside of the specification limits if they are both less than 1.0. 99. A value on the abscissa of the t distribution is 1.093. What is the area to the right of this value if the sample size is 11? b) 0.15 1. Degrees of freedom = n-1 = 11-1=10 2. Assuming that the t-table provided by QReview is used, go to the left column find 10 df and then across to the table value of 1.093. 3. Read the probability value on the top row, p=.3. The probabilities represent areas. 4. Since the table is set up for two tails (see picture on top), the value must be divided by 2. .3/2=.15

Statistics

73

100. The spread of individual observations from a normal process capability distribution may be expressed numerically as a) 6 R /d2 If only control chart information is available, the standard deviation may be calculated by R /d2, where d2 is a factor corresponding to the sample size. The process capability is defined as 6σ, therefore the expression may be defined as 6 R /d2. 101. What percentage of data will normally fall within a process capability? b) 99.73% Process capability is defined as 6σ. The area under the normal curve from -3σ to +3σ, which is 6σ, is 99.73% of the total area. Use the following information to answer problems 102 - 105. A winding machine wraps wire around a metal core to make small transformers. The design engineers have determined that the nominal number of windings is to be 10,060 with a minimum of 10,025 and a maximum of 10,095. A sample of 300 transformers was selected in a three month period and the wire was unwrapped on each part to determine the number of windings. The results were: x = 10,052 windings and s = 10 windings 102. What is the process capability? c) 10022 - 10082 The process capability equals x ± 3σ , where s is an estimate of σ. This formula will yield a spread of 6σ centered at the mean. x + 3σ = 10,052 + 3(10) = 10082 x − 3σ = 10,052 − 3(10) = 10022 103. Compute the value of Cp? d) 1.17 Cp =

USL − LSL 10095 − 10025 70 . = = = 117 6σ 6(10) 60

74


104. Compute the value of Cpk? a) .90 x − nearest spec lim it 10052 − 10025 = =.90 3σ 3(10) 105. What is the expected percent defective? C pk =

b) 0.35% The area outside of the specification limits is where defective product can be found. x i − x 10025 − 10052 = = −2.7 10 s x − x 10095 − 10052 z2 = i = = +4.3 10 s 10022 10082 z1 =

Area ≈ 0

area = .0035

10025 -2.7

10052 0

10095 +4.3

x z

Statistical Inference

75

STATISTICAL INFERENCE

106. Which of the following cannot be a null hypothesis? c) The sample means are equal. A hypothesis is a statement or inference about a population concerning either variables or attributes data. For example, sample means may be used in a hypothesis test to determine if the data is representative of a population but not to determine if two sample means are equal. 107. In a sampling distribution which of the following represents the critical region? a) α The critical region of a sampling distribution is represented as α, otherwise known as a type I error. This is the area under the distribution curve where one will reject a true null hypothesis. 108. In a hypothesis test which of the following represents the acceptance region? d) 1 - α The acceptance region is denoted as 1 - α or one minus the critical region. This can also be termed as the area under the distribution curve where one would accept a true null hypothesis. 109. The Chi Square distribution is b) a distribution of variances. The chi square distribution is a special case of the gamma distribution. The variance of the chi square distribution is known as two times the number of degrees of freedom, therefore the variance can be substituted back into the density function to give a distribution of variances. 110. Which of the following is a number derived from sample data that describes the data in some useful way? b) statistic Any function of the random variables constituting a random sample is called a statistic. 111. A null hypothesis assumes that a process is producing no more than the maximum allowable rate of defective items. What does the type II error conclude about the process?

76


b) It is not producing too many defectives when it actually is. The type II error or β is the acceptance of a false null hypothesis. If the null hypothesis is false in this case then the process will actually be producing too many defectives and the type II error will conclude that the process is acceptable. 112. A quality engineer wants to determine whether or not there is any difference between the means of the convolute paperboard cans supplied by two vendors, A and B. A random sample of 100 cans is selected from the output of each vendor. The sample from vendor A yielded a mean of 13.59 with a standard deviation of 5.94. The sample from vendor B yielded a mean of 14.43 with a standard deviation of 5.61. Which of the following would be a suitable null hypothesis? a) µA = µB The null hypothesis is the statement we are trying to prove or disprove and the alternate hypothesis is the statement we will accept if the null hypothesis is not accepted. Typically, if the null hypothesis is not accepted we will accept a range of values as provided in the wrong answers to this question. 113. A chi square test for independence has a 4 x 3 contingency table and a calculated chi square value of 11.5. At a .05 level of significance, which of the following are true about the decision regarding the null hypothesis? c) 6 df, accept Ho The degrees of freedom for a contingency table are (r - 1)(c - 1), therefore df = 3 x 2 = 6. From the chi square table (α = .05), the critical value is 12.59. The calculated value of 11.5 falls within the acceptance region, therefore accept Ho. 114. A supplier states that the average diameter of optical glass plates is 37.50 mm. The specifications are 37.50 mm ± .40 mm. A shipment of 500 plates is received and a sample of 10 plates is selected and the diameters of each are measured. The following data are obtained: 37.42 mm 37.63

