1) Some degree of variability is present in almost all processes.

2) The purpose of process control is to detect when natural causes of variation are present.

3) A normal distribution is generally described by its two parameters: the mean and the range.

4) A process is said to be in statistical control when assignable causes are the only sources of variation.

5) Mistakes stemming from workers’ inadequate training represent an assignable cause of variation.

6) Averages of small samples, not individual measurements, are generally used in statistical process control.

7) The x-bar chart indicates that a gain or loss of uniformity has occurred in dispersion of a production process.

8) The Central Limit Theorem states that when the sample size increases, the distribution of the sample means will approach the normal distribution.

9) In statistical process control, the range often substitutes for the standard deviation.

10) If the process average is in control, then the process range must also be in control.

11) A process range chart illustrates the amount of variation within the samples.

12) Mean charts and range charts complement one another, one detecting shifts in process average, the other detecting shifts in process dispersion.

13) X-bar charts are used when we are sampling attributes.

14) To measure the voltage of batteries, one would sample by attributes.

15) A p-chart is appropriate to plot the number of typographic errors per page of text.

16) A c-chart is appropriate to plot the number of flaws in a bolt of fabric.

17) The x-bar chart, like the c-chart, is based on the exponential distribution.

18) A process that is in statistical control will always yield products that meet their design specifications.

19) The higher the process capability ratio, the greater the likelihood that process will be within design specifications.

20) The Cpk index measures the difference between desired and actual dimensions of goods or services produced.

21) Acceptance sampling accepts or rejects an entire lot based on the information contained in the sample.

22) A lot that is accepted by acceptance sampling is certified to be free of defects.

23) In acceptance sampling, a manager can reach the wrong conclusion if the sample is not representative of the population it was drawn from.

24) The probability of rejecting a good lot is known as consumer’s risk.

25) An acceptance sampling plan must define “good lots” and “bad lots” and specify the risk level associated with each one.

26) The acceptable quality level (AQL) is the average level of quality we are willing to accept.

27) The steeper an OC curve, the better it discriminates between good and bad lots.

**Multiple Choice**

1) If a sample of items is taken and the mean of the sample is outside the control limits the process is

2) The causes of variation in statistical process control are

3) Natural variations

4) Natural variations

5) Assignable variation

6) Assignable causes

7) Control charts for variables are based on data that come from

8) The purpose of an x-bar chart is to determine whether there has been a

9) The number of defects after a hotel room cleaning (sheets not straight, smears on mirror, missed debris on carpet, etc) should be measured using a(n)

10) The number of late insurance claim payouts per 100 should be measured with a

11) The upper and lower limits for diving ring diameters made by John’s Swimming are 40 and 39 cm. 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?

12) 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, 8. What is the upper control limit for the c-chart?

13) A process that is assumed to be in control with limits of 89 +/- 2 had sample averages of the following 87.1, 87, 87.2, 89, 90, 89.5, 88.5, and 88. Is the process in control?

14) Which of the following was Unisys Corp.’s failed quality measure in its management of Florida Health Care Services?

15) 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, 11. Find the standard deviation of the sampling distribution. (Hint- Use p-bar formula)

16) An x-bar control chart was examined and no data points fell outside of the limits. Can this process be considered in control?

17) Statistical process control charts

18) Consumer’s risk is the probability of

19) The Central Limit Theorem

20) For an x-bar chart where the standard deviation is known, the Upper Control Limit

21) Up to three standard deviations above or below the centerline is the amount of variation that statistical process control allows for

22) A manager wants to build 3-sigma 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, the UCL and LCL will be

23) The type of inspection that classifies items as being either good or defective is

24) The p-chart tells us whether there has been a

25) The mean and standard deviation for a process for which we have a substantial history are

µ = 120 and σ = 2. For the x-bar chart, a sample size of 16 will be used. What is the mean of the sampling distribution?

26) 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?

27) If = 23 ounces, σ = 0.4 ounces, and n = 16, the ±3σ control limits will be

28) The usual purpose of an R-chart is to signal whether there has been a

29) A manager wishes to build a 3-sigma 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. The UCL and LCL for this range chart are

30) Plots of sample ranges indicate that the most recent value is below the lower control limit. What course of action would you recommend?

31) To set -chart upper and lower control limits, one must know the process central line, which is the

32) According to the text, the most common choice of limits for control charts is usually

33) Which of the following is true of a p-chart?

34) The normal application of a p-chart is in

35) The statistical process chart used to control the number of defects per unit of output is the

36) The c-chart signals whether there has been a

37) The local newspaper receives several complaints per day about typographic errors. Over a seven-day period, the publisher has received calls from readers reporting the following 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?

