Operations Management, 12e (Heizer/Render/Munson)
Supplement 6 Statistical Process Control
Section 1 Statistical Process Control (SPC)
1) Some degree of variability is present in almost all processes.
Answer: TRUE
Diff: 1
Key Term: Natural variations
Objective: LO S6.1 Explain the purpose of a control chart
Learning Outcome: Apply basic statistical process control (SPC) methods
2) The purpose of process control is to detect when natural causes of variation are present.
Answer: FALSE
Diff: 2
Key Term: Statistical Process Control (SPC)
Objective: LO S6.1 Explain the purpose of a control chart
Learning Outcome: Apply basic statistical process control (SPC) methods
3) A normal distribution is generally described by its two parameters: the mean and the range.
Answer: FALSE
Diff: 1
Key Term: Central limit theorem
Objective: LO S6.2 Explain the role of the central limit theorem in SPC
Learning Outcome: Apply basic statistical process control (SPC) methods
4) A process is said to be in statistical control when assignable causes are the only sources of variation.
Answer: FALSE
Diff: 2
Key Term: Statistical Process Control (SPC)
Objective: LO S6.1 Explain the purpose of a control chart
Learning Outcome: Apply basic statistical process control (SPC) methods
5) Mistakes stemming from workers' inadequate training represent an assignable cause of variation.
Answer: TRUE
Diff: 1
Key Term: Assignable variation
Objective: LO S6.1 Explain the purpose of a control chart
Learning Outcome: Apply basic statistical process control (SPC) methods
6) Averages of small samples, not individual measurements, are generally used in statistical process control.
Answer: TRUE
Diff: 2
Key Term: Statistical Process Control (SPC)
Objective: LO S6.1 Explain the purpose of a control chart
Learning Outcome: Apply basic statistical process control (SPC) methods
7) The xbar chart indicates that a gain or loss of uniformity has occurred in dispersion of a production process.
Answer: FALSE
Diff: 2
Key Term: xbar chart
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
8) The Central Limit Theorem states that when the sample size increases, the distribution of the sample means will approach the normal distribution.
Answer: TRUE
Diff: 2
Key Term: Central limit theorem
Objective: LO S6.2 Explain the role of the central limit theorem in SPC
Learning Outcome: Apply basic statistical process control (SPC) methods
9) In statistical process control, the range is often used as a substitute for the standard deviation.
Answer: TRUE
Diff: 2
Key Term: Control charts
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
10) If the process average is in control, then the process range must also be in control.
Answer: FALSE
Diff: 2
Key Term: Control charts
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
11) A process range chart illustrates the amount of dispersion within the samples.
Answer: TRUE
Diff: 2
Key Term: Rchart
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
12) Mean charts and range charts complement one another, one detecting shifts in process average, the other detecting shifts in process dispersion.
Answer: TRUE
Diff: 2
Key Term: Control charts
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
13) An xbar chart is used when we are sampling attributes.
Answer: FALSE
Diff: 1
Key Term: xbar chart
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
14) To measure the voltage of batteries, one would sample by attributes.
Answer: FALSE
Diff: 1
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
15) A pchart is appropriate to plot the number of typographic errors per page of text.
Answer: FALSE
Diff: 2
Key Term: pchart
Objective: LO S6.5 Build pcharts and ccharts
Learning Outcome: Apply basic statistical process control (SPC) methods
16) A cchart is appropriate to plot the number of flaws in a bolt of fabric.
Answer: TRUE
Diff: 2
Key Term: cchart
Objective: LO S6.5 Build pcharts and ccharts
Learning Outcome: Apply basic statistical process control (SPC) methods
17) The xbar chart, like the cchart, is based on the exponential distribution.
Answer: FALSE
Diff: 3
Key Term: Control charts
Learning Outcome: Apply basic statistical process control (SPC) methods
18) If a sample of items is taken and the mean of the sample is outside the control limits, the process is:
 A) likely out of control and the cause should be investigated.
 B) in control, but not capable of producing within the established control limits.
 C) within the established control limits with only natural causes of variation.
 D) monitored closely to see if the next sample mean will also fall outside the control limits.
 E) producing high quality products.
Answer: A
Diff: 2
Key Term: Control charts
Objective: LO S6.1 Explain the purpose of a control chart
Learning Outcome: Apply basic statistical process control (SPC) methods
19) The causes of variation in statistical process control are:
 A) cycles, trends, seasonality, and random variations.
 B) producer's causes and consumer's causes.
 C) mean and range.
 D) natural causes and assignable causes.
 E) Type I and Type II.
Answer: D
Diff: 2
Key Term: Statistical Process Control (SPC)
Objective: LO S6.1 Explain the purpose of a control chart
Learning Outcome: Apply basic statistical process control (SPC) methods
20) Natural variations:
 A) affect almost every production process.
 B) are the many sources of variation that occur when a process is under control.
 C) when grouped, form a pattern, or distribution.
 D) are tolerated, within limits, when a process is under control.
 E) All of the above are true.
Answer: E
Diff: 2
Key Term: Natural variations
Objective: LO S6.1 Explain the purpose of a control chart
Learning Outcome: Apply basic statistical process control (SPC) methods
21) Natural variations:
 A) are variations that are to be identified and investigated.
 B) are variations that can be traced to a specific cause.
 C) are the same as assignable variations.
 D) lead to occasional false findings that processes are out of control.
 E) play no role in statistical process control.
Answer: D
Diff: 2
Key Term: Natural variations
AACSB: Reflective thinking
Objective: LO S6.1 Explain the purpose of a control chart
Learning Outcome: Apply basic statistical process control (SPC) methods
22) Assignable variation:
 A) is a sign that a process is under control.
 B) is to be identified and investigated.
 C) is the same as random variation.
 D) is variation that cannot be traced to a specific cause.
 E) leads to a steep OC curve.
Answer: B
Diff: 2
Key Term: Assignable variation
Objective: LO S6.1 Explain the purpose of a control chart
Learning Outcome: Apply basic statistical process control (SPC) methods
23) Assignable causes:
 A) are not as important as natural causes.
 B) are within the limits of a control chart.
 C) depend on the inspector assigned to the job.
 D) are also referred to as "chance" causes.
 E) are causes of variation that can be identified and investigated.
Answer: E
Diff: 2
Key Term: Assignable variation
Objective: LO S6.1 Explain the purpose of a control chart
Learning Outcome: Apply basic statistical process control (SPC) methods
24) Control charts for variables are based on data that come from:
 A) acceptance sampling.
 B) individual items.
 C) averages of small samples.
 D) averages of large samples.
 E) the entire lot.
Answer: C
Diff: 2
Key Term: Control charts
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
25) The purpose of an xbar chart is to determine whether there has been a:
 A) change in the dispersion of the process output.
 B) change in the percent defective in a sample.
 C) change in the central tendency of the process output.
 D) change in the number of defects in a sample.
 E) change in the AOQ.
Answer: C
Diff: 2
Key Term: xbar chart
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
26) The number of defects after a hotel room cleaning (sheets not straight, smears on mirror, missed debris on carpet, etc.) should be measured using what type of control chart?
 A) xbar chart
 B) Rchart
 C) pchart
 D) cchart
 E) either xbar chart or R chart
Answer: D
Diff: 2
Key Term: cchart
AACSB: Reflective thinking
Objective: LO S6.5 Build pcharts and ccharts
Learning Outcome: Apply basic statistical process control (SPC) methods
27) The number of late insurance claim payouts per 100 should be measured with what type of control chart?
 A) xbar chart
 B) Rchart
 C) pchart
 D) cchart
 E) either pchart or cchart
Answer: C
Diff: 2
Key Term: pchart
AACSB: Reflective thinking
Objective: LO S6.5 Build pcharts and ccharts
Learning Outcome: Apply basic statistical process control (SPC) methods
28) The upper and lower limits for diving ring diameters made by John's Swimming Co. are 40 and 39 cm., respectively. John took 11 samples with the following average diameters (39, 39.1, 39.2, 39.3, 39.4, 39.5 39.6, 39.7, 39.8, 39.9, 40). Is the process in control?
