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Lesson 14 Statistical Process Control Solutions
Solved Problem #2: see textbook
Solved Problem #4: see textbook
Solved Problem #5: see textbook
Solved Problem #6: see textbook (manual problem)

#1: Checkout time at a supermarket is monitored using a range and mean chart. Six samples which contain
20 observations per sample have been collected and the sample means and sample ranges have been computed as shown below.
Sample
1
2
3
4
5
6

Mean
3.06
3.15
3.11
3.13
3.06
3.09

Range
0.42
0.50
0.41
0.46
0.46
0.45

Perform the manual calculations necessary to answer the following questions.
a.

What is the sample size?
20

b.

Is the variability of the checkout time known or unknown?
Unknown

c.

Which chart is used to analyze the checkout time variability?
Range chart

d.

Which chart is used to analyze the checkout time?
Mean chart

e.

Should the range or mean chart be analyzed first? Explain your answer.
Range chart should be analyzed before the mean chart because the variability of the process must be in control before the mean chart can be analyzed.

Analyze the range chart by answering the following questions.
f.

What is the grand range?
.45

g.

Calculate the centerline, upper and lower control limits for the range chart?

1

Centerline = grand range = R = .45
LCL = R * D3 = .45 * .415 = .18675
UCL = R * D4 = .45 * 1.585 = .71325
h.

Plot the range chart showing the centerline, upper and lower control limits.

0.8
UCL

0.7
0.6
0.5

Centerline

Sample Range
0.4
0.3
0.2

LCL

0.1
0
0

i.

1

2

3

4

5

6

7

Are the sample ranges within the control limits?
Yes

j.

Count the number of A/B and U/D runs for the range chart.
A/B
U/D

k.

BABAAB
UDUDD

5 runs
4 runs

What is the expected number of A/B runs?

E (r ) A / B =
l.

# Observations
6
+1 = +1 = 4
2
2

What is the variability of the expected number of A/B runs?

σ (r ) A / B =

#Observations − 1
=
4

6 −1
=1.118
4

m. What is the test statistic for the expected number of runs?

Z (r ) A / B =
n.

# runs − E (r ) A / B 5 − 4
=
= .89
1.118
σ (r ) A / B

Based on 3-sigma control limits for the runs test, does the pattern analysis indicate the A/B runs are random? Why?

2

Yes, because it is between -3 and 3.
o.

What is the expected number of U/D runs?

E (r )U / D =
p.

What is the variability of the expected number of A/B runs?

σ (r )U / D =
q.

2(# Observations) − 1 2 * 6 − 1
=
= 3.67
3
3

16(#Observations ) − 29
16 * 6 − 29
=
= .86
90
90

What is the test statistic for the expected number of runs?

Z (r )U / D =
r.

# runs − E (r )U / D 4 − 3.67
=
= .38 σ (r )U / D
.86

Based on 3-sigma control limits for the runs test, does the pattern analysis indicate the U/D runs are random? Why?
Yes, because it is between -3 and 3

s.

Is the variability of the checkout times in control (i.e is the range chart in control)? Why?
Yes. The sample ranges are within the limits, the A/B runs are random, and the U/D runs are random.

Analyze the mean chart by answering the following questions.
t. What is the grand mean?
3.10
u.

Calculate the centerline, upper and lower control limits for the mean chart?

Centerline = grand average = X = 3.10
LCL = X − A2 R = 3.10 − .18 * .45 = 3.019
UCL = X + A2 R = .45 + .18 * .45 = 3.181
v.

Plot the mean chart showing the centerline, upper and lower control limits.

3

3.2
UCL

3.18
3.16
3.14
3.12
3.1

Centerline

3.08
3.06
Sample Mean
3.04
3.02

LCL

3
0

1

2

3

4

5

6

7

w. Are the sample means within the control limits?
Yes
x.

Count the number of A/B and U/D runs for the mean chart.
A/B
U/D

y.

BAAABB
UDUDU

3 runs
5 runs

What is the expected number of A/B runs?

E (r ) A / B =
z.

# Observations
+1 = 4
2

What is the variability of the expected number of A/B runs?

σ (r ) A / B =

#Observations − 1
= 1.118
4

aa. What is the test statistic for the expected number of runs?