37.84 37.82

37.50 37.78

37.48 37.79

37.75 37.51

Does the entire population have an average diameter of 37.5 mm? Test at a level of significance of .05. a) t = +2.97, Reject Ho The null and alternate hypotheses are H0 : x = 37.5 and H1: x = 37.5 . The t statistic must be computed to determine if it falls in the critical region of the t distribution. The mean and standard deviation need to be computed to determine the t statistic.


t=

77

x − µ 37.652 − 37.5 = = +2.967 s .162 n 10

.025

.025

37.5

2.967

x

-2.262

0

2.262

t

The t statistic falls in the critical region, therefore reject Ho in favor of H1. 115. Given: n = 40, p = .20 defects/unit Could this sample have possibly come from a process whose process average is .14 defects/unit? Test at a level of significance of .01. c) z = +1.01, Do not reject Ho The null and alternate hypotheses are H0: p = .14 and H1: p > .14. The z statistic must be computed to determine if it falls within the critical region. The critical z value, from the normal curve table, is +2.325. The computation of the z statistic is as follows: np (40 )(.14 ) σ= = =.059 n 40 p − p .20 −.14 z= = = +101 . σ .059

.01

P=.14 0 1.01 2.325

x z

The z statistic falls in the acceptance region, therefore do not reject Ho. 116. A study of 26 families across a certain state indicated that the average family income during 1989 was $33,400 with a Standard Deviation of $3680. Test the hypothesis that the true average income in this state during 1989 was $32,000 against the alternative that it was not $32,000. Test at a level of significance of .05. b) t = +1.94, Accept Ho

78


The null and alternate hypotheses are H0: µ = 32,000 and H1: µ ≠ 32,000. The t statistic must be computed to determine if it falls within the critical region. The critical t values, from the student t table, are ±2.060. The computation of the t statistic is as follows: x − µ 33,400 − 32,000 t= = = +194 . σ 3,680 n 26 1.94 .025

.025

x

32,000

-2.060

0

+2.060

t

The t statistic falls in the acceptance region, therefore accept Ho. 117. An inspection yields the following results: n = 20,

x = 10.5 lb.,

s = 1.2 lb.

Could this data have reasonably come from a process which usually has an average of 11.3 lb. or greater? Test at a level of significance of .01. c) t = -2.98, Reject Ho The null and alternate hypotheses are H0: µ = 11.3 and H1: µ < 11.3. The t statistic must be computed to determine if it falls within the critical region. The critical t value, from the student t table, is -2.539. The computation of the t statistic is as follows: t=

x − µ 10.5 − 113 . = = −2.98 σ 12 . 20 n

.01 -2.98 11.3

x

-2.539

t 0

The t statistic falls in the critical region, therefore reject Ho in favor of H1. 118. In a hypothesis test what determines whether the test is one tailed or two tailed? b) Alternate Hypothesis If the alternate hypothesis is written so that the population being tested is less than (e.g. µ < .90) or greater than (e.g. µ > .90) some value then the test will be one tailed.


79

If it is stated so that the population is not equal to some value (e.g. µA ≠ µB) then the test will be two tailed. 119. Pooled variance is best described as b) the combination of variances of two or more samples. The pooled variance is used when two or more samples are considered to be independent. Independent is defined as when the individual data in one sample cannot be related to or associated with the other in any meaningful manner. 120. Goodness of Fit is best described as d) the comparison of an observed sample distribution with a theoretical distribution. If some sample data are taken, one might want to know what distribution the data follows. Tests for goodness of fit can be performed to determine if the data follows any particular distribution.

80


Sampling

81

SAMPLING

121. The Dodge-Romig Tables are designed to minimize which parameter? c) ATI Dodge and Romig define ATI as Average Total Inspection. The sampling plans in the Doge Romig tables attempt to minimize the ATI. 122. What are the two unique quantities that determine a single sampling attributes plan? b) Sample Size and Acceptance Number An OC curve that describes the sampling plan may be developed using the sample size, acceptance number and various values of incoming quality. 123. Characteristics for which 100% inspection may be practicable include all of the following except d) ultimate physical properties (tensile strength, viscosity). Typically, testing these types of properties can take a long time and are usually unnecessary. They also usually require some type of destructive test. If destructive testing is performed on 100% of the product then no product will ever be available for use. 124. The term AQL as used in sampling inspection, means c) the maximum percent defective that can be considered satisfactory as a process average. AQL is the acceptable quality level. If the incoming quality level is at or below the AQL, the lots have a high probability of being accepted. 125. Which of the following best describes what an operating characteristic curve shows? a) The probability of accepting lots of various quality levels by sampling methods. 1

The Producer's Risk (α)

Probability of Acceptance

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0%

Approximately two times the AOQL

The Consumer’s Risk (β)