 A) Yes, no diameters exceeded the control limits.
 B) No, some diameters exceeded the control limits.
 C) No, there is a distinguishable pattern to the samples.
 D) No, the range is not in control.
 E) There is not enough information to make a decision.
Answer: C
Diff: 3
Key Term: xbar chart
AACSB: Analytical thinking
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
29) Red Top Cab Company receives multiple complaints per day about driver behavior. Over 9 days the owner recorded the number of calls to be 3, 0, 8, 9, 6, 7, 4, 9, and 8. What is the upper control limit for the 3sigma cchart?
 A) 13.35
 B) 8.45
 C) 24.00
 D) 0.00
 E) 9.03
Answer: A
Diff: 2
Key Term: cchart
AACSB: Analytical thinking
Objective: LO S6.5 Build pcharts and ccharts
Learning Outcome: Apply basic statistical process control (SPC) methods
30) A process that is assumed to be in control with limits of 89 ± 2 had sample averages for the xbar chart of the following: 87.1, 87, 87.2, 89, 90, 88.5, 89.5, and 88. Is the process in control?
 A) Yes.
 B) No, one or more averages exceeded the limits.
 C) Not enough information to tell.
 D) No, there is a distinguishable trend.
 E) No, two or more consecutive points are very near the lower (or upper) limit.
Answer: E
Diff: 2
Key Term: xbar chart
AACSB: Analytical thinking
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
31) Ten samples of a process measuring the number of returns per 100 receipts were taken for a local retail store. The number of returns were 10, 9, 11, 7, 3, 12, 8, 4, 6, and 11. Find the standard deviation of the sampling distribution for the pbar chart.
 A) There is not enough information to answer the question.
 B) .081
 C) 8.1
 D) .0273
 E) .0863
Answer: D
Diff: 2
Key Term: pchart
AACSB: Analytical thinking
Objective: LO S6.5 Build pcharts and ccharts
Learning Outcome: Apply basic statistical process control (SPC) methods
32) An xbar control chart was examined and no data points fell outside of the limits. Can this process be considered in control?
 A) Not yet, there could be a pattern to the points.
 B) Not yet, the Rchart must be checked.
 C) Not yet, the number of samples must be known.
 D) Yes.
 E) Both A and B
Answer: E
Diff: 2
Key Term: xbar chart
AACSB: Reflective thinking
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
33) Statistical process control charts:
 A) display the measurements on every item being produced.
 B) display upper and lower limits for process variables or attributes and signal when a process is no longer in control.
 C) indicate to the process operator the average outgoing quality of each lot.
 D) indicate to the operator the true quality of material leaving the process.
 E) are a graphic way of classifying problems by their level of importance, often referred to as the 8020 rule.
Answer: B
Diff: 2
Key Term: Control charts
Objective: LO S6.1 Explain the purpose of a control chart
Learning Outcome: Apply basic statistical process control (SPC) methods
34) The Central Limit Theorem:
 A) is the theoretical foundation of the cchart.
 B) states that the average of assignable variations is zero.
 C) allows managers to use the normal distribution as the basis for building some control charts.
 D) states that the average range can be used as a proxy for the standard deviation.
 E) controls the steepness of an operating characteristic curve.
Answer: C
Diff: 3
Key Term: Central limit theorem
Objective: LO S6.2 Explain the role of the central limit theorem in SPC
Learning Outcome: Apply basic statistical process control (SPC) methods
35) For a 3sigma xbar chart where the process standard deviation is known, the upper control limit:
 A) is 3σ below the mean of sample means for a 3σ control chart.
 B) is 3σ above the mean of sample means for a 3σ control chart.
 C) is 3σ/below the mean of sample means for a 3σ control chart.
 D) is 3σ/above the mean of sample means for a 3σ control chart.
 E) cannot be calculated unless the average range is known.
Answer: D
Diff: 2
Key Term: xbar chart
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
36) Up to three standard deviations above or below the centerline is the amount of variation that statistical process control allows for:
 A) Type I errors.
 B) about 95.5% variation.
 C) natural variation.
 D) all types of variation.
 E) assignable variation.
Answer: C
Diff: 2
Key Term: Natural variations
Objective: LO S6.1 Explain the purpose of a control chart
Learning Outcome: Apply basic statistical process control (SPC) methods
37) A manager wants to build 3sigma xbar control limits for a process. The target value for the mean of the process is 10 units, and the standard deviation of the process is 6. If samples of size 9 are to be taken, what will be the upper and lower control limits, respectively?
 A) 8 and 28
 B) 16 and 4
 C) 12 and 8
 D) 4 and 16
 E) 8 and 12
Answer: B
Diff: 2
Key Term: xbar chart
AACSB: Analytical thinking
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
38) Jars of pickles are sampled and weighed. Sample measures are plotted on control charts. The ideal weight should be precisely 11 oz. Which type of chart(s) would you recommend?
 A) pchart
 B) cchart
 C) both an xbar chart and an Rchart
 D) an xbar chart, but not an Rchart
 E) both a pchart and a cchart
Answer: C
Diff: 2
Key Term: Control charts
AACSB: Reflective thinking
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
39) If = 23 ounces, σ = 0.4 ounces, and n = 16, what will be the ±3σ control limits for the xbar chart?
 A) 21.8 to 24.2 ounces
 B) 23 ounces
 C) 22.70 to 23.30 ounces
 D) 22.25 to 23.75 ounces
 E) 22.90 to 23.10 ounces
Answer: C
Diff: 2
Key Term: xbar chart
AACSB: Analytical thinking
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
40) The usual purpose of an Rchart is to signal whether there has been a:
 A) gain or loss in dispersion.
 B) change in the percent defective in a sample.
 C) change in the central tendency of the process output.
 D) change in the number of defects in a sample.
 E) change in the consumer's risk.
Answer: A
Diff: 2
Key Term: Rchart
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
41) A manager wishes to build a 3sigma range chart for a process. The sample size is five, the mean of sample means is 16.01, and the average range is 5.3. From Table S6.1, the appropriate value of D3 is 0, and D4 is 2.115. What are the UCL and LCL, respectively, for this range chart?
 A) 33.9 and 11.2
 B) 33.9 and 0
 C) 11.2 and 0
 D) 6.3 and 0
 E) 31.91 and 0.11
Answer: C
Diff: 2
Key Term: Rchart
AACSB: Analytical thinking
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
42) Plots of sample ranges indicate that the most recent value is below the lower control limit. What course of action would you recommend?
 A) Since there is no obvious pattern in the measurements, variability is in control.
 B) One value outside the control limits is insufficient to warrant any action.
 C) Lower than expected dispersion is a desirable condition; there is no reason to investigate.
 D) The process is out of control; reject the last units produced.
 E) Variation is not in control; investigate what created this condition.
Answer: E
Diff: 3
Key Term: Rchart
AACSB: Reflective thinking
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
43) To set xbar chart upper and lower control limits, one must know the process central line, which is the:
 A) average of the sample means.
 B) total number of defects in the population.
 C) percent defects in the population.
 D) size of the population.
 E) average range.
Answer: A
Diff: 2
Key Term: xbar chart
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
44) According to the text, what is the most common choice of limits for control charts?
 A) ±1 standard deviation
 B) ±2 standard deviations
 C) ±3 standard deviations
 D) ±3 standard deviations for means and ± 2 standard deviations for ranges
 E) ±6 standard deviations
Answer: C
Diff: 2
Key Term: Control charts
Objective: LO S6.1 Explain the purpose of a control chart
Learning Outcome: Apply basic statistical process control (SPC) methods
45) Which of the following is true of a pchart?
 A) The lower control limit is found by subtracting a fraction from the average number of defects.
 B) The lower control limit indicates the minimum acceptable number of defects.
 C) The lower control limit equals D3times pbar.
 D) The lower control limit may be at zero.
 E) The lower control limit is the same as the lot tolerance percent defective.
Answer: D
Diff: 2
Key Term: pchart
AACSB: Reflective thinking
Objective: LO S6.5 Build pcharts and ccharts
Learning Outcome: Apply basic statistical process control (SPC) methods
46) The normal application of a pchart is in:
 A) process sampling by variables.