38) 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?

39) 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?

40) A run test is used

41) The process capability measures Cp and Cpk differ because

42) A Cp of 1.33 indicates how many sigma limits

43) Which of the following is true regarding the process capability index Cpk?

44) If the Cpk index exceeds 1

45) The statistical definition of Six Sigma allows for 3.4 defects per million. This is achieved by a Cpk index of

46) A Cpk index of 1.00 equates to a defect rate of

47) Acceptance sampling

48) Acceptance sampling’s primary purpose is to

49) An acceptance sampling plan’s ability to discriminate between low quality lots and high quality lots is described by

50) Acceptance sampling

51) Which of the following statements on acceptance sampling is true?

52) Acceptance sampling is usually used to control

53) An operating characteristic (OC) curve describes

54) An operating characteristics curve shows

55) Producer’s risk is the probability of

56) Which of the following is true regarding the relationship between AOQ and the true population percent defective?

57) Average outgoing quality (AOQ) usually

58) A Type I error occurs when

59) A Type II error occurs when

60) 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

61) 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?

62) When a lot has been accepted by acceptance sampling, we know that

63) Which of the following statements about acceptance sampling is true?

64) Which of the following is true regarding the average outgoing quality level?

**Topic: Acceptance sampling**

**Objective: LO6-Supplement-7**

1) __________ is variation in a production process that can be traced to specific causes.

2) The __________ is the chief way to control attributes.

3) If a process has only natural variations, __________ percent of the time the sample averages will fall inside the (or ) control limits.

4) The __________ is a quality control chart that indicates when changes occur in the central tendency of a production process.

5) The __________ are calculated to show how much allowance should be made for natural variation.

6) The __________ is a quality control chart used to control the number of defects per unit of output.

7) The term __________ is used to describe how well a process makes units within design specifications (or tolerances).

8) A Cpk index greater than __________ is a capable process, one that generates fewer than 2.7 defects per 1000 at the ±3σ level.

9) __________ is a method of measuring samples of lots or batches of product against predetermined standards.

10) A(n) __________ is a graph that describes how well an acceptance plan discriminates between good and bad lots.

11) The __________ is the lowest level of quality that we are willing to accept.

12) The __________ is the percent defective in an average lot of goods inspected through acceptance sampling.

13) What is the basic objective of a process control system?

14) Briefly explain what the Central Limit Theorem has to do with control charts.

15) What are the three possible results (or findings) from the use of control charts?

16) Why do range charts exist? Aren’t x-bar charts enough?

17) Examine the Statistical Process Control outputs below. Answer the following questions.

18) Can “in control” and “capable” be shown on the same chart?

19) What is the difference between natural and assignable causes of variation?

20) Why are x-bar and R-charts usually used hand in hand?

21) What does it mean for a process to be “capable”?

22) What is the difference between the process capability ratio Cp and the process capability index Cpk?

23) 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 x-bar on the LSL, the formula guarantees a Cpk of zero.

24) What is acceptance sampling?

25) Why doesn’t acceptance sampling remove all defects from a batch?

26) What is the purpose of the Operating Characteristics curve?

27) What is the AOQ of an acceptance sampling plan?

28) Define consumer’s risk. How does it relate to the errors of hypothesis testing? What is the symbol for its value?

29) What four elements determine the value of average outgoing quality? Why does this curve rise, peak, and fall?

30) What do the terms producer’s risk and consumer’s risk mean?

31) 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.

32) 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.

33) 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.

**Topic: Acceptance sampling**

**Objective: LO6-Supplement-7**

1) 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 appears 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 control limits that allow for natural variations.

(c) Is this process in control?

2) 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 appears 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, what is the probability of concluding that this process is out of control when it is actually in control?

(b) With the UCL and LCL of part a, what do you conclude about this process–is it in control?

3) 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 x-bar and R chart?

4) The defect rate for a product has historically been about 1.6%. What are the upper and lower control chart limits for a p-chart, if you wish to use a sample size of 100 and 3-sigma limits?

5) A small, independent amusement park collects data on the number of cars with out-of-state license plates. The sample size is fixed at n=25 each day. Data from the previous 10 days indicate the following number of out-of-state license plates:

Day Out-of-state 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 out-of-state 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.

6) 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.

SampleMean 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

7) McDaniel Shipyards wants to develop control charts 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 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

8) The mean and standard deviations for a process are μ= 90 and σ = 9. For the variable control chart, a sample size of 16 will be used. Calculate the standard deviation of the sampling distribution.

9) If μ = 9 ounces, σ = 0.5 ounces, and n = 9, calculate the 3-sigma control limits.