Rejectable Quality Level (RQL)

Acceptable Quality Level (AQL)

1%

2%

3%

4%

5%

6%

p - Quality of lots submitted to inspection (AIQ)

7%

82


126. In ANSI/ASQ Z1.4, the AQL is always determined at what Pa on the OC curve? d) None of the above The probability of acceptance is not set at a specific value. AQL's may have a different Pa depending on what sample code is chosen. 127. Which of the following best describes what an average outgoing quality curve shows? b) The outgoing quality level versus the incoming quality level. 0.03

AOQL = .028

Average Outgoing Quality (AOQ)

0.025

0.02

0.015

0.01

0.005

0 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

Average Incoming Quality (AIQ)

Use the following information to answer problems 128 - 134. A sampling plan is needed which will satisfy the following requirements: • • • •

Accept acceptable quality product almost all (90%) of the time. Accept rejectable quality product almost none (12%) of the time. Acceptable quality is defined as a process average of 1% defective. Rejectable quality is defined as a process average of 7% defective.

128. What is the value of α? d) 10% Alpha is known as the producer's risk or when acceptable quality product is rejected and is 1 - .90. 129. What is the value of β? b) 12% Beta is known as the consumer's risk or when unacceptable quality product is accepted and is given. 130. What is the value of the AQL? c) 1% AQL is defined as acceptable quality level and is given.

Sampling

83

131. What is the value of the LTPD? a) 7% The lot tolerance percent defective is the same as RQL expressed as percent defective and is given. 132. What is the sample size that will satisfy the requirements stated above? d) 50 See Binomial Nomograph below. 133. What is the acceptance number that will satisfy the requirements stated above? a) 1 See Binomial Nomograph below. 134. What is the approximate value of the AOQL? a) .0165 The Binomial Nomograph may not be drawn to scale. AOQL ≈ .033 / 2 = .0165 The point where the lines meet is where the sample size and acceptance number can be found.

.01 .033 .07

.12 .50 .90

Use the following data to answer problems 135 - 137.

84


Given: AQL = .015 (1.5% defective), RQL = .050 (5.0% defective), α = .05, β = .10 135. What is the sample size of the sampling plan? d) 200 See Binomial Nomograph below. 136. What is the acceptance number of the sampling plan? c) 6 See Binomial Nomograph below. 137. What is the value of AOQL? b) .0160 The Binomial Nomograph may not be drawn to scale. AOQL ≈ .032 / 2 = .0160 The point where the lines meet is where the sample size and acceptance number can be found. .015 .032 .05

.05

.50

.95

138. A sampling plan specifies a sample size of 50 and an acceptance number of 3. What is the value of the AQL if α = .05? d) .0280 See the Binomial Nomograph on the following page.

139. A sampling plan specifies a sample size of 50 and an acceptance number of 3. What is the approximate value of the AOQL?

Sampling

85

c) .0370 The Binomial Nomograph may not be drawn to scale. AOQL ≈ .0614 / 2 = .037 The Nomograph must be constructed and the lines must drawn through the sample size and acceptance number so that the AQL and AOQL can be obtained. .028

n = 50 c=3

.0614

.50

.95

140. All of the following are methods for checking product except d) safeguard checking. Checking product is a safegaurd but it is not a particular method for checking product. Constant percentage sampling, spot checking and no checking at all are among the many methods for checking product. 141. All of the following are characteristics of lot by lot single sampling except c) rejected lots are immediately scrapped. In lot by lot single sampling, rejected lots are usually screened 100% for the cause of rejection. In some cases, the lot may be scrapped. 142. All of the following are characteristics of continuous sampling except a) it is only used where product flow is discrete. Continuous sampling is used where product flow is continuous and it is not feasible to be formed into lots. 143. The following parameters are required for sampling plan construction using the Binomial Nomograph except d) Sample size. The sample size and acceptance number are the two values that describe a sampling plan and can be obtained by using the Binomial Nomograph. Input values include AQL, RQL, α and β.

86


CONTROL CHARTS 144. The primary use of a control chart is to a) detect assignable causes of variation in the process. When assignable causes of variation are detected, changes in the inputs can be made to bring the process back in control. Control charts also provide feedback to operators and engineers to help reduce process variability. 145. np charts are based on which distribution? b) Binomial The p in np is a percentage. The Binomial distribution is used to calculate probabilities associated with "percentages", therefore the np chart is based the Binomial distribution. 146. c and u charts are based on which distribution? a) Poisson The c and u values are the number of defects in a sample. The Poisson distribution is used to calculate probabilities associated with "defects." The Binomial cannot be used to calculate the probability of a certain number of defects. 147. Why do x charts always follow a normal distribution? c) The Central Limit Theorem The central limit theorem states that the distribution of averages always tend to follow a normal distribution even though the distribution of individual data values may be skewed. 148. When used together for variables data, which of the following pair of quantities is the most useful in preparing control charts? c)

x, R

These two statistics are derived from variable data, where x is the average of a sample of measurements and R is the range or difference between the highest and lowest value in the sample.