 B) acceptance sampling by variables.
 C) process sampling by attributes.
 D) acceptance sampling by attributes.
 E) process capability ratio computations.
Answer: C
Diff: 2
Key Term: pchart
Objective: LO S6.5 Build pcharts and ccharts
Learning Outcome: Apply basic statistical process control (SPC) methods
47) What is the statistical process chart used to control the number of defects per unit of output?
 A) xbar chart
 B) Rchart
 C) pchart
 D) AOQ chart
 E) cchart
Answer: E
Diff: 2
Key Term: cchart
Objective: LO S6.5 Build pcharts and ccharts
Learning Outcome: Apply basic statistical process control (SPC) methods
48) The cchart signals whether there has been a:
 A) gain or loss in uniformity.
 B) change in the number of defects per unit.
 C) change in the central tendency of the process output.
 D) change in the percent defective in a sample.
 E) change in the AOQ.
Answer: B
Diff: 2
Key Term: cchart
Objective: LO S6.5 Build pcharts and ccharts
Learning Outcome: Apply basic statistical process control (SPC) methods
49) The local newspaper receives several complaints per day about typographic errors. Over a sevenday period, the publisher has received calls from readers reporting the following total daily number of errors: 4, 3, 2, 6, 7, 3, and 9. Based on these data alone, what type of control chart(s) should the publisher use?
 A) pchart
 B) cchart
 C) xbar chart
 D) Rchart
 E) xbar chart and Rchart
Answer: B
Diff: 2
Key Term: cchart
AACSB: Reflective thinking
Objective: LO S6.5 Build pcharts and ccharts
Learning Outcome: Apply basic statistical process control (SPC) methods
50) A manufacturer uses statistical process control to control the quality of the firm's products. Samples of 50 of Product A are taken, and a defective/acceptable decision is made on each unit sampled. For Product B, the number of flaws per unit is counted. What type(s) of control charts should be used?
 A) pcharts for both A and B
 B) pchart for A, cchart for B
 C) ccharts for both A and B
 D) pchart for A, mean and range charts for B
 E) cchart for A, mean and range charts for B
Answer: B
Diff: 3
Key Term: Control charts
AACSB: Reflective thinking
Objective: LO S6.5 Build pcharts and ccharts
Learning Outcome: Apply basic statistical process control (SPC) methods
51) A nationwide parcel delivery service keeps track of the number of late deliveries (more than 30 minutes past the time promised to clients) per day. They plan on using a control chart to plot their results. Which type of control chart(s) would you recommend?
 A) both xbar chart and Rchart
 B) pchart
 C) cchart
 D) xbar chart
 E) both pchart and cchart
Answer: C
Diff: 2
Key Term: cchart
AACSB: Reflective thinking
Objective: LO S6.5 Build pcharts and ccharts
Learning Outcome: Apply basic statistical process control (SPC) methods
52) A run test is used:
 A) to examine variability in acceptance sampling plans.
 B) in acceptance sampling to establish control.
 C) to examine points in a control chart to check for natural variability.
 D) to examine points in a control chart to check for nonrandom variability.
 E) to test the validity of the Central Limit Theorem.
Answer: D
Diff: 2
Key Term: Run test
Objective: LO S6.1 Explain the purpose of a control chart
Learning Outcome: Apply basic statistical process control (SPC) methods
53) ________ is variation in a production process that can be traced to specific causes.
Answer: Assignable variation
Diff: 2
Key Term: Assignable variation
Objective: LO S6.1 Explain the purpose of a control chart
Learning Outcome: Apply basic statistical process control (SPC) methods
54) The ________ chart is the chief way to control attributes.
Answer: p
Diff: 3
Key Term: pchart
Objective: LO S6.5 Build pcharts and ccharts
Learning Outcome: Apply basic statistical process control (SPC) methods
55) If a process has only natural variations, ________ percent of the time the sample averages will fall inside the ±3sigma control limits.
Answer: 99.73
Diff: 2
Key Term: Natural variations
Objective: LO S6.1 Explain the purpose of a control chart
Learning Outcome: Apply basic statistical process control (SPC) methods
56) The ________ is a quality control chart that indicates when changes occur in the central tendency of a production process.
Answer: xbar chart
Diff: 2
Key Term: xbar chart
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
57) The ________ is a quality control chart used to control the number of defects per unit of output.
Answer: cchart
Diff: 2
Key Term: cchart
Objective: LO S6.5 Build pcharts and ccharts
Learning Outcome: Apply basic statistical process control (SPC) methods
58) What is the basic objective of a process control system?
Answer: It is to provide a statistical signal when assignable causes of variation are present.
Diff: 2
Key Term: Statistical Process Control (SPC)
Objective: LO S6.1 Explain the purpose of a control chart
Learning Outcome: Apply basic statistical process control (SPC) methods
59) Briefly explain what the Central Limit Theorem has to do with control charts.
Answer: It provides the theoretical foundation for xbar charts. It leads to the usability of the normal distribution in control charts because regardless of the distribution of the population of all parts or services, the distribution of the samples means tends to follow a normal curve as the number of samples increases.
Diff: 3
Key Term: Central limit theorem
Objective: LO S6.2 Explain the role of the central limit theorem in SPC
Learning Outcome: Apply basic statistical process control (SPC) methods
60) What are the three possible results (or findings) from the use of control charts?
Answer: The results of a control chart can indicate (a) in control and capable, (b) in control but not capable, and (c) out of control.
Diff: 2
Key Term: Control charts
Objective: LO S6.1 Explain the purpose of a control chart
Learning Outcome: Apply basic statistical process control (SPC) methods
61) Why do range charts exist? Aren't xbar charts enough?
Answer: Range charts and mean charts perform different functions. The mean chart is used to detect changes in the average of a process. But that average might stay the same while output is getting more scattered. The purpose of the range chart is to detect changes in the uniformity (dispersion) of a process.
Diff: 2
Key Term: Control charts
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
62) Examine the Statistical Process Control outputs below. Answer the following questions.
 What is the sample size?
 What is the number of samples?
 What is the mean of sample 8; what is the range of sample 10?
d.. Is this process in control? Explaina simple Yes or No is insufficient.
 What additional steps should the quality assurance team take?
Answer: a. The sample size is 4. b. ten samples were taken. c. The mean of sample 8 is 12.175; the range of sample 10 is 0.5. d. The process is not in control–while all means are within limits, the range for sample 3 is too large. e. Investigate for assignable cause and eliminate that cause.
Diff: 3
AACSB: Analytical thinking
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
63) What is the difference between natural and assignable causes of variation?
Answer: Natural variations are those variations that are inherent in the process and for which there is no identifiable cause. These variations fall in a natural pattern. Assignable causes are variations beyond those that can be expected to occur because of natural variation. These variations can be traced to a specific cause.
Diff: 2
Objective: LO S6.1 Explain the purpose of a control chart
Learning Outcome: Apply basic statistical process control (SPC) methods
64) A quality analyst wants to construct a sample mean chart for controlling a packaging process. He knows from past experience that the process standard deviation is two ounces. Each day last week, he randomly selected four packages and weighed each. The data from that activity appear below.

Weight 

Day 
Package 1 
Package 2 
Package 3 
Package 4 
Monday 
23 
22 
23 
24 
Tuesday 
23 
21 
19 
21 
Wednesday 
20 
19 
20 
21 
Thursday 
18 
19 
20 
19 
Friday 
18 
20 
22 
20 
(a) Calculate all sample means and the mean of all sample means.
(b) Calculate upper and lower 2sigma xbar chart control limits that allow for natural variations.
(c) Based on the xbar chart, is this process in control?
Answer:
(a) The five sample means are 23, 21, 20, 19, and 20. The mean of all sample means is 20.6
(b) UCL = 20.6 + 2(2/) = 22.6; LCL = 20.6  2(2/) = 18.6
(c) Sample 1 is above the UCL; all others are within limits. The process is out of control.