10) A hospital-billing 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). Among the last 100 bills inspected, the defect rate has been 16%. Calculate the upper and lower limits for the billing process for 99.7% confidence.

11) 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.

12) 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 3-sigma control limits for this R-chart?

13) A woodworker is concerned about the quality of the finished appearance of her work. In sampling units of a split-willow hand-woven basket, she has found the following number of finish defects in ten units sampled: 4, 0, 3, 1, 2, 0, 1, 2, 0, 2.

a. Calculate the average number of defects per basket

b. If 3-sigma control limits are used, calculate the lower control limit, centerline, and upper control limit.

14) The width of a bronze bar is intended to be one-eighth 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 x-bar and R-chart for this process, using 3-sigma limits?

15) A part that connects two levels should have a distance between the two holes of 4″. It has been determined that x-bar and R-charts 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

16) 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 x-bar and R-chart using this data. Is the process in control?

SampleMean 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

17) Larry’s boat shop wants to monitor the number of blemishes in the paint of each boat. Construct a 3-sigma c-chart 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

18) 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. What are the upper and lower specification limits for this product? The process is currently operating at a mean thickness of 2.50 mm. (a) What is the Cp for this process? (b) About what percent of all units of this liner will meet specifications? Does this meet the technical definition of Six Sigma?

19) 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. What are the upper and lower specification limits for this product? The process is currently operating at a mean thickness of 2.50 mm. (a) What is the Cp for this process? (b) The purchaser of these parts requires a capability index of 1.50. Is this process capable? Is this process good enough for the supplier? (c) 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?

20) 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?

21) 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 is the Cp and Cpk for this process? Is this process capable of producing the desired part?

22) In the table below are selected values for the OC curve for the acceptance sampling plan n=210, c=6. Upon failed inspection, defective items are replaced. Calculate the AOQ for each data point. (You may assume that the population is much larger than the sample.) Plot the AOQ curve. At approximately what population defective rate is the AOQ at its worst? Explain how this happens. How well does this plan meet the specifications of AQL=0.015, α=0.05; LTPD=0.05, β=0.10? Discuss.

Population percent defective Probability of acceptance

0.00 1.00000

0.01 0.99408

0.02 0.86650

0.03 0.55623

0.04 0.26516

0.05 0.10056

0.06 0.03217

0.07 0.00905

0.08 0.00231

0.09 0.00054

0.10 0.00012

23) In the table below are selected values for the OC curve associated with the acceptance sampling plan n=50, c=1. (Watch out–the points are not evenly spaced.) Assume that upon failed inspection, defective items are replaced. Calculate the AOQ for each data point. (You may assume that the population is much larger than the sample.) Plot the AOQ curve. At approximately what population defective rate is the AOQ at its worst? Explain how this happens. How well does this plan meet the specifications of AQL=0.0050, α =0.05; LTPD=0.05, β =0.10? Discuss.

Population percent defective Probability of acceptance

0.005 0.97387

0.01 0.91056

0.02 0.73577

0.03 0.55528

0.04 0.40048

0.05 0.27943

0.06 0.19000

0.08 0.08271

24) 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.

25) 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 p-chart at the 95% confidence level. Is the process in control?

26) 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?

27) 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.

28) A car mechanic is thinking of guaranteeing customers that an oil change will take no more than 15 minutes with a 99.73% confidence level. He takes a few samples of size 5 and finds the process mean to be 13 minutes with a standard deviation of .2 minutes and average sample range of 1.2 minutes. Find the A2, D4, and D3 values and use them to compute the upper and lower limits for an x-bar chart. Use the upper limit to determine if the mechanic can offer a 15 minute guarantee. Assume the mechanic plots the samples on the x-bar control chart and finds the process is in control, is there anything else the mechanic is missing to ensure the process is in control?

29) 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 limits of 20.5 and 19.5 oz, respectively, with a mean of 19.8 oz

and standard deviation of .3 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

or if you should be looking for assignable variation.

30) A retail store manager is trying to improve and control the rate at which cashiers sign customers up for store credit cards. Suppose that the manager wants the maximum standard deviation of the sampling distribution to be 5% and he cannot estimate p-bar. How many observations per sample would this require?

31) 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. P-bar

is found to be .05, and the manager does not want a lower limit below .0064. What z-value would

this imply, and how confident can he be that the true lower limit is greater than or equal to .0064?

32) A retail store manager is trying to improve and control the rate at which cashiers sign customers up

for store credit cards. After posting a p-chart of the store’s credit card sign-ups the manager takes

new samples of size 50 three weeks later. He finds that each sample of 50 contained 5 credit card

signups on average. Find p-bar and 99.73% control limits.

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