149. A process is in control at x = 100, R = 7.3 with n = 4. If the process level shifts to

Control Charts

87

101.5, with the same R , what is the probability that the next x point will fall outside the old control limits? a) .016 Original control limits: UCL = x + A2 R = 100 + (.729)(7.3) = 105.329 LCL = x - A2 R = 100 - (.729)(7.3) = 94.671 Revised control limits: UCL = x + A2 R = 101.5 + .73(7.3) = 106.829 UCL = x - A2 R = 101.5 - .73(7.3) = 96.171 Probability of exceeding old control limits = probability xbar > 105.329.

UCL = 105.329

106.829 101.5

x = 100 LCL = 94.671

96.171

Standard error = se = A2 R /3 = 5.329/3 = 1.776 Z=

x i − x 105.329 − 101.5 = = +2.15 se 1.776

From the normal curve table, P( x > 105.329) = P(Z > 2.15) = .5000 - .4842 = 0158 ≈ .016 150. A process is checked by inspection at random samples of 4 shafts after a polishing operation, x and R charts are maintained. A person making a spot check picks out 2 shafts, measures them accurately, and plots the value of each on the x chart. Both points fall just outside the control limits. He advises the department foreman to stop the process. This decision indicates that d) the person is not using the chart correctly. The plotted points must be the average of the sample or x . Individual data points are not to be plotted on a x chart.

151. The hardness of rivets is normally distributed with µ = 60.0 and standard deviation σ = 1.2. What are the 3 sigma control limits for the x chart using samples of size 5?

88


c) 58.38,

61.62

R = sd 2 = 1.2 (2.33) = 2.79 UCL x = x + A 2 R = 60 +.58(2.79 ) = 6162 . LCL x = x − A 2 R = 60 −.58(2.79 ) = 58.38 152. A possible cause of a cycle pattern in a control chart include all of the following except d) new supplier. A new supplier may cause a sudden shift in the pattern. The shift will appear exactly when the new supplier's products are used. Use the following information to answer problems 153 - 158. The following data are obtained from measuring the length of a metal bracket. Sample #1 x = 1.51" R = .03" n=5

Sample #2 x = 1.50" R = .02" n=5

Sample #3 x = 1.52" R = .04" n=5

153. What are the values of x , UCL x and LCL x for an x chart? b) 1.515", 1.532", 1.498" x=

151 . + 150 . + 152 . + 153 . = 1515 . 4

UCL x = x + A 2 R = 1515 . + 0.58(0.03) = 1532 . LCL x = x − A 2 R = 1515 . − 0.58(0.03) = 1498 .

154. What are the values of R , UCLR, LCLR for an R chart? c) .030", .0634", 0 0.03 + 0.02 + 0.04 + 0.03 = 0.03 4 UCL R = D 4 R = 2.114(0.03) = 0.063

R=

LCL R = D 3 R = 0(0.03) = 0 155. What are the values of the sample size and number of samples? c) 5, 4

Sample #4 x = 1.53" R = .03" n=5

Control Charts

89

Four samples were taken. Each sample consisted of 5 brackets, therefore the sample size is 5. 156. What is the standard error? b) .0058" S tan dard Error = S.E. =

A 2 R (0.58)(0.03) = =.0058 3 3

157. What is the standard deviation of the individual data points? a) .013" S tan dard Deviation = s =

.030 R = =.013 d 2 2.326

158. What is the process capability for the individual data points? d) 1.476" to 1.554" The process capability = x + 3σ where s is an estimate of σ. x + 3σ = 1515 . + 3(.013) = 1554 . x − 3σ = 1515 . − 3(.013) = 1476 .

90


Use the following information to answer problems 159 - 162. The following data are the result of inspecting aircraft seat belt buckles. Number of buckles in the sample

Number of defective buckles

50 60 40 50 55 55 50 60

2 3 1 2 0 1 0 1

159. What is the average percent defective ( p ) ? b) .024 p=

number of defective buckles 10 = =.024 total sample 420

160. What are the upper and lower control limits? d) .087,

0 p (1 − p ) (.024 )(.9760 ) =.024 + 3 =.087 n 52.5 p(1 − p ) (.024)(.9760) LCL = p − 3 =.024 − 3 =0 n 52.5

UCL = p + 3

161. What is the appropriate type of control chart? b) p chart The points plotted are values of fraction defective or percent defective. These types of data require the use of p charts. 162. An additional sample of 75 buckles contains 4 defectives. Is this sample point in control? a) Yes, p = .053 The new sample point (p = 4 / 75 = .053) value does not exceed the upper control limit, therefore the sample point is in control.