Diff: 3
Key Term: xbar chart
AACSB: Analytical thinking
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
65) A quality analyst wants to construct a sample mean chart for controlling a packaging process. He knows from past experience that when the process is operating as intended, packaging weight is normally distributed with a mean of twenty ounces, and a process standard deviation of two ounces. Each day last week, he randomly selected four packages and weighed each. The data from that activity appear below.

Weight 

Day 
Package 1 
Package 2 
Package 3 
Package 4 
Monday 
23 
22 
23 
24 
Tuesday 
23 
21 
19 
21 
Wednesday 
20 
19 
20 
21 
Thursday 
18 
19 
20 
19 
Friday 
18 
20 
22 
20 
(a) If he sets an upper control limit of 21 and a lower control limit of 19 around the target value of twenty ounces, the control chart is based on what value of z?
(b) With the UCL and LCL of part a, what do you conclude about this process–is it in control?
Answer:
(a) These control limits are one standard error away from the centerline.
(b) The mean of sample 1 lies outside the control limits. All other points are on or within the limits. The process is not in control based on the 1sigma limits, although there's a significant chance with 1sigma limits that the chart is sending a false positive outofcontrol signal.
Diff: 2
Key Term: xbar chart
AACSB: Analytical thinking
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
66) An operator trainee is attempting to monitor a filling process that has an overall average of 705 cc. The average range is 17 cc. If you use a sample size of 6, what are the upper and lower control limits for the xbar chart and Rchart?
Answer: From the table, A2 = 0.483, D4 = 2.004, D3 = 0
UCL = + A2 × LCL =  A2 × UCLR = D4 ×
= 705 + 0.483 × 17 = 705  0.483 × 17 = 2.004 × 17
= 713.211 = 696.789 = 34.068
LCLR = D3 ×
= 0 × 17
= 0
Diff: 2
AACSB: Analytical thinking
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
67) The defect rate for a product has historically been about 1.6%. What are the upper and lower control chart limits for a pchart, if you wish to use a sample size of 100 and 3sigma limits?
Answer:
UCLp = + 3 = 0.016 + 3 . = .0536
UCLp =  3 = 0.016  3 . = 0.0216, or zero
Diff: 2
Key Term: pchart
AACSB: Analytical thinking
Objective: LO S6.5 Build pcharts and ccharts
Learning Outcome: Apply basic statistical process control (SPC) methods
68) A small, independent amusement park collects data on the number of cars with outofstate license plates. The sample size is fixed at n=25 each day. Data from the previous 10 days indicate the following number of outofstate license plates:
Day 
Outofstate Plates 
1 
6 
2 
4 
3 
5 
4 
7 
5 
8 
6 
3 
7 
4 
8 
5 
9 
3 
10 
11 
(a) Calculate the overall proportion of "tourists" (cars with outofstate plates) and the standard deviation of proportions.
(b) Using ±3σ limits, calculate the LCL and UCL for these data.
(c) Is the process under control? Explain.
Answer:
(a) pbar is 56/250 = 0.224; the standard deviation of proportions is the square root of
(.224)(.776)/ 25 = 0.0834
(b) UCL = .224 + (3)(0.834) = .4742; LCL = .224 (3)(.0834) which is negative, so the LCL = 0
(c) The largest percentage of tourists (day 10) is 11/25 = .44, which is still below the UCL. Thus, all the points are within the control limits, so the process is under control.
Diff: 2
Key Term: pchart
AACSB: Analytical thinking
Objective: LO S6.5 Build pcharts and ccharts
Learning Outcome: Apply basic statistical process control (SPC) methods
69) Cartons of Plaster of Paris are supposed to weigh exactly 32 oz. Inspectors want to develop process control charts. They take ten samples of six boxes each and weigh them. Based on the following data, compute the lower and upper control limits and determine whether the process is in control.
Sample 
Mean 
Range 
1 
33.8 
1.1 
2 
34.6 
0.3 
3 
34.7 
0.4 
4 
34.1 
0.7 
5 
34.2 
0.3 
6 
34.3 
0.4 
7 
33.9 
0.5 
8 
34.1 
0.8 
9 
34.2 
0.4 
10 
34.4 
0.3 
Answer: n = 6; overall mean = 34.23; = 0.52.
Upper control limit 
34.48116 
1.04208 
Center line 
34.23 
0.52 
Lower control limit 
33.97884 
0 
The mean values for samples 1, 2, 3, and 7 fall outside the control limits on the xbar chart and sample 1 falls outside the upper limit on the Rchart. Therefore, the process is out of control.
Diff: 2
AACSB: Analytical thinking
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
70) McDaniel Shipyards wants to develop a control chart to assess the quality of its steel plate. They take ten sheets of 1" steel plate and compute the number of cosmetic flaws on each roll. Each sheet is 20' by 100'. Based on the following data, develop limits for the 3sigma control chart, plot the control chart, and determine whether the process is in control.
Sheet 
Number of flaws 
1 
1 
2 
1 
3 
2 
4 
0 
5 
1 
6 
5 
7 
0 
8 
2 
9 
0 
10 
2 
Answer:
Total units sampled 
10 
Total defects 
14 
Defect rate, cbar 
1.4 
Standard deviation 
1.183216 
z value 
3 


Upper Control Limit 
4.949648 
Center Line 
1.4 
Lower Control Limit 
0 
Sample six is above the control limits; therefore, the process is out of control.
Diff: 2
Key Term: cchart
AACSB: Analytical thinking
Objective: LO S6.5 Build pcharts and ccharts
Learning Outcome: Apply basic statistical process control (SPC) methods
71) The mean and standard deviations for a process are μ = 90 and σ = 9, respectively. For the variable control chart, a sample size of 16 will be used. Calculate the standard deviation of the sample means.
Answer: Sigma xbar = 9/ = 2.25
Diff: 2
Key Term: xbar chart
AACSB: Analytical thinking
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
72) If μ = 9 ounces, σ = 0.5 ounces, and n = 9, calculate the 3sigma control limits for the xbar chart.
Answer: 8.50 to 9.50 ounces
Diff: 2
Key Term: xbar chart
AACSB: Analytical thinking
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
73) A hospitalbilling auditor has been inspecting patient bills. While almost all bills contain some errors, the auditor is looking now for large errors (errors in excess of $250). Each day last week, the auditor examined 100 bills and found an average defect rate of 16%. Calculate the upper and lower limits for the billing process for 99.73% confidence.
Answer: 0.16 plus or minus (3)(0.03667): LCL = .050 and UCL = 0.270
Diff: 2
Key Term: pchart
AACSB: Analytical thinking
Objective: LO S6.5 Build pcharts and ccharts
Learning Outcome: Apply basic statistical process control (SPC) methods
74) A local manufacturer supplies you with parts, and you would like to install a quality monitoring system at his factory for these parts. Historically, the defect rate for these parts has been 1.25 percent (You've observed this from your acceptance sampling procedures, which you would like to discontinue). Develop ± 3σ control limits for this process. Assume the sample size will be 200 items.
Answer: pbar is 0.0125; the standard error of the proportion is = 0.00786
The upper control limit is 0.0125 + (3)(0.00786) = 0.03608; the lower control limit is
0.0125  (3)(0.00786) which is negative, so the LCL is 0.
Diff: 2
Key Term: pchart
AACSB: Analytical thinking
Objective: LO S6.5 Build pcharts and ccharts
Learning Outcome: Apply basic statistical process control (SPC) methods
75) Repeated sampling of a certain process shows the average of all sample ranges to be 1.0 cm. The sample size has been constant at n = 5. What are the 3sigma control limits for this Rchart?
Answer: LCL = 0, and UCL = (1.0)(2.115) = 2.115
Diff: 2
Key Term: Rchart
AACSB: Analytical thinking
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
76) A woodworker is concerned about the quality of the finished appearance of her work. In sampling units of a splitwillow handwoven basket, she has found the following number of finish defects in ten units sampled: 4, 0, 3, 1, 2, 0, 1, 2, 0, 2.
 Calculate the average number of defects per basket.
 If 3sigma control limits are used, calculate the lower control limit and upper control limit.
Answer: (a) 1.5; (b) LCL = 0, and UCL = 5.2.