Z (r ) A / B =

# runs − E (r ) A / B 3 − 4
=
= −.89 σ (r ) A / B
1.118

bb. Based on 3-sigma control limits for the runs test, does the pattern analysis indicate the A/B runs are random? Why?
Yes, because it is between -3 and 3. cc. What is the expected number of U/D runs?

E (r )U / D =

2(# Observations) − 1
= 3.67
3

dd. What is the variability of the expected number of A/B runs?

4

σ (r )U / D =

16(#Observations ) − 29
16 * 6 − 29
=
= .86
90
90

ee. What is the test statistic for the expected number of runs?

Z (r )U / D =

# runs − E (r )U / D 5 − 3.67
=
= 1.55 σ (r )U / D
.86

ff. Based on 3-sigma control limits for the runs test, does the pattern analysis indicate the U/D runs are random? Why?
Yes, because it is between -3 and 3 gg. Are the average checkout times in control (i.e. is the mean chart in control)? Why?
Yes. The sample means are within the limits and the A/B runs are random, and the U/D runs are random.
Based on your analysis is the checkout process in control? Why?
Yes, both the range and mean charts indicate the process is in control.

#2: The Bayfield Mud Company supplies railroad boxcars of 50 pound bags of mud treating agents to the
Wet-Land Drilling Company. Mud treating agents are used to control the pH and other chemical properties of the cone during oil drilling operations. Wet-Land has complained to Bayfield that it’s most recent shipment of bags was underweight by 5%. This is a problem which Wet-Land needs corrected because the use of under weight bags may result in poor chemical control during drilling which may hurt drilling efficiency resulting in serious economic consequences.
Afraid of losing a long time customer, Bayfield immediately began investigating their production process.
Management suspected that the causes of the problem were their recently added third shift and the fact that all three of the shifts were under pressure to increase output to meet increasing demand for the product.
Their quality control staff began taking random samples of the output of 3 bags per hour. They observed the process for 36 hours beginning at 6am.
To avoid manually entering the data into the templates it can be copied and pasted from Data Sets on the
Lesson Page. Use “copy, paste special, values” to transfer the data to the appropriate SPC template worksheet.. 5

Sample
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36

Time
6am
7am
8am
9am
10am
11am
12noon
1pm
2pm
3pm
4pm
5pm
6pm
7pm
8pm
9pm
10pm
11pm
12pm
1am
2am
3am
4am
5am
6am
7am
8am
9am
10am
11am
12noon
1pm
2pm
3pm
4pm
5pm

Bag 1
49.60
50.20
50.60
50.80
49.90
50.30
48.60
49.00
49.00
49.80
50.30
51.40
51.60
51.80
51.00
50.50
49.20
49.00
48.40
47.60
47.40
48.20
48.00
48.40
48.60
50.00
49.80
50.30
50.20
50.00
50.00
50.10
49.70
48.40
47.20
46.80

Bag 2
48.70
49.10
49.60
50.20
49.20
48.60
46.20
46.40
46.00
48.20
49.20
50.00
49.20
50.00
48.60
49.40
46.10
46.30
45.40
44.30
44.10
45.20
45.50
47.10
47.40
49.20
49.00
49.40
49.60
49.00
48.80
49.40
48.60
47.20
45.30
44.10

Bag 3
50.70
51.20
51.40
51.80
52.30
51.70
50.40
50.00
50.60
50.80
52.70
55.30
54.70
55.60
53.20
52.40
50.70
50.80
50.20
49.70
49.60
49.00
49.10
49.60
52.00
52.20
52.40
51.70
51.80
52.30
52.40
53.60
51.00
51.70
50.90
49.00

Answer the following questions.
a.

What is the sample size?
3

b.

Is the variability of the checkout time known or unknown?
Unknown

c.

Which chart is used to analyze the variability in the bag filling process?

6

Range chart
d.

Which chart is used to analyze the amount of fill in each bag?
Mean chart

e.

Should the range or mean chart be analyzed first? Explain your answer.
Range chart should be analyzed before the mean chart because the variability of the process must be in control before the mean chart can be analyzed.

Analyze the range chart by answering the following questions.
f.