91

RELIABILITY

163. A reliability data system usually implies collecting data on b) product failures and operating time. Reliability is defined as the probability of a device performing its intended function under given operating conditions for a specified length of time. When collecting data, product failures and operating time are the information required to perform a good reliability analysis. 164. Which of the following best describes the failure rate in the wear out phase of the bathtub curve? d) Gets Worse The bathtub curve has three phases: infant mortality, useful life and wear out. Typically, if a product fails in its infant mortality phase it will never reach the customer. As the product goes through its useful life phase it will begin and continue to wear out, increasing its failure rate through time. 165. The greatest contribution of a reliability effort is made in the a) design area. A product will not become more reliable after it has been designed. Certain characteristics must be designed into products to ensure the smallest probability of failure. Various statistical techniques can be used to help determine designs that will meet the required expectations. 166. Reliability prediction is b) the process of estimating the probability that a product will perform its intended function for a stated time. Reliability prediction is the implementation of various statistical techniques to determine the reliability of a product before the product actually fails. 167. Maintainability is a) the probability of a system being restored to functional operation within a given period of time. One of the biggest problems with system failures is the downtime while a system is being fixed. An important factor in designing a system is to minimize the time to restore the system to functional operation. 168. A set of components has a MTBF of 1000 hours. What percentage will fail if the

92


components are tested for 500 hours? b) 39% 1 1 = =.001 MTBF 1000 Re liability = e − λt = e −(.001)( 500 ) =.6065 probability of failure = 1 − Re liability = 1−.6065 =.3900 λ=

169. What is the reliability of a system at 850 hours, if the average usage on the system was 400 hours for 1650 items and the total number of failures was 145? Assume an exponential distribution. d) 83% failure rate = λ =

145 =.0002196 (400 )(1650 )

reliability = e − λt = e − (.0002196 )( 850 ) =.8297 ≈ 83%

Use the following information to answer problems 170, 171 and 172. λ A = .001,

λ B = .002,

λ C = .003,

λ D = .0025,

A

C

B

D

t = 100 hours

170. What is the reliability of the system? c) .8805 R A = e − λ A t = e −.001(100 ) = e −.10 =.9048 RB = e − λB t = e −.002(100 ) = e −.20 =.8187 R C = e − λ C t = e −.003(100 ) = e −.30 =.7408 RD = e − λD t = e −.0025(100 ) = e −.25 =.7788 R AC = R A xR C =.9048 x.7408 =.6703 R AB = RB xRD =.8187 x.7788 =.6376 R sys = (1 − probability of failure) = 1 − [(1−.6093)(1−.6376)] =.8805

171. What is the MTBF for components A, B, C and D respectively?


d) 1000, 500, 333, 400 MTBFA =

1 1 = = 1000 λ A .001

MTBFB =

1 1 = = 500 λ B .002

MTBFC =

1 1 = = 333.33 λ C .003

MTBFD =

1 1 = = 400 λ D .0025

172. What is the probability that component C will fail before 100 hours. b) .2592 The probability that component C will work for at least 100 hours is R C = e −λ C t = e.003(100 ) =.7408 therefore, the probability that component C will fail before 100 hours is 1 - RC = 1 - .7408 = .2592 or 25.9% 173. The MTBF of a complex piece of repairable radar equipment is determined to be 950 hours. The equipment has been in continuous operation for 150 hours. What is the probability that the equipment will continue to operate without failure for at least another 375 hours? a) 0.5754 1 1 = =.0010526 MTBF 950 t = 150 + 375 = 525 hours

λ=

R = e − λt = e −(.0010526 )( 525 ) =.5754

93

94


REGRESSION AND CORRELATION

174. All of the following about multiple regression are true except c) that it involves one independent and two or more dependant variables. Multiple regression implies that more than one independent variable is being used to observe only one dependent variable. 175. All of the following statements are true about a correlation coefficient except d) a correlation coefficient of ±1 indicates a cause and effect relationship. A correlation coefficient of ±1 indicates perfect correlation, but it does not imply that the independent variables are related, as to cause and effect, to the dependant variable. 176. A study was conducted on the relationship between the speed of different cars and their gasoline mileage. The correlation coefficient was found to be 0.35 from the study. Later, it was discovered that there was a defect in the speedometers and they had all been set 5 miles per hour too fast. The correlation coefficient was computed using the correct data. What is the new correlation coefficient? b) 0.35 The correlation coefficient is a mathematical measure of the degree of correlation between the different cars and their gas mileage. If all of the data is changed so that the relative difference between the data does not change then the amount of correlation between the cars will not change. Use the following information to answer problems 177 - 182. Quality Training (Costs per employee per year)

Quality Cost Savings per employee per year

$500

$2700

800

3500

1000

4600

1200

7000

1400

9500

2000

11500

177. What is the intercept of the regression line?


c) -1027.4 x 500 800 1,000 1,200 1,400 2,000 6,900

x2 250,000 640,000 1,000,000 1,440,000 1,960,000 4,000,000 9,290,000

Y 2,700 3,500 4,600 7,000 9,500 11,500 38,800

xY 1,350,000 2,800,000 4,600,000 8,400,000 13,160,000 23,000,000 53,450,000

(∑ x)(∑ xY) − ∑ Y(∑ x ) int ercept = a = (∑ x) − n(∑ x ) 2

2

a =

2

6,900(53,450,000) − 38,800(9,290,000 ) 6900 2 − 6(9,290,000)