Diff: 2
Key Term: cchart
AACSB: Analytical thinking
Objective: LO S6.5 Build pcharts and ccharts
Learning Outcome: Apply basic statistical process control (SPC) methods
77) The width of a bronze bar is intended to be oneeighth of an inch (0.125 inches). Inspection samples contain five bars each. The average range of these samples is 0.01 inches. What are the upper and lower control limits for the xbar and Rchart for this process, using 3sigma limits?
Answer: xbar chart: LCL = .119; UCL =.131. Rchart: LCL = 0.0; UCL =.021
Diff: 2
AACSB: Analytical thinking
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
78) A part that connects two levels should have a distance between the two holes of 4". It has been determined that xbar and Rcharts should be set up to determine if the process is in statistical control. The following ten samples of size four were collected. Calculate the control limits, plot the control charts, and determine if the process is in control.

Mean 
Range 
Sample 1 
4.01 
0.04 
Sample 2 
3.98 
0.06 
Sample 3 
4.00 
0.02 
Sample 4 
3.99 
0.05 
Sample 5 
4.03 
0.06 
Sample 6 
3.97 
0.02 
Sample 7 
4.02 
0.02 
Sample 8 
3.99 
0.04 
Sample 9 
3.98 
0.05 
Sample 10 
4.01 
0.06 
Answer:

Xbar 
Range 
xdoublebar value 
3.998 

R bar 
0.042 

Upper control limit 
4.029 
0.096 
Center line 
3.998 
0.042 
Lower control limit 
3.967 
0 
The process is out of control because of sample 5 on the xbar chart.
Diff: 3
AACSB: Analytical thinking
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
79) Ten samples of size four were taken from a process, and their weights measured. The sample averages and sample ranges are in the following table. Construct and plot an xbar and Rchart using these data. Is the process in control?
Sample 
Mean 
Range 
1 
20.01 
0.45 
2 
19.98 
0.67 
3 
20.25 
0.30 
4 
19.90 
0.30 
5 
20.35 
0.36 
6 
19.23 
0.49 
7 
20.01 
0.53 
8 
19.98 
0.40 
9 
20.56 
0.95 
10 
19.97 
0.79 
Answer:

Xbar 
Range 
xdoublebar value 
20.024 




R bar 
0.524 




Upper control limit 
20.406 
1.196 
Center line 
20.024 
0.524 
Lower control limit 
19.642 
0 
The xbar chart is out of control because samples 6 and 9 are outside of the control limit, and therefore the process is out of control.
Diff: 3
AACSB: Analytical thinking
Objective: LO S6.3 Build xcharts and Rcharts
Learning Outcome: Apply basic statistical process control (SPC) methods
80) Larry's boat shop wants to monitor the number of blemishes in the paint of each boat. Construct a 3sigma cchart to determine if their paint process is in control using the following data.
Sample Number 
Number of Defects 
1 
3 
2 
4 
3 
2 
4 
1 
5 
3 
6 
2 
7 
1 
8 
4 
9 
2 
10 
3 
Answer:
Total units sampled 
10 
Total defects 
25 
Defect rate, cbar 
2.5 
Standard deviation 
1.581 
z value 
3 


Upper Control Limit 
7.243 
Center Line 
2.5 
Lower Control Limit 
0 
The process is in control.
Diff: 2
Key Term: cchart
AACSB: Analytical thinking
Objective: LO S6.5 Build pcharts and ccharts
Learning Outcome: Apply basic statistical process control (SPC) methods
81) A bank's manager has videotaped 20 different teller transactions to observe the number of mistakes being made. Ten transactions had no mistakes, five had one mistake and five had two mistakes. Compute proper control limits at the 90% confidence level.
Answer: A cchart should be used, and from Table S6.2, the zvalue = 1.65
The mean cbar = [10(0) + 5(1) + 5(2)]/20 = 0.75
UCLc = 0.75 + 1.65 = 2.18
LCLc = 0.75  1.65 = 0.68 (or 0)
Diff: 2
Key Term: cchart
AACSB: Analytical thinking
Objective: LO S6.5 Build pcharts and ccharts
Learning Outcome: Apply basic statistical process control (SPC) methods
82) A department chair wants to monitor the percentage of failing students in classes in her department. Each class had an enrollment of 50 students last spring. The number of failing students in the 10 classes offered that term were 1, 4, 2, 0, 0, 0, 0, 0, 0, and 3, respectively. Compute the control limits for a pchart at the 95% confidence level. Is the process in control?
Answer: From Table S6.2, the zvalue = 1.96.
The mean pbar = [1 + 4 + 2 + 0 + 0 + 0 + 0 + 0 + 0 + 3]/(50 × 10) = 0.02.
σp = 0.0198
UCLp = 0.02 + 1.96(.0198) = 0.0589
LCLp = 0.02  1.96(.0198) = 0.0189 (or 0)
Since the percent defects in classes 2 and 10 both exceeded 5.89%, the percentage of failing students is not in statistical control. The department chair should investigate.
Diff: 2
Key Term: pchart
AACSB: Analytical thinking
Objective: LO S6.5 Build pcharts and ccharts
Learning Outcome: Apply basic statistical process control (SPC) methods
83) A city police chief decides to do an annual review of the police department by checking the number of monthly complaints. If the total number of complaints in each of the 12 months were 15, 18, 13, 12, 16, 20, 5, 10, 9, 11, 8, and 3 and the police chief wants a 90% confidence level, are the complaints in control?
Answer: From Table S6.2 z = 1.65
cbar = (15 + 18 + 13 + 12 + 16 + 20 + 5 + 10 + 9 + 11 + 8 +3)/12 = 11.67 complaints
UCL= 11.67+ 5.6366 = 17.307 complaints
LCL= 11.67  5.6366 = 6.033 complaints
Since 18 and 20 complaints fall above the UCL, while 5 and 3 complaints fall below the LCL, the complaints are not in control.
Diff: 2
Key Term: cchart
AACSB: Analytical thinking
Objective: LO S6.5 Build pcharts and ccharts
Learning Outcome: Apply basic statistical process control (SPC) methods
84) A retail store manager is trying to improve and control the rate at which cashiers sign customers up
for store credit cards. Suppose the manager takes 10 samples, each with 100 observations. The pbar
value is found to be .05, and the manager does not want a lower limit below .0064. What zvalue would this imply, and how confident can she be that the true lower limit is greater than or equal to .0064?
Answer: Sigmap = sqrt (.05*(1  .05)/100) = .0217945. Using the equation that LCL = pbar  z*sigmap gives .0064 = .05  z*(.0217945). Solving gives z=2. Using table S6.2 shows that for z = 2 the manager can be 95.45% confident.
Diff: 3
Key Term: pchart
AACSB: Analytical thinking
Objective: LO S6.5 Build pcharts and ccharts
Learning Outcome: Apply basic statistical process control (SPC) methods
85) A retail store manager is trying to improve and control the rate at which cashiers sign customers up for store credit cards. The manager takes several samples of size 50. He finds that each sample of 50 contained 5 credit card signups on average. Find pbar and 99.73% control limits.
Answer: Pbar = 5/50 = .1 or 10%. Sigmap = sqrt(.1(1.1)/50) = .042426. z = 3 for the given confidence level. Using the equations for control limits gives
UCL= .1 + 3(.042426) = .22728
LCL= .1  3(.042426) = .02728 and since a control limit cannot be negative round up to 0.
Diff: 2
Key Term: pchart
AACSB: Analytical thinking
Objective: LO S6.5 Build pcharts and ccharts
Learning Outcome: Apply basic statistical process control (SPC) methods
Section 2 Process Capability
1) A process that is in statistical control will always yield products that meet their design specifications.
Answer: FALSE
Diff: 2
Key Term: Process capability ratio
Objective: LO S6.6 Explain process capability and compute Cp and Cpk
Learning Outcome: Apply basic statistical process control (SPC) methods
2) The higher the process capability ratio, the greater the likelihood that process will be within design specifications.
