Lesson 14 Statistical Process Control Solutions

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?

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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

0.7

UCL

0.6

0.5 0S.4ample Range

Centerline

0.3

0.2

LCL

0.1

0

0

1

2

3

4

5

6

7

i. 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 B A B A A B 5 runs

U/D

U D U D D 4 runs

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

E(r) A/ B

=

# Observations 2

+1=

6 2

+1=

4

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

(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

=

# runs - E(r) A/ B (r) A/ B

= 5-4 1.118

= .89

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

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Yes, because it is between -3 and 3.

o. What is the expected number of U/D runs?

E(r)U / D

=

2(#

Observations) -1 3

=

2*6 -1 3

=

3.67

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

(r)U / D =

16(#Observations) - 29 = 90

16 * 6 - 29 = .86 90

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

Z (r)U / D

=

# runs - E(r)U / D (r)U / D

=

4 - 3.67 .86

= .38

r. 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

3.18

UCL

3.16

3.14

3.12

3.1

Centerline

3.08

3.0S6ample 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 B A A A B B 3 runs

U/D

U D U D U 5 runs

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

E(r) A/ B

=

# Observations 2

+1=

4

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

(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 (r) A/ B

= 3-4 1.118

= -.89

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

=

3.67

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

4

 (r)U / D =

16(#Observations) - 29 = 90

16 * 6 - 29 = .86 90

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

Z (r)U / D

=

# runs - E(r)U / D (r)U / D

= 5 - 3.67 .86

= 1.55

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..

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