=

8.353 x10 9 8,130,000

= −1027.4

178. What is the slope of the regression line? a) 6.52

∑ slope = b =

 xY −   

∑

(∑ x)(∑ Y) 

 x −   2

 

n

(∑ x) n

2

   

 (6,900)( 38,800)  53,450,000 −   8,830,000 6  = = 6.52 b= 1335 , ,000  6900 2  9,290,000 −    6 

179. What is the correlation coefficient? d) .97

95

96


X

(x − x )

Y

( Y − Y)

(x − x ) ( Y − Y)

500 800 1,000 1,200 1,400 2,000 6,900

-650 -350 -150 50 250 850

2,700 3,500 4,600 7,000 9,500 11,500 38,800

-3,766.67 -2,966.67 -1,866.67 533.33 3033.33 5033.33

2,448,335.50 1,038,334.50 280,000.50 26,666.50 758,332.50 4,278,330.50 8,829,999.50

1 R= n

∑ [( x − x)( Y − Y)]

σ xσ Y σ x = 475.22, σ Y = 3,196.18 1 (8,830,000) 1471666 , , .7 6 R= = =.97 (475.22)(3,196.18) 1518 , ,887.8 180. What is the formula for the regression line? d) Y = -1027.42 + 6.52x The formula for the regression line is Y = a + bx. 181. What will the projected quality costs savings be when $1700 per employee is invested in quality training? b) $10,056.58 Y = projected quality cost savings x = $1700 Y = -1027.42 + 6.52(1700) Y = $10,056.58 182. If the cost savings were $8000 per employee, what are the probable training costs per employee? c) $1,384.57 Manipulating the regression equation to solve for x yields, x = (1027.42 + Y) / 6.52 Y = cost savings per employee = $8,000 x = probable training costs per employee x = (1027.42 + 8000) / 6.52 = $1,384.57

Cost of Quality

97

COST OF QUALITY 183. The basic objective of a quality cost program is to c) improve the profit of your company. A quality cost program can be used to formally evaluate the costs associated quality. Having this financial control enables management to control these costs and identify opportunities to reduce costs. 184. Analysis of quality costs consists of d) examining each cost element in relation to other elements and the total. Quality costs are broken down into different areas: prevention costs, appraisals costs, internal failure costs and external failure costs. These areas are compared to one another and to the total so that the relationship can be seen and a balance between the costs can be obtained. 185. In selecting a base for measuring quality costs, which of the following should be considered? d) All of the above. Sensitivity to increases and decreases in production schedules, affects by seasonal product sales and sensitivity to material price fluctuations could result in dramatic effects when comparing quality costs to a single base. For example, if sales triple during the Christmas season then quality costs may be insignificant to the total costs. Appropriate analysis is required to account for these fluctuations. 186. Which of the following quality cost indices is likely to have the greatest appeal to top management as an indicator of relative cost? d) Quality cost per unit of sales Comparing quality costs directly to sales is the only true relative measure. Comparing quality costs to revenue received from sales provides a good measure to determine if the appropriate amount is being spent quality. 187. If prevention costs are increased to pay for engineering work in quality control, and this results in a reduction in the number of product defects, this yields a reduction in c) failure costs. All costs associated with parts failing to meet quality standards or that cause manufacturing loss are considered failure costs.

98


188. A process that sorts good product from defective product falls into which of the following quality cost categories? c) Internal failure Any costs associated with defective products are considered failure costs. Any failure costs discovered before the customer receives the product are considered internal failure costs. 189. Cost of calibrating test and inspection equipment would be included in a) prevention costs. Prevention costs are costs associated with designing, implementing and maintaining a quality system. These are costs incurred to reduce other quality costs. 190. The cost of writing instructions and operating procedures for inspection and testing should be charged to a) prevention costs. Writing instructions and operating procedures are prevention costs because they may significantly reduce problems and the costs associated with resolving the problems, if properly adhered to. 191. Failure costs include costs due to d) supplier analysis of non-conforming hardware. The fact that the analysis was done on non-conforming hardware and not on the hardware before it was known to be non-conforming constitutes this as a failure cost. 192. Which of the following is least likely to be reported as a failure related cost? b) Downtime caused by late delivery of a purchased part rejected by the supplier's final inspection. Typically, this cost is overlooked because there is no direct method to obtain the cost associated with the late delivery. Often times, other events can take place while waiting for the part, therefore resulting in a smaller failure cost. 193. Which of the following activities is not normally charged as a preventive cost? d) Laboratory Acceptance Testing Laboratory acceptance testing is considered an appraisal cost because it is a cost associated with ensuring that the product meets certain requirements or standards.

