# Statistical Process Control Questions

Soft Drink Data for Problem 5.21Sample Number

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

X1

15.8

16.3

16.1

16.3

16.1

16.1

16.1

16.2

16.3

16.6

16.2

15.9

16.4

16.5

16.4

16

16.4

16

16.4

16.4

X2

16.3

15.9

16.2

16.2

16.1

15.8

16.3

16.1

16.2

16.3

16.4

16.6

16.1

16.3

16.1

16.2

16.2

16.2

16

16.4

X3

16.2

15.9

16.5

15.9

16.4

16.7

16.5

16.2

16.4

16.4

15.9

16.7

16.6

16.2

16.3

16.3

16.4

16.4

16.3

16.5

X4

16.1

16.2

16.4

16.4

16.5

16.6

16.1

16.1

16.3

16.1

16.3

16.2

16.4

16.3

16.2

16.3

16.3

16.5

16.4

16

X5

16.6

16.4

16.3

16.2

16

16.4

16.5

16.3

16.5

16.5

16.4

16.5

16.1

16.4

16.2

16.2

16.2

16.1

16.4

15.8

ISE 131 HW #2

Problem #1 (Central Tendency): Rediscovery of the Central Tendency with Summing

Two Simple Independent Random Variables β A Slightly More Complex Case Than the

Case Shown in the Class.

The simple example illustrated in the class is documented in the file mean & s.d. of linear

combination of RVs.pdf. Find and plot the probability function of X1 and that of (X1+X2)/2

as what was done in that file except that the two independent random variables of this HW

exercise have this slightly more complex distribution, i = 1 and 2:

-1 with probability 1/3

X i = 0 with probability 1/3

1 with probability 1/3.

Problem #2 (X-bar and R charts): Answer the following question first (i.e., Part (0)),

before answering Part (a) and Part (b) of Exercise 5.21 on page 222. (You do not need to

do Part (c), now for now.) Use Excel to answer the questions, and submit your Excel files

for grading.

Part (0): Assume that both the mean and the standard deviation are known, and they are

16.268 and 0.20145, respectively. Use the formulae for the control limits for the πΜ
and R

charts we discussed in the class for the case of known parameters (i.e., known mean and

known variance) to calculate the control limits for the πΜ
and R charts of sample size 3. Use

Excel to solve this problem, and submit your Excel file for grading.

Now, do Part (a) and Part (b) of Exercise 5.21, as stated in the Exercise statement. (You

do not need to do Part (c), now for now.) For your convenience, the data contained in Table

5E.2 have been entered into a companion Excel file and has been posted on this site.

Problem #3 (X-bar and S charts): Exercise 5.23 on page 224, except Part (c).

Repeat Part (0), Part (a), and Part (b) of Problem #2. However, for Part (0), just calculate

the centerline, the upper control limit and lower control limit, for each of the two charts;

there is no need to plot the charts.

Problem #4 (Alpha and Beta risks, ARL): An x-bar chart has a center line of 100, uses

three-sigma control limits, and is based on a sample size of four. The process standard

deviation is known to be six. If the process mean shifts from 100 to 92, what is the

probability of detecting this shift on the first sample following the shift? What is the

average run length for the chart to detect the shift?

Problem #5 (X-bar and R charts with variable sample size): Assume we have some

historical samples, and these samples have variable sample size. The data for the samples

can be found below. Estimate the π and π using X-bar and R method. What will be control

limits of X-bar and R charts if the new samples have sample size = 4?

Sample #1

11

9

10

Sample #2

10

10

Sample #3

12

11

10

12

Sample #4

12

11

Problem #1 (Central Tendency): Rediscovery of the Central Tendency with Summing

Two Simple Independent Random Variables β A Slightly More Complex Case Than the

Case Shown in the Class.

The simple example illustrated in the class is documented in the file mean & s.d. of linear

combination of RVs.pdf. Find and plot the probability function of X, and that of (Xi+X2)/2

as what was done in that file except that the two independent random variables of this HW

exercise have this slightly more complex distribution, i = 1 and 2:

-1 with probability 1/3

X; – = 0 with probability 1/3

1 with probability 1/3.

Problem #2 (X-bar and R charts): Answer the following question first (i.e., Part (0)),

before answering Part (a) and Part (b) of Exercise 5.21 on page 222. (You do not need to

do Part (c), now for now.) Use Excel to answer the questions, and submit your Excel files

for grading

Part (0): Assume that both the mean and the standard deviation are known, and they are

16.268 and 0.20145, respectively. Use the formulae for the control limits for the X and R

charts we discussed in the class for the case of known parameters (i.e., known mean and

known variance) to calculate the control limits for the X and R charts of sample size 3. Use

Excel to solve this problem, and submit your Excel file for grading.