What is the grand range?
3.7472

g.

Calculate the centerline, upper and lower control limits for the range chart?
3σ Control Chart Settings
UCL
9.6454
Centerline
3.7472
LCL
0.0000

h.

Plot the range chart showing the centerline, upper and lower control limits.
Control Chart

12

10

UCL, 9.6454

8

6

4

Centerline, 3.7472

2

LCL, 0.0000

0
0

10

20

30

40

50

60

70

Range

i.

Are the sample ranges within the control limits?
Yes

7

j.

Based on 3-sigma control limits for the runs test, does the pattern analysis indicate the A/B runs are random?
Yes

k.

Based on 3-sigma control limits for the runs test, does the pattern analysis indicate the U/D runs are random?
Yes

l.

Is the variability of the checkout times in control (i.e is the range chart in control)? Why?
Yes. The sample ranges are within the limits, the A/B runs are random, and the U/D runs are random.

Analyze the mean chart by answering the following questions.
m. What is the grand mean?
49.5907
n.

Calculate the centerline, upper and lower control limits for the mean chart?
3σ Control Chart Settings
UCL
53.4241
Centerline
49.5907
LCL
45.7573

o.

Plot the mean chart showing the centerline, upper and lower control limits.
Control Chart

54

UCL, 53.4241
53

52

51

50

Centerline, 49.5907
49

48

47

46

LCL, 45.7573
45
0

10

20

30

40

50

60

70

Mean Unknow n Variation

p.

Are the sample means within the control limits?

8

Yes
q.

Based on 3-sigma control limits for the runs test, does the pattern analysis indicate the A/B runs are random?
No

r.

Based on 3-sigma control limits for the runs test, does the pattern analysis indicate the U/D runs are random?
No

s.

Are the average checkout times in control (i.e. is the mean chart in control)? Why?
No. While the sample means are within the limits, the A/B runs are not random, and the
U/D runs are not random.

Based on your analysis is the bag filling process in control? Why? What should management do?
No, while the range chart indicates the variability of the bag process is in control, the mean chart indicates the mean chart is not in control. Management should take immediate corrective action and discuss with the people who work in the bag filling process to identify potential causes for the problem. A TQM analysis such as the fishbone (Ishikawa) diagram could be used to identify areas for improvement to get the process under control.
Based on the mean control chart, provide an explanation of what may be occurring.
It appears the pressure to increase production (fill more bags per hour) is creating the under filled bags.

#3: Optical scanners are used in all supermarkets to speed the checkout process. Whenever the scanner fails to read the bar code on the product, the cashier is required to manually punch the code into the register. Obviously, unreadable bar codes slow the checkout process. SPC is used to determine whether the scanner is working properly. Once a day at the check out counter, a sample of 500 scans is taken and the number of times the scanner is unable to read the bar code is determined. The sampling process is performed automatically by the cash register.
To avoid manually entering the data into the templates it can be copied and pasted from Data Sets on the
Lesson Page. Use “copy, paste special, values” to transfer the data to the appropriate SPC template worksheet.. 9

Sample
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25

Sample
Size
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500

Incorrect
1
5
9
4
5
6
3
5
7
4
3
5
2
7
7
10
5
4
5
5
9
10
11
16
9

Answer the following questions.
a.

What is the average sample size?
500

b.

How many samples were taken?
25

c.

Which chart should be used to determine whether the scanning process is in control? Explain your answer. Proportion (P) chart, because it is counted (attribute) data.

d.

What is the estimate of the expected proportion of scans which are unable to read the bar code?
.0126

10

e.

Calculate the centerline, upper and lower control limits for the P chart?
3σ Control Chart Settings
UCL
0.0275
Centerline
0.0126
LCL
0.0000

f.

Plot the P chart showing the centerline, upper and lower control limits.
Control Chart

0.035

0.03

UCL, 0.0275
0.025

0.02

0.015

Centerline, 0.0126
0.01

0.005

LCL, 0.0000

0
0

10

20

30

40

50

60

70

Proportion

g.

Are the sample proportions within the control limits?
No

h.

Based on 3-sigma control limits for the runs test, does the pattern analysis indicate the A/B runs are random?
Yes

i.