Cost of Quality

99

194. In deciding whether sampling inspection of parts would be more economical than 100% inspection, you need to determine all of the following except d) cost of improving the production process. The cost of improving a production process may be of interest after inspection has been completed and it has been determined that the process needs to be modified to increase the quality of product. 195. Quality cost trend analysis is facilitated by comparing quality costs to c) appropriate measurement bases. A measurement base is developed using historical data. Measurement bases should take into account various factors that could affect the quality costs such as seasonal trends. This data can then be used to compare against future data. 196. For a typical month, the 3D Manufacturing Company identified and reported the following quality costs: Inspection wages…………………………………………….. Quality planning………………………………………………. Source inspection…………………………………………….. In-plant scrap and rework………………………………….… Final product test…………………………….…..…………… Retest and troubleshooting………………………………….. Field warranty cost…………………………………………… Evaluation and processing of deviation requests…………

$2,000 $4,000 $2,000 $88,000 $110,000 $39,000 $205,000 $6,000

What is the total failure cost from this data? d) $332,000

Failure Costs In-plant scrap and rework Retest and troubleshooting Field warranty cost Total

88,000 39,000 205,000 332,000

197. Which of the following is a typical external failure cost? a) Material Repair This cost is primarily the value of time spent on the return and repair of goods that have been in the customer's hands. Use the following information to answer problems 198 - 201. A manufacturer produces an electronic memory device. The following costs are incurred on a yearly basis.

100


Activity Research and Development Manufacturing Engineering Quality Engineering Plant Facilities Training Direct Labor Process Control (Labor) Supervision Materials Inspection Scrap Costs Rework Costs

Cost $50,000 40,000 5,000 60,000 1,000 80,000 5,000 30,000 90,000 13,000 21,000 12,000

198. What are the annual Prevention costs? b) $6,000

Prevention Costs Quality Engineering Training Total

5,000 1,000 6,000

199. What are the annual Appraisal costs? a) $18,000

Appraisal Costs Process Control Inspection Total

5,000 13,000 18,000

200. What are the annual Failure costs? c) $33,000

Failure Costs Scrap Rework Total

21,000 12,000 33,000

201. What is the percentage of quality costs to total costs d) 14% Total costs = $407,000 Total quality costs = prevention costs + appraisal costs + failure costs = 6,000 + 18,000 + 33,000 = $57,000 % of quality costs to total costs = 57,000 / 407,000 = .14 or 14%

Design of Experiments

101

DESIGN OF EXPERIMENTS 202. In performing an Analysis of Variance for a single factor experiment, a fundamental assumption which is made is that the c) variances are equal. The model errors are assumed to be normally and independently distributed random variables with mean zero and variance σ2. The variance σ2 is also assumed constant for all levels of the factor, which is necessary to test means. 203. To state that a model in an experimental design is fixed indicates that a) the levels used for each factor are the only ones of interest. The conclusions of the hypothesis test will only apply to the factor levels considered. The conclusions cannot be extended to similar factor levels that were not considered. 204. An experiment with two factors, in which all levels of one variable are run at each level of the second variable, is called a c) factorial experiment. Factorial experiments are used when several factors are of interest. More specifically, in each complete trial or replicate of the experiment combinations of the levels of the factors are investigated. 205. A two-way Analysis of Variance has r levels for the first variable and c levels for the second variable with 2 observations per cell. The degrees of freedom for interaction is b)

(r - 1)(c - 1)

206. An analysis of variance results in a calculated F value of F(10, 12) = 2.75. What is the level of significance? c) p = .05 The F.05 table will yield 2.75 for 10 degrees of freedom in the numerator and 12 degrees of freedom for the denominator.

102


207. A 32 experiment means that we are considering d) three levels of two factors. The Xy notation is commonly used in experimental designs. The base (X) indicates the number of levels. The exponent (y) indicates the number of factors. 208. Which of the following distributions is needed to evaluate the results of analysis of variance (ANOVA)? c) F distribution The F distribution is a ratio of two independent chi square random variables. The chi square distribution is a distribution of sample variances. 209. The primary advantage of the Latin Square design compared to the factorial design, is that a) it requires less data. A Latin square is an experimental design in which each level of each is combined only once with each level of two other factors or variables. Also, no interactions exist between the variables. 210. Consider the SS and MS columns of an Analysis of Variance table for a single factor design. The appropriate ratio for testing the null hypothesis of no treatment effect is b) MS treatments divided by MS residual. The MS residual estimates the variance of the experimental error. The MS treatments will estimate σ2 only if all factor level means are equal, but the value of MS treatments will be greater than σ2 if the factor level means are different. Dividing one by the other will provide an F ratio that can be tested against a critical F value to determine if the factor level means are different. 211. A completely randomized design is best described as c) a design in which all treatments are assigned to the experimental units in a completely random manner. It is often necessary to design experiments so that variability arising from nuisance variables can be controlled. This type of design helps to ensure the removal of this undesired variation.

