15-41

15.8.4 The concentration of a chemical product is measured by taking four samples from each batch of material. The average

concentration of these measurements for the last 20 batches is shown in the following table:

Use σ = 8 and assume that the desired process target is 100.

Batch

Concentration

Batch

Concentration

1

104.5

11

95.4

2

99.9

12

94.5

3

106.7

13

104.5

4

105.2

14

99.7

5

94.8

15

97.7

6

94.6

16

97

7

104.4

17

95.8

8

99.4

18

97.4

9

100.3

19

99

10

100.3

20

102.6

(a) Construct an EWMA control chart with

= 0.2. Does the process appear to be in control?

(b) Construct an EWMA control chart with

= 0.5. Compare your results to part (a).

(c) If the concentration shifted to 104, would you prefer the chart in part (a) or (b)? Explain.

(a) The process appears to be in control.

Applied Statistics and Probability for Engineers, 7th edition 2017

15-42

(b) The process appears to be in control.

15.8.5 Heart rate (in counts/minute) is measured every 30 minutes. The results of 20 consecutive measurements are as follows:

Use μ = 70 and σ = 3.

Sample

Heart Rate

Sample

Heart Rate

1

68

11

79

2

71

12

79

3

67

13

78

4

69

14

78

5

71

15

78

6

70

16

79

7

69

17

79

8

67

18

82

9

70

19

82

10

70

20

81

(a) Construct an EWMA control chart with

= 0.1. Use L = 2.81. Does the process appear to be in control?

(b) Construct an EWMA control chart with

= 0.5. Use L = 3.07. Compare your results to those in part (a).

(c) If the heart rate mean shifts to 76, approximate the ARLs for the charts in parts (a) and (b).

Applied Statistics and Probability for Engineers, 7th edition 2017

(a) UCL = 71.93 and LCL = 68.07, the chart signals at observation 13.

(b) UCL = 75.32 and LCL = 64.68, the chart signals at observation 19.

15-44

15.8.6 The number of influenza patients (in thousands) visiting hospitals weekly is shown in the following table.

Use μ = 160 and σ = 2.

Sample

Number of Patients

Sample

Number of Patients

1

162.27

13

159.989

2

157.47

14

159.09

3

157.065

15

162.699

4

160.45

16

163.89

5

157.993

17

164.247

6

162.27

18

162.7

7

160.652

19

164.859

8

159.09

20

163.65

9

157.442

21

165.99

10

160.78

22

163.22

11

159.138

23

164.338

12

161.08

24

164.83

(a) Construct an EWMA control chart with

= 0.1. Use L = 2.81. Does the process appear to be in control?

(b) Construct an EWMA control chart with

= 0.5. Use L = 3.07. Compare your results to those in part (a).

Applied Statistics and Probability for Engineers, 7th edition 2017

Section 15-10

15.10.1 Suppose that the cost of a major repair without the extended warranty in Example 15-8 is changed to $1000.

Determine the decision selected based on the minimax, most probable, and expected cost criteria.

15.10.2 Reconsider the extended warranty decision in Example 15-8. Suppose that the probabilities of the major, minor,

and no repair states are changed to 0.2, 0.4, and 0.4, respectively. Determine the decision selected based on the

minimax, most probable, and expected cost criteria.

15-46

15.10.3 Analyze Example 15-9 based on the most probable criterion and determine the actions that are selected at each decision

node. Do any actions differ from those selected in the example?

Decisions:

1. When a new product is developed and a unique product is achieved, the most probable outcomes for the high and

15.10.4 Analyze Example 15-9 based on the expected profit criterion and determine the actions that are selected at each

decision node. Do any actions differ from those selected in the example?

Decisions:

1. When a new product is developed and a unique product is achieved, the expected outcomes for the high

2. When a new product is developed and a unique product is not achieved, the expected outcomes for the high and low

Applied Statistics and Probability for Engineers, 7th edition 2017

15-47

Supplementary Exercises

15.S7 The diameter of fuse pins used in an aircraft engine application is an important quality characteristic. Twenty-five

samples of three pins each are shown as follows:

