# 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
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 πΜ 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 πΜ 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
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
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

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