Based on 3-sigma control limits for the runs test, does the pattern analysis indicate the U/D runs are random?
Yes

j.

Is the optical scanning process in control? Why?
No. The sample ranges are not within the limits, the A/B runs are random, and the U/D runs are random.

What should management do?

11

Management should take corrective action to see what could have been the causes for the one observation which made the process appear to be out of control. Using the computer records, they could identify the items which were scanned during this time to see if there were any unusual products that may have caused the problem. A TQM summary chart and
Pareto analysis might identify the potential causes of the unreadable scans.

#4: Given the following control chart.
Control Chart

80

60

UCL, 58.2692

40

20

0

Centerline, -1.7308
0

10

20

30

40

50

60

70

-20

-40

-60

LCL, -61.7308

-80
Mean Know n Variation

Answer the following questions.
a. How many observations are in the control chart?
26
b.

Count the number of A/B and U/D runs for the control chart.
A/B
U/D

c.

AAAABAABBBBBABBBBBBBBBBBBB
UDUDUDDUDUDUDUDDUDUDUDUDD

What is the expected number of A/B runs?

E (r ) A / B =
d.

6 runs
22 runs

# Observations
26
+1 =
+ 1 = 14
2
2

What is the variability of the expected number of A/B runs?

σ (r ) A / B =

#Observations − 1
=
4

26 − 1
= 2 .5
4

12

e.

What is the test statistic for the expected number of runs?

Z (r ) A / B =
f.

# runs − E (r ) A / B 6 − 14
=
= − 3 .2 σ (r ) A / B
12.5

Based on 3-sigma control limits for the runs test, does the pattern analysis indicate the A/B runs are random? Why? Are three too few, just the right amount, or too many runs?
No, because it is not between -3 and 3. There are too few runs.

g.

What is the expected number of U/D runs?

E (r )U / D =
h.

What is the variability of the expected number of A/B runs?

σ (r )U / D =
i.

2(# Observations) − 1 2 * 26 − 1
=
= 17
3
3

16(#Observations ) − 29
16 * 26 − 29
=
= 2.07
90
90

What is the test statistic for the expected number of runs?

Z (r )U / D =
j.

# runs − E (r )U / D 22 − 17
=
= 2.415 σ (r )U / D
2.07

Based on 3-sigma control limits for the runs test, does the pattern analysis indicate the U/D runs are random? Why?
Yes, because it is between -3 and 3

k.

Is the process in control? Why?
No. While the sample ranges are within the limits, the U/D runs are random, the A/B runs are not random indicating the process is out of control.

#5: An appliance manufacturer wants to contract with a repair shop to handle authorize repairs in
Indianapolis. The company has an acceptable range of repair time of 50 minutes to 90 minutes. Two vendors have submitted bids for the work: A and B. In test trials, vendor A had a mean repair time 74 minutes with a standard deviation of 4.0 minutes, and vendor B had a repair time of 72 minutes with a standard deviation of 5.1 minutes.
a.

What is the tolerance range the manufacturer will accept?
40 minutes (90 – 50)

13

Answer the following questions assuming the appliance manufacture wants the repair shop to meet 3-sigma quality control.
b.

What is the process capability for vendor A? Vendor B?
Vendor A:
Vendor B:

c.

Between 62 minutes (74 – 3*4) and 86 minutes (74+3*4)
Between 56.7 minutes (72-3*5.1) and 87.3 minutes (72+3*5.1)

Which company meets the appliance manufacturer’s tolerance specifications? Why?
Both meet the tolerance specifications for repair times because they are between the required 50 minutes and 90 minutes

d.

Assuming the appliance manufacturer will only choose one vendor and all other factors are equal
(e.g. price, delivery time, etc), which vendor should the company choose? Why?
Vendor A because the variability of their process is less than the variability of vendor B

Answer the following questions assuming the appliance manufacture wants the repair shop to meet 6-sigma quality control.
e.

What is the process variability a vendor should have if their average repair time is 70 minutes?
Why?
3.3 minutes
6 times the process variability (standard deviation) should be within the allowable range of
20 minutes (90 – 70 or 70 -50). For this to occur 6*s<20 implies s<3.33… minutes.

14

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