Sample Number

Diameter

1

64.030

64.002

64.019

2

63.995

63.992

64.001

3

63.988

64.024

64.021

4

64.002

63.996

63.993

5

63.992

64.007

64.015

6

64.009

63.994

63.997

7

63.995

64.006

63.994

8

63.985

64.003

63.993

9

64.008

63.995

64.009

10

63.998

74.000

63.990

11

63.994

63.998

63.994

12

64.004

64.000

64.007

13

63.983

64.002

63.998

14

64.006

63.967

63.994

15

64.012

64.014

63.998

16

64.000

63.984

64.005

17

63.994

64.012

63.986

18

64.006

64.010

64.018

19

63.984

64.002

64.003

20

64.000

64.010

64.013

21

63.988

64.001

64.009

22

64.004

63.999

63.990

23

64.010

63.989

63.990

24

64.015

64.008

63.993

25

63.982

63.984

63.995

(a) Set up

X

and R charts for this process. If necessary, revise limits so that no observations are out of control.

(b) Estimate the process mean and standard deviation.

(c) Suppose that the process specifications are at 64 ± 0.02. Calculate an estimate of PCR. Does the process meet a

minimum capability level of PCR ≥ 1.33?

(d) Calculate an estimate of PCRk. Use this ratio to draw conclusions about process capability.

(e) To make this process a 6-sigma process, the variance

2 would have to be decreased such that PCRk = 2.0. What

should this new variance value be?

(f) Suppose that the mean shifts to 64.01. What is the probability that this shift is detected on the next sample? What is

the ARL after the shift?

Applied Statistics and Probability for Engineers, 7th edition 2017

(a) The process is not in control. The control chart follows.

Sample 10 is removed to obtain the following chart.

15-49

15.S8 Rework Exercise 15.S1 with

X

and S charts.

(a)

The following chart is obtained with subgroup 10 excluded:

Applied Statistics and Probability for Engineers, 7th edition 2017

15-50

(e) Same as the referenced exercise

(f)

− − −

=

63.9818 64.01 64.0187 64.01

(63.98 64.02) 0.01065 / 3 0.01065 / 3

x

X

P X P

15.S9 Plastic bottles for liquid laundry detergent are formed by blow molding. Twenty samples of n = 100 bottles are

inspected in time order of production, and the fraction defective in each sample is reported. The data are as follows:

Sample

Fraction Defective

1

0.12

2

0.15

3

0.18

4

0.10

5

0.12

6

0.11

7

0.05

8

0.09

9

0.13

10

0.13

11

0.10

12

0.07

13

0.12

14

0.08

15

0.09

16

0.15

17

0.10

18

0.06

19

0.12

20

0.13

(a) Set up a P chart for this process. Is the process in statistical control?

Applied Statistics and Probability for Engineers, 7th edition 2017

15-51

(b) Suppose that instead of n = 100, n = 200. Use the data given to set up a P chart for this process. Revise the control

limits if necessary.

(c) Compare your control limits for the P charts in parts (a) and (b) Explain why they differ. Also, explain why your

assessment about statistical control differs for the two sizes of n.

(a)

(b)

Applied Statistics and Probability for Engineers, 7th edition 2017

15-52

15.S10 The following data from the U.S. Department of Energy Web site (www.eia.doe.gov) reported the total U.S. renewable

energy consumption by year (trillion BTU) from 1973 to 2015.

(a) Using all the data, find calculate control limits for a control chart for individual measurements, construct the chart,

and plot the data.

(b) Do the data appear to be generated from an in-control process? Comment on any patterns on the chart.

(a) A control chart for individuals

A control chart for moving ranges

(b) The data does not appear to be generated from an in-control process. Observations at the year 1973–1981, 2001,

Applied Statistics and Probability for Engineers, 7th edition 2017

15-53

15.S11 An article in Quality Engineering [“Is the Process Capable? Tables and Graphs in Assessing Cpm” (1992,

Vol. 4(4)]. Considered manufacturing data. Specifications for the outer diameter of the hubs were 60.3265 ± 0.001 mm.

A random sample of 20 hubs was taken and the data are shown in the following table:

Sample

x

Sample

x

1

60.3262

11

60.3262

2

60.3262

12

60.3262

3

60.3262

13

60.3269

4

60.3266

14

60.3261

5

60.3263

15

60.3265

6

60.3260

16

60.3266

7

60.3262

17

60.3265

8

60.3267

18

60.3268

9

60.3263

19

60.3262

10

60.3269

20

60.3266

(a) Construct a control chart for individual measurements. Revise the control limits if necessary.