Answer: TRUE
Diff: 2
Key Term: Process capability ratio
Objective: LO S6.6 Explain process capability and compute Cp and Cpk
Learning Outcome: Apply basic statistical process control (SPC) methods
3) The Cpk index measures the difference between the desired and actual dimensions of goods or services produced.
Answer: TRUE
Diff: 2
Key Term: Process capability index
Objective: LO S6.6 Explain process capability and compute Cp and Cpk
Learning Outcome: Apply basic statistical process control (SPC) methods
4) The process capability measures Cp and Cpk differ because:
 A) only one ensures the process mean is centered within the limits.
 B) Cpvalues above 1 indicate a capable process, while Cpkvalues above 2 indicate a capable process.
 C) They do not differ: both are identical.
 D) Cpvalues for a given process will always be greater than or equal to Cpk
 E) Both A and D are correct.
Answer: E
Diff: 3
Key Term: Process capability
Objective: LO S6.6 Explain process capability and compute Cp and Cpk
Learning Outcome: Apply basic statistical process control (SPC) methods
5) A Cp value of 1.33 indicates a standard of how many standard deviations (sigmas)?
 A) 6
 B) 1.33
 C) 2
 D) 3
 E) 4
Answer: E
Diff: 2
Key Term: Process capability ratio
Objective: LO S6.6 Explain process capability and compute Cp and Cpk
Learning Outcome: Apply basic statistical process control (SPC) methods
6) Which of the following is true regarding the process capability index Cpk?
 A) A Cpkindex value of 1 is the highest possible.
 B) The larger the Cpk, the more units meet specifications.
 C) The Cpkindex can only be used when the process centerline is also the specification centerline.
 D) Positive values of the Cpkindex are good; negative values are bad.
 E) Its value will always be at least as large as the Cp value for the same process.
Answer: B
Diff: 2
Key Term: Process capability index
Objective: LO S6.6 Explain process capability and compute Cp and Cpk
Learning Outcome: Apply basic statistical process control (SPC) methods
7) If the Cpk index exceeds 1:
 A) the AQL must be smaller than the LTPD.
 B) σ must be less than onethird of the absolute value of the difference between each specification limit and the process mean.
 C) the xbar chart must indicate that the process is in control.
 D) the process is capable of Six Sigma quality.
 E) the process is characterized as "not capable."
Answer: B
Diff: 2
Key Term: Process capability index
AACSB: Analytical thinking
Objective: LO S6.6 Explain process capability and compute Cp and Cpk
Learning Outcome: Apply basic statistical process control (SPC) methods
8) The statistical definition of Six Sigma allows for 3.4 defects per million. This is achieved by what Cpk index value?
 A) 6
 B) 1
 C) 1.33
 D) 1.67
 E) 2
Answer: E
Diff: 2
Key Term: Process capability index
AACSB: Analytical thinking
Objective: LO S6.6 Explain process capability and compute Cp and Cpk
Learning Outcome: Apply basic statistical process control (SPC) methods
9) A Cpk index of 1.00 equates to what defect rate?
 A) five percent
 B) 3.4 defects per million items
 C) 2.7 defects per 1,000 items
 D) 97.23 percent
 E) one percent
Answer: C
Diff: 3
Key Term: Process capability index
AACSB: Analytical thinking
Objective: LO S6.6 Explain process capability and compute Cp and Cpk
Learning Outcome: Apply basic statistical process control (SPC) methods
10) The term ________ is used to describe how well a process makes units within design specifications (or tolerances).
Answer: process capability
Diff: 2
Key Term: Process capability
Objective: LO S6.6 Explain process capability and compute Cp and Cpk
Learning Outcome: Apply basic statistical process control (SPC) methods
11) A Cpk index greater than ________ is a capable process, one that generates fewer than 2.7 defects per 1000 at the ±3σ level.
Answer: unity, or 1
Diff: 2
Key Term: Process capability index
Objective: LO S6.6 Explain process capability and compute Cp and Cpk
Learning Outcome: Apply basic statistical process control (SPC) methods
12) What does it mean for a process to be "capable"?
Answer: Process capability refers to the ability of a process to meet design specifications, which are set by engineering design or customer requirements.
Diff: 2
Key Term: Process capability
Objective: LO S6.6 Explain process capability and compute Cp and Cpk
Learning Outcome: Apply basic statistical process control (SPC) methods
13) What is the difference between the process capability ratio Cp and the process capability index Cpk?
Answer: The Cp ratio does not consider how well the process average is centered on the target value. However, Cpk does consider how well the process is centered. In particular, the Cp value will always be greater than or equal to the Cpk value, and a Cp value of 1 will not actually indicate ±3sigma capability if the process is not centered. There's really no reason to use Cp instead of Cpk.
Diff: 3
Key Term: Process capability
AACSB: Reflective thinking
Objective: LO S6.6 Explain process capability and compute Cp and Cpk
Learning Outcome: Apply basic statistical process control (SPC) methods
14) A process is operating in such a manner that the mean of the process is exactly on the lower specification limit. What must be true about the two measures of capability for this process?
Answer: The Cp ratio does not consider how well the process average is centered on the target value; its value is unaffected by the setting for the process mean. However, Cpk does consider how well the process is centered; with xbar on the LSL, the formula guarantees a Cpk of zero.
Diff: 2
Key Term: Process capability
AACSB: Analytical thinking
Objective: LO S6.6 Explain process capability and compute Cp and Cpk
Learning Outcome: Apply basic statistical process control (SPC) methods
15) The specifications for a manifold gasket that installs between two engine parts calls for a thickness of 2.500 mm ± .020 mm. The standard deviation of the process is estimated to be 0.004 mm. The process is currently operating at a mean thickness of 2.50 mm. (a) What are the upper and lower specification limits for this product? (b) What is the Cp for this process? (c) About what percent of all units of this gasket will meet specifications? Does this meet the technical definition of Six Sigma?
Answer: (a) LSL = 2.48 mm, USL = 2.52 mm. (b) Cp = (2.52  2.48)/(6 ∗ 0.004) = 1.67. (b) Each specification limit lies 5 standard deviations from the centerline, so practically 100 percent of units will meet specifications. However, this percentage is not quite as high as Six Sigma would call for.
Diff: 2
Key Term: Process capability ratio
AACSB: Analytical thinking
Objective: LO S6.6 Explain process capability and compute Cp and Cpk
Learning Outcome: Apply basic statistical process control (SPC) methods
16) The specifications for a manifold gasket that installs between two engine parts calls for a thickness of 2.500 mm ± .020 mm. The standard deviation of the process is estimated to be 0.004 mm. The process is currently operating at a mean thickness of 2.50 mm. (a) What are the upper and lower specification limits for this product? (b) What is the Cp for this process? (c) The purchaser of these parts requires a capability index of 1.50. Is this process capable? Is this process good enough for the supplier? (d) If the process mean were to drift from its setting of 2.500 mm to a new mean of 2.497, would the process still be good enough for the supplier's needs?
Answer: (a) LSL = 2.48 mm, USL = 2.52 mm. (b) Cp = (2.52  2.48)/(6 ∗ 0.004) = 1.67. (c) Yes to both parts of the question. (d) The Cpk index is now relevant, and its value is the lesser of 1.917 and 1.417. The process is still capable, but not to the supplier's needs.
Diff: 2
Key Term: Process capability
AACSB: Analytical thinking
Objective: LO S6.6 Explain process capability and compute Cp and Cpk
Learning Outcome: Apply basic statistical process control (SPC) methods
17) The specification for a plastic liner for concrete highway projects calls for a thickness of 6.0 mm ± 0.1 mm. The standard deviation of the process is estimated to be 0.02 mm. What are the upper and lower specification limits for this product? The process is known to operate at a mean thickness of 6.03 mm. What is the Cp and Cpk for this process? About what percent of all units of this liner will meet specifications?
Answer: LSL = 5.9 mm, USL = 6.1 mm. Cp is (6.1  5.9)/6(.02) = 1.67. Cpk is the lesser of = 1.17 and (6.03  5.9)/(3 ∗ 0.02) = 2.17; therefore, 1.17. The upper specification limit lies more than 3 standard deviations from the centerline, and the lower specification limit is further away, so practically all units will meet specifications.