Now, do Part (a) and Part (b) of Exercise 5.21, as stated in the Exercise statement. (You

do not need to do Part (c), now for now.) For your convenience, the data contained in Table

58.2 have been entered into a companion Excel file and has been posted on this site.

Problem #3 (X-bar and S charts): Exercise 5.23 on page 224, except Part (C).

Repeat Part (0), Part (a), and Part (b) of Problem #2. However, for Part (0), just calculate

the centerline, the upper control limit and lower control limit, for each of the two charts;

there is no need to plot the charts.

Problem #4 (Alpha and Beta risks, ARL): An x-bar chart has a center line of 100, uses

three-sigma control limits, and is based on a sample size of four. The process standard

deviation is known to be six. If the process mean shifts from 100 to 92, what is the

probability of detecting this shift on the first sample following the shift? What is the

average run length for the chart to detect the shift?

Problem #5 (X-bar and R charts with variable sample size): Assume we have some

historical samples, and these samples have variable sample size. The data for the samples

can be found below. Estimate the u and o using X-bar and R method. What will be control

limits of X-bar and R charts if the new samples have sample size = 4?

Sample #1

11

9

10

Sample #2

10

10

Sample #3

12

11

10

12

Sample #4

12

11

the Western Electric rules to the control

ited in Exercise 5.16. Would these rules

out-of-control signals?

ider the time-varying process behavior

w. Match each of these several patterns of

rformance to the corresponding and R

in in Figures (a) to (e) below.

(in inches) are given in Table 5E.1 for 25 samples di

three boards each.

a. Set up X and R control charts. Is the process is

statistical control?

b. Estimate the process standard deviation

c. What are the limits that you would expect to con

tain nearly all the process measurements?

5.21 The net weight of a soft drink is to be mon:

tored by X and R control charts using a sample sa

thickness of a printed circuit board is an

quality parameter. Data on board thickness

Behavior

Control Chart

UCL

2

ww

T

LCL

UCL

R

M

R

Table 5E,3

Continued

of n – 5 Data for 20 preliminary samples are shown

in Table SE 2

Set up and control charts using these data

Does the process exhibit statistical control?

b. Estimate the process mean and standard devia-

tion

Sample

Number

X1

X2

X3

8

132.7 151.1 124.

c. Does fill weight seem to follow a normal distri-

bution?

9

136.4 126.2 154.7

10

135.0 115.4 149.1

Table 5.2

Soft Drink Data for Problem 5.21

11

139.6 127.9 151.1

12

Sample Number

5.24 Samples of six items are taken from a serie

process at regular intervals. A quality characteristics

measured and Y and R values are calculated for el

sample. After 50 groups of size six have been taken wy

haver-40 and R – 4. The data is normally distributed

a. Compute control limits for the X and R contul

charts. Do all points fall within the control limits?

b. Estimate the mean and standard deviation of the

process. What are the 3 standard deviation lim-

its for the individual data?

c. If the specification limits are 41 +5, do you think

the process is capable of producing within these

specifications?

5.25 Table 55.3 presents 20 subgroups of five med

urements on the time it takes to service a customer

a. Set up X and R control charts for this process and

verify that it is in statistical control.

b. Following establishing of the control charts in

part (a), 10 new samples have been provided in

Table 5E.4. Plot the new X and R values using

the control chart limits you established in part

C. Suppose that the assignable cause responsible for

the action signals generated in part (b) has been

identified and adjustments made to the proces

125.3 160.2 130.4

1

15.8

13

16.3

16.2

145.7 101.8 149.5

16.1

16.6

2

163

15.9

15.9

14

16.2

138.6 139.0 131.9

16.4

3

16.1

16.2

16.5

16.4

15

16.3

110.1 114.6 165.1

4

16.3

16.2

15.9

16.4

16

5

16.2

145.2 101.0 154.6

16.1

16.1

16.4

16.5

6

17

16.1

16.0

125.9 135.3 121.5

15.8

16.7

2

16.6

16.1

164

18

16.3

16.5

(a) and draw conclusions.

129.7

97.3 130.5

8

16.1

16.2

16.5

16.1

19

162

123.4 150.0 161.6

9

16.1

163

16.2

16.3

20

164

10

16.6

163

144.8 138.3 119.6

16.3

16.5

to co

16

16.0

16.2

16.3

16.3

16.2

2

149.3 142.1 105

17

16.4

16.2

16.4

16.3

16.2

3

115.9 135.6 124

18

16.0

16.2

16.4

16.5

16.1

4

118.5 116.5 130.

19

16.4

16.0

16.3

16.4

16.4

20

16.4

16.4

16.5

16.0

15.8

5

108.2 123.8 117.

6

102.8 112.0 135.

5.22 Rework Exercise 5.20 using an X-S chart.

5.23 Rework Exercise 5.21 using an X-S chart.

N

120.4

84.3

112.

224 Chapter 5 Control Charts for Variables