(b) Compare your chart in part (a) to one that uses only the last (least significant) digit of each diameter as the

measurement. Explain your conclusion.

(c) Estimate μ and

from the moving range of the revised chart and use this value to estimate PCR and PCRk and

interpret these ratios.

(a) Using I-MR chart.

Applied Statistics and Probability for Engineers, 7th edition 2017

15-54

(b) The chart is identical to the chart in part (a) except for the scale of the individuals chart.

15.S12 Suppose that an

X

control chart with 2-sigma limits is used to control a process. Find the probability that a false out-of-

control signal is produced on the next sample. Compare this with the corresponding probability for the chart with 3-

sigma limits and discuss. Comment on when you would prefer to use 2-sigma limits instead of

3-sigma limits.

Applied Statistics and Probability for Engineers, 7th edition 2017

15-55

15.S13 The following dataset was considered in Quality Engineering [“Analytic Examination of Variance Components”

(1994–1995, Vol. 7(2)]. A quality characteristic for cement mortar briquettes was monitored. Samples of size

n = 6 were taken from the process, and 25 samples from the process are shown in the following table:

(a) Using all the data, calculate trial control limits for X and S charts. Is the process in control?

Batch

X

s

1

572.00

73.25

2

583.83

79.30

3

720.50

86.44

4

368.67

98.62

5

374.00

92.36

6

580.33

93.50

7

388.33

110.23

8

559.33

74.79

9

562.00

76.53

10

729.00

49.80

11

469.00

40.52

12

566.67

113.82

13

578.33

58.03

14

485.67

103.33

15

746.33

107.88

16

436.33

98.69

17

556.83

99.25

18

390.33

17.35

19

562.33

75.69

20

675.00

90.10

21

416.50

89.27

22

568.33

61.36

23

762.67

105.94

24

786.17

65.05

25

530.67

99.42

(b) Suppose that the specifications are at 580 ± 250. What statements can you make about process capability? Compute

estimates of the appropriate process capability ratios.

(c) To make this process a “6-sigma process,” the variance

2 would have to be decreased such that PCRk = 2.0. What

should this new variance value be?

(d) Suppose the mean shifts to 600. What is the probability that this shift is detected on the next sample? What is the

ARL after the shift?

Applied Statistics and Probability for Engineers, 7th edition 2017

15-56

(a)Trial control limits:

(b) An estimate of

is given by

==

4

/ 86.4208/ 0.9515 90.8259Sc

....

3

(d) The probability that

X

falls within the control limits is

Applied Statistics and Probability for Engineers, 7th edition 2017

15-57

15.S14 Suppose that a process is in control and an

X

chart is used with a sample size of 4 to monitor the process. Suddenly

there is a mean shift of 1.5

.

(a) If 3-sigma control limits are used on the

X

chart, what is the probability that this shift remains undetected for three

consecutive samples?

(b) If 2-sigma control limits are in use on the

X

chart, what is the probability that this shift remains undetected for three

consecutive samples?

(c) Compare your answers to parts (a) and (b) and explain why they differ. Also, which limits you would recommend

using and why?

(a) Let p denote the probability that a point plots outside of the control limits when the mean has shifted from

Applied Statistics and Probability for Engineers, 7th edition 2017

15-58

(b) If 2-sigma control limits were used, then

15.S15 Consider the diameter data in Exercise 15.S1.

(a) Construct an EWMA control chart with

= 0.2 and L = 3. Comment on process control.

(b) Construct an EWMA control chart with

= 0.5 and L = 3 and compare your conclusion to part (a).

(a) The following control chart use the average range from 25 subgroups of size 3 to estimate the process standard

Applied Statistics and Probability for Engineers, 7th edition 2017

(b) The following control chart use the average range from 25 subgroups of size 3 to estimate the process standard

15.S16 Consider a control chart for individuals applied to a continuous 24-hour chemical process with observations taken

every hour.

(a) If the chart has 3-sigma limits, how many false alarms would occur each 30-day month, on the average, with this

chart?

(b) Suppose that the chart has 2-sigma limits. Does this reduce the ARL for detecting a shift in the mean of magnitude

?

(c) Find the in-control ARL if 2-sigma limits are used on the chart. How many false alarms would occur each month

with this chart? Is this in-control ARL performance satisfactory? Explain your answer.

(b) With 2-sigma limits the probability of a point plotting out of control is determined as follows, when μ = μ0 + σ

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