Design of Experiments

103

Use the following information to answer problems 212 - 216. The following factorial experiment was conducted to determine the effect of study time and study material used on test scores. The numbers in the box represent the test scores. You will need to construct the ANOVA table to complete all of the questions. Hours of Study Study Material Used

4 60, 85 55, 65

1 2

6 77, 92 60, 80

212. How many factors are used in this experiment? a) 2 The two factors are hours of study and study material used. 213. How many levels are being examined for each factor? a) 2 The number of levels is the number of settings for being analyzed for each factor. 214. What is the sum of squares for the residual? a) 675.0 A complete ANOVA table will be constructed to answer this problem and the following problems.

(∑ x) CM =

2

=

nT

574 2 = 41184.5 8

∑ x − CM = 42468 − 41184.5 = 1283.5 ∑ (∑ columns) − CM = 265 + 309 − 41184.5 = 242.0 = 2

SS total =

2

SShours

SSmaterial

2

4

n col

∑ (∑ rows) = n rows

SSint eraction

2

∑ (∑ cells) = n cells

− CM =

2

4

314 2 260 2 + − 41184.5 = 364.5 4 4

2

− SShours − SSint eraction − CM

145 2 + 169 2 + 120 2 + 140 2 − 242.0 − 364.5 − 41184.5 = 2.0 2 = SS total − SShours − SSmaterial − SSint eraction =

SSresidual

= 1283.5 − 242.0 − 364.5 − 2.0 = 675.0

104


Source Hours Material Interaction Residual

SS 242.0 364.5 2.0 675.0

df 1 1 1 4

MS 242.0 364.5 2.0 168.75

F 1.43 2.16 0.01

Fcritical 7.71 7.71 7.71

SS = Sum of Squares df = Degrees of Freedom MS = Mean Square = SS / df = Variance F = F Ratio = MSsource / MSresidual 215. Which factor has the greatest impact on the outcome? b) Study Material Used The computed F value of the study material used is the largest of any of the factors, therefore it has the biggest impact on the outcome. 216. Which of the following factors are significant? d) None of the above None of the computed F values are larger than the critical value of F, therefore none of the factors are significant.


105

METROLOGY AND CALIBRATION 217. The key to designing an effective calibration program is to balance b) cost and quality considerations. The concept is to manufacture the highest quality product in the most efficient manner possible. For example, a program which requires daily testing of highly accurate electronic test equipment would probably not be necessary, would be very expensive to implement and therefore would substantially increase production costs. 218. Reliability metrics can be used to measure equipment c) performance. Reliability metrics can be used to obtain levels of confidence in equipment performance. These metrics are useful because they can provide a foundation for implementing changes which balance cost and quality considerations. 219. Which of the following terms is defined as the probability that measurement equipment will be found to be in tolerance after a specified period of time? d) Reliability If a certain class of equipment is tested every six months, and 95% of the units pass the full verification test, then this class is said to be 95% reliable on a six month interval. 220. Suppose a test procedure is performed in order to determine the value of a product measurement. What is the order of equipment type, from lowest to highest accuracy, if NIST traceability is to be achieved? d) Working standards, transfer standards, primary standards Working standards can be calibrated with more accurate transfer standards. Transfer standards are then calibrated with primary standards of the highest known accuracy, i. e. NIST standards. The reason for the use of transfer standards is to lower overall test costs. It could be very expensive to have a complete inventory of working standards periodically calibrated against a primary standard.

221. Which procedures are performed on product and test equipment in order to determine if predefined specifications have been met?

106


a) Inspection A "calibration" can consist of several components. The first step is the "inspection" or verification portion of the procedure. An adjustment and retest may or may not be necessary based upon the results of the initial inspection, which normally requires compliance to predefined specifications. 222. Which of the following is used to qualify measurement instruments in reference to national or internationally recognized standards? b) Traceability Traceability is the chain that ensures measurements have been made in relation to recognized standards. This effort, which is an ISO standard requirement, is made in order to promote consistency of measurement and therefore increase overall product quality. 223. A measurement standard with a stated uncertainty of 0.1 V, 95% confidence level, is used to test a device that has a nominal value of 100 V, and a tolerance of ±10 V. What is the TUR for this measurement? b) 100 The TUR of 100 is the uncertainty (0.1 V) divided by the allowable tolerance (10 V). TUR = 0.1 V /10 V = 100 The 95% confidence level reflects an industry consensus to express the uncertainty portion (0.1 V) of the calculation at a 2σ confidence level. The nominal value of 100 V has no bearing on the calculation. 224. A requirement of the ISO 9000 series standards is that all M&TE used for product inspection c) is identified and labeled appropriately. This requirement is made in order to ensure consistent adherence to your own calibration system. 225. What ISO 9001 element addresses equipment calibration? c) 4.11 This ISO element requires identifying inspection, measuring and test equipment. It also requires establishing, documenting and maintaining calibration procedures and records.

CQE Study Questions,Answers and Solutions

Recommend Documents