Diff: 2
Key Term: Process capability
AACSB: Analytical thinking
Objective: LO S6.6 Explain process capability and compute Cp and Cpk
Learning Outcome: Apply basic statistical process control (SPC) methods
18) The specification for a plastic handle calls for a length of 6.0 inches ± .2 inches. The standard deviation of the process is estimated to be 0.05 inches. What are the upper and lower specification limits for this product? The process is known to operate at a mean thickness of 6.1 inches. What are the Cp and Cpk values for this process? Is this process capable of producing the desired part?
Answer: LSL = 5.8 inches, USL = 6.2 inches. Cp is (6.25.8)/6(.05) = 1.33. Cpk is the lesser of = .67 and (6.1  5.8)/(3 ∗ 0.02) = 2.00; therefore, .67. The process is capable based upon the Cp. However, the process is not centered (based upon its Cpk) and based upon its current center is not producing parts that are of an acceptable quality.
Diff: 2
Key Term: Process capability
AACSB: Analytical thinking
Objective: LO S6.6 Explain process capability and compute Cp and Cpk
Learning Outcome: Apply basic statistical process control (SPC) methods
19) A consultant has been brought in to a manufacturing plant to help apply Six Sigma principles. Her first task is to work on the production of rubber balls. The upper and lower spec limits are 21 and 19 cm, respectively. The consultant takes ten samples of size five and computes the sample standard deviation to be .7 cm and the sample mean to be 19.89 cm. Compute Cp and Cpk for the process. Give the consultant advice on what to do with the process based on your findings.
Answer: Cp = (21  19)/(6 ∗ .7) = .476
Cpk = min [(21  19.89)/(3 ∗ .7), (19.89  19)/(3 ∗ .7)] = .424
The very low capability metrics mean the process is not capable. The variability must be reduced.
Diff: 2
Key Term: Process capability
AACSB: Analytical thinking
Objective: LO S6.6 Explain process capability and compute Cp and Cpk
Learning Outcome: Apply basic statistical process control (SPC) methods
20) At your first job out of college you have been assigned to the production of bottled 20 oz. soda.
The process has upper and lower spec limits of 20.5 and 19.5 oz, respectively, with a mean of 19.8 oz
and standard deviation of .1 oz. Your manager has requested the process produce no more than 3.4
defects per 1 million bottles produced. Calculate Cpk and then determine if the process is capable according to the manager's standard.
Answer: Cpk = minimum [(20.5  19.8)/(3 ∗ .1),(19.8  .5)/(3 ∗ .1)] = 1.0 oz
The process is capable according to the typical 3sigma standard of needing a Cpk value ≥ 1; however, it is not capable according to the manager's sixsigma standard because Cpk is less than the 2.0 required for under 3.4 defects per million.
Diff: 3
Key Term: Process capability index
AACSB: Analytical thinking
Objective: LO S6.6 Explain process capability and compute Cp and Cpk
Learning Outcome: Apply basic statistical process control (SPC) methods
Section 3 Acceptance Sampling
1) Acceptance sampling accepts or rejects an entire lot based on the information contained in the sample.
Answer: TRUE
Diff: 2
Key Term: Acceptance sampling
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
2) A lot that is accepted by acceptance sampling is certified to be free of defects.
Answer: FALSE
Diff: 2
Key Term: Acceptance sampling
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
3) In acceptance sampling, a manager can reach the wrong conclusion if the sample is not representative of the population it was drawn from.
Answer: TRUE
Diff: 2
Key Term: Acceptance sampling
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
4) The probability of rejecting a good lot is known as consumer's risk.
Answer: FALSE
Diff: 2
Key Term: Producer's risk
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
5) An acceptance sampling plan must define "good lots" and "bad lots" and specify the risk level associated with each one.
Answer: TRUE
Diff: 2
Key Term: Acceptance sampling
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
6) The acceptable quality level (AQL) is the average level of quality we are willing to accept.
Answer: FALSE
Diff: 2
Key Term: Acceptable quality level (AQL)
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
7) The steeper an OC curve, the better it discriminates between good and bad lots.
Answer: TRUE
Diff: 2
Key Term: Operating characteristic (OC) curve
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
8) Consumer's risk is the probability of:
 A) accepting a good lot.
 B) rejecting a good lot.
 C) rejecting a bad lot.
 D) accepting a bad lot.
 E) none of the above
Answer: D
Diff: 2
Key Term: Consumer's risk
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
9) Acceptance sampling:
 A) is the application of statistical techniques to the control of processes.
 B) was developed by Walter Shewhart of Bell Laboratories.
 C) is used to determine whether to accept or reject a lot of material based on the evaluation of a sample.
 D) separates the natural and assignable causes of variation.
 E) is another name for 100% inspection.
Answer: C
Diff: 2
Key Term: Acceptance sampling
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
10) Acceptance sampling's primary purpose is to:
 A) estimate process quality.
 B) identify processes that are out of control.
 C) detect and eliminate defectives.
 D) decide if a lot meets predetermined standards.
 E) determine whether defective items found in sampling should be replaced.
Answer: D
Diff: 3
Key Term: Acceptance sampling
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
11) An acceptance sampling plan's ability to discriminate between low quality lots and high quality lots is described by:
 A) a Gantt chart.
 B) the Central Limit Theorem.
 C) a process control chart.
 D) an operating characteristic curve.
 E) a range chart.
Answer: D
Diff: 2
Key Term: Operating characteristic (OC) curve
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
12) Acceptance sampling:
 A) may involve inspectors taking random samples (or batches) of finished products and measuring them against predetermined standards.
 B) may involve inspectors taking random samples (or batches) of incoming raw materials and measuring them against predetermined standards.
 C) is more economical than 100% inspection.
 D) may be either of a variable or attribute type, although attribute inspection is more common in the business environment.
 E) All of the above are true.
Answer: E
Diff: 2
Key Term: Acceptance sampling
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
13) Which of the following statements about acceptance sampling is true?
 A) Acceptance sampling draws a sample from a population of items, tests the sample, accepts the entire population if the sample is good enough, and rejects it if the sample is poor enough.
 B) The sampling plan contains information about the sample size to be drawn and the critical acceptance or rejection numbers for that sample size.
 C) The steeper an operating characteristic curve, the better its ability to discriminate between good and bad lots.
 D) All of the above are true.
 E) All of the above are false.
Answer: D
Diff: 2
Key Term: Acceptance sampling
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
14) Acceptance sampling is usually used to control:
 A) the number of units of output from one stage of a process that are then sent to the next stage.
 B) the number of units delivered to the customer.
 C) the quality of workinprocess inventory.
 D) incoming lots of purchased products.
 E) all of the above.
Answer: D
Diff: 2
Key Term: Acceptance sampling
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
15) An operating characteristic (OC) curve describes:
 A) how many defects per unit are permitted before rejection occurs.
 B) the sample size necessary to distinguish between good and bad lots.
 C) the most appropriate sampling plan for a given incoming product quality level.
 D) how well an acceptance sampling plan discriminates between good and bad lots.
 E) none of the above.
Answer: D
Diff: 2
Key Term: Operating characteristic (OC) curve
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
16) An operating characteristics curve shows:
 A) upper and lower product specifications.
 B) product quality under different manufacturing conditions.
 C) how the probability of accepting a lot varies with the population percent defective.
 D) when product specifications don't match process control limits.
 E) how operations affect certain characteristics of a product.
Answer: C
Diff: 3
Key Term: Operating characteristic (OC) curve
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
17) Producer's risk is the probability of:
 A) accepting a good lot.
 B) rejecting a good lot.
 C) rejecting a bad lot.
 D) accepting a bad lot.
 E) none of the above.
Answer: B
Diff: 2
Key Term: Producer's risk
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
18) Which of the following is true regarding the relationship between AOQ and the true population percent defective?
 A) AOQ is greater than the true percent defective.
 B) AOQ is the same as the true percent defective.
 C) AOQ is less than the true percent defective.
 D) There is no relationship between AOQ and the true percent defective.
 E) The relationship between these two cannot be determined.
Answer: C
Diff: 3
Key Term: Average outgoing quality (AOQ)
AACSB: Analytical thinking
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
19) Under which of the following situations will the average outgoing quality (AOQ) decrease?
 A) The true percentage defective of the lot increases.
 B) The number of items in the sample decreases.
 C) The number of items in the lot decreases.
 D) The probability of accepting the lot for a given sample size and quantity defective increases.
 E) The difference between the number of items in the lot and the number of items in the sample increases.
Answer: C
Diff: 2
Key Term: Average outgoing quality (AOQ)
AACSB: Analytical thinking
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
20) A Type I error occurs when:
 A) a good lot is rejected.
 B) a bad lot is accepted.
 C) the number of defectives is very large.
 D) the population is worse than the AQL.
 E) none of the above
Answer: A
Diff: 2
Key Term: Type I error
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
21) A Type II error occurs when:
 A) a good lot is rejected.
 B) a bad lot is accepted.
 C) the population is worse than the LTPD.
 D) the proportion of defectives is very small.
 E) none of the above
Answer: B
Diff: 2
Key Term: Type II error
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
22) In most acceptance sampling plans, when a lot is rejected, the entire lot is inspected and all defective items are replaced. When using this technique the AOQ:
 A) worsens (AOQ becomes a larger fraction).
 B) improves (AOQ becomes a smaller fraction).
 C) is not affected, but the AQL is improved.
 D) is not affected.
 E) falls to zero.
Answer: B
Diff: 3
Key Term: Average outgoing quality (AOQ)
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
23) An acceptance sampling plan is to be designed to meet the organization's targets for product quality and risk levels. Which of the following is true?
 A) n and c determine the AQL.
 B) AQL, LTPD, α and β collectively determine n and c.
 C) n and c are determined from the values of AQL and LTPD.
 D) α and β are determined from the values of AQL and LTPD.
 E) None of the above is true.
Answer: B
Diff: 3
Key Term: Acceptance sampling
AACSB: Reflective thinking
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
24) When a lot has been accepted by acceptance sampling, we know that:
 A) it has more defects than existed before the sampling.
 B) it has had all its defects removed by 100% inspection.
 C) it will have the same defect percentage as the LTPD.
 D) it has no defects present.
 E) All of the above are false.
Answer: E
Diff: 2
Key Term: Acceptance sampling
AACSB: Reflective thinking
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
25) Which of the following statements about acceptance sampling is TRUE?
 A) The steeper an OC curve, the better it discriminates between good and bad lots.
 B) Acceptance sampling removes all defective items.
 C) Acceptance sampling of incoming lots is replacing statistical process control at the supplier.
 D) Acceptance sampling occurs continuously along the assembly line.
 E) All of the above are true.
Answer: A
Diff: 2
Key Term: Acceptance sampling
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
26) Which of the following is TRUE regarding the average outgoing quality level?
 A) An AOQ value of 1 is ideal, because all defects have been removed.
 B) AOQ is always greater than AQL but less than LTPD.
 C) AOQ rises (worsens) following inspection of failed lots.
 D) AOQ is very low (very good) for extremely poor quality lots.
 E) None of the above is true.
Answer: D
Diff: 3
Key Term: Average outgoing quality (AOQ)
AACSB: Analytical thinking
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
27) ________ is a method of measuring samples of lots or batches of product against predetermined standards.
Answer: Acceptance sampling
Diff: 2
Key Term: Acceptance sampling
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
28) A(n) ________ is a graph that describes how well an acceptance plan discriminates between good and bad lots.
Answer: OC or operating characteristic curve
Diff: 2
Key Term: Operating characteristic (OC) curve
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
29) The ________ is the lowest level of quality that we are willing to accept.
Answer: AQL or acceptable quality level
Diff: 2
Key Term: Acceptable quality level (AQL)
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
30) The ________ is the percentage defective in an average lot of goods inspected through acceptance sampling.
Answer: AOQ or average outgoing quality
Diff: 2
Key Term: Average outgoing quality (AOQ)
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
31) What is acceptance sampling?
Answer: Acceptance sampling is a method of measuring random samples of lots or batches of products against predetermined standards.
Diff: 2
Key Term: Acceptance sampling
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
32) What is the purpose of the Operating Characteristics curve?
Answer: An OC curve plots the probability of acceptance against the percentage of defects in the lot. It therefore shows how well an acceptance sampling plan discriminates between good and bad lots.
Diff: 2
Key Term: Operating characteristic (OC) curve
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
33) What is the AOQ of an acceptance sampling plan?
Answer: The AOQ is the average outgoing quality. It is the percentage defective in an average lot of goods inspected through acceptance sampling.
Diff: 2
Key Term: Average outgoing quality (AOQ)
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
34) Define consumer's risk. Is it a Type I or Type II error? What is the symbol for its value?
Answer: The consumer's risk is the probability of accepting a bad lot. It is a Type II error. Its value is beta.
Diff: 2
Key Term: Consumer's risk
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
35) What four elements determine the value of average outgoing quality?
Answer: The four elements are the true percentage defective of the lot, the probability of accepting the lot for a given sample size and quantity defective, the number of items in the lot, and the number of items in the sample.
Diff: 2
Key Term: Average outgoing quality (AOQ)
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
36) What do the terms producer's risk and consumer's risk mean?
Answer: Producer's risk: the risk of rejecting a good lot; Consumer's risk: the risk of accepting a defective lot.
Diff: 2
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
37) Pierre's Motorized Pirogues and Mudboats is setting up an acceptance sampling plan for the special air cleaners he manufactures for his boats. His specifications, and the resulting plan, are shown on the POM for Windows output below. In relatively plain English (someone else will translate for Pierre), explain exactly what he will do when performing the acceptance sampling procedure, and what actions he might take based on the results.
Answer: Pierre should select samples of size 175 from his lots of air cleaners. He should count the number of defects in each sample. If there are 4 or fewer defects, the lot passes inspection. If there are 5 or more defects, the lot fails inspection. Lots that fail can be handled several ways: they can be 100% inspected to remove defects; they can be sold at a discount; they can be destroyed; they can be sent back for rework, etc.
Diff: 2
Key Term: Acceptance sampling
AACSB: Analytical thinking
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
38) Pierre's Motorized Pirogues and Mudboats is setting up an acceptance sampling plan for the special air cleaners he manufactures for his boats. His specifications, and the resulting plan, are shown on the POM for Windows output below. Pierre is a bit confused. He mistakenly thinks that acceptance sampling will reject all bad lots and accept all good lots. Explain why this will not happen.
Answer: Acceptance sampling cannot discriminate perfectly between good and bad lots; this is illustrated by the OC curve that is not straight up and down. In this example, "good" lots will still be rejected almost 5% of the time. "Bad" lots will still be accepted almost 5% of the time.
Diff: 3
Key Term: Operating characteristic (OC) curve
AACSB: Analytical thinking
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods
39) Pierre's Motorized Pirogues and Mudboats is setting up an acceptance sampling plan for the special air cleaners he manufactures for his boats. His specifications, and the resulting plan, are shown on the POM for Windows output below. Pierre wants acceptance sampling to remove ALL defects from his production of air cleaners. Explain carefully why this won't happen.
Answer: Acceptance sampling is not intended to remove all defects, nor will it. Consider a lot with a defect rate of 0.005 in this example. If the sample is representative, the lot will pass inspectionwhich means that no one will inspect the lot for defects. The defects that were present before sampling are still there. Generally, acceptance sampling passes some lots and rejects others. Defects can only be removed from those lots that fail inspection.
Diff: 3
Key Term: Average outgoing quality (AOQ)
AACSB: Analytical thinking
Objective: LO S6.7 Explain acceptance sampling
Learning Outcome: Apply basic statistical process control (SPC) methods

Operations Management: Sustainability and Supply Chain Management, 12th Edition (Free Ebooks Download)
Jay Heizer, Texas Lutheran University
Barry Render, Graduate School of Business, Rollins College
Chuck Munson, Carson College of Business, Washington State University
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