Concept explainers
a)
To determine: The control limits for the mean and range chart and the overall means
Introduction: Control charts used to determine whether the process is under control or not. Attributes and variables are the factors under the control charts.
a)

Answer to Problem 11P
Hence, the UCL for
Explanation of Solution
Given information:
The following information is given:
Sample | Sample mean (in.) | Range (in.) |
1 | 10.002 | 0.011 |
2 | 10.002 | 0.014 |
3 | 9.991 | 0.007 |
4 | 10.006 | 0.022 |
5 | 9.997 | 0.013 |
6 | 9.999 | 0.012 |
7 | 10.001 | 0.008 |
8 | 10.005 | 0.013 |
9 | 9.995 | 0.004 |
10 | 10.001 | 0.011 |
11 | 10.001 | 0.014 |
12 | 10.006 | 0.009 |
Twelve samples that contain five parts each were taken.
Calculate the average for sample and range:
Sample | Sample mean (in.) | Range (in.) |
1 | 10.002 | 0.011 |
2 | 10.002 | 0.014 |
3 | 9.991 | 0.007 |
4 | 10.006 | 0.022 |
5 | 9.997 | 0.013 |
6 | 9.999 | 0.012 |
7 | 10.001 | 0.008 |
8 | 10.005 | 0.013 |
9 | 9.995 | 0.004 |
10 | 10.001 | 0.011 |
11 | 10.001 | 0.014 |
12 | 10.006 | 0.009 |
Total | 120.006 | 0.138 |
Average | 10.0005 | 0.0115 |
Working note:
Average for sample:
It is calculated by dividing the total of sample and number of samples. Hence, the value of
Average for range:
It is calculated by dividing the total of range and number of samples. Hence, the value of
Determine the UCL and LCL for mean:
Formulae to calculate control limits:
Here, the overall mean
Substitute the values in equation (1) to determine the value of UCL as follows:
Hence, the UCL value is 10.00714.
Substitute the values in equation (2) to determine the value of LCL as follows:
Hence, the LCL value is 9.993865.
Therefore for the
Determine the UCL and LCL for range:
Formulae to calculate control limits:
Here,
The average range is
Substitute the values in equation (3) to determine the value of UCL as follows:
Hence, the UCL value is 0.024323.
Substitute the values in equation (4) to determine the value of LCL as follows:
Hence, the LCL value is 0.
Therefore, for the R-chart, the upper control limit is
b)
To plot: The values of sample means and ranges in the chart.
Introduction: Control charts used to determine whether the process is under control or not. Attributes and variables are the factors under the control charts.
b)

Answer to Problem 11P
Control chart has been plotted for sample means and ranges.
Explanation of Solution
Given information:
The following information is given:
Sample | Sample mean (in.) | Range (in.) |
1 | 10.002 | 0.011 |
2 | 10.002 | 0.014 |
3 | 9.991 | 0.007 |
4 | 10.006 | 0.022 |
5 | 9.997 | 0.013 |
6 | 9.999 | 0.012 |
7 | 10.001 | 0.008 |
8 | 10.005 | 0.013 |
9 | 9.995 | 0.004 |
10 | 10.001 | 0.011 |
11 | 10.001 | 0.014 |
12 | 10.006 | 0.009 |
Twelve samples that contain five parts each were taken.
Plot the sample mean values in the
Note: Observe that sample 3 mean value is lower than the
Plot the sample mean values in theR-control chart where
c)
To determine: Whether the process is in control
Introduction: Control charts used to determine whether the process is under control or not. Attributes and variables are the factors under the control charts.
c)

Answer to Problem 11P
Process is out of control during sample #3 in
Explanation of Solution
Given information:
The following information is given:
Sample | Sample mean (in.) | Range (in.) |
1 | 10.002 | 0.011 |
2 | 10.002 | 0.014 |
3 | 9.991 | 0.007 |
4 | 10.006 | 0.022 |
5 | 9.997 | 0.013 |
6 | 9.999 | 0.012 |
7 | 10.001 | 0.008 |
8 | 10.005 | 0.013 |
9 | 9.995 | 0.004 |
10 | 10.001 | 0.011 |
11 | 10.001 | 0.014 |
12 | 10.006 | 0.009 |
Twelve samples that contain five parts each were taken.
Determine whether the process is in control:
Since, the mean for sample #3 is outside the control limits of the
d)
To determine: Why the process is not in control
Introduction: Control charts used to determine whether the process is under control or not. Attributes and variables are the factors under the control charts.
d)

Explanation of Solution
Given information:
The following information is given:
Sample | Sample mean (in.) | Range (in.) |
1 | 10.002 | 0.011 |
2 | 10.002 | 0.014 |
3 | 9.991 | 0.007 |
4 | 10.006 | 0.022 |
5 | 9.997 | 0.013 |
6 | 9.999 | 0.012 |
7 | 10.001 | 0.008 |
8 | 10.005 | 0.013 |
9 | 9.995 | 0.004 |
10 | 10.001 | 0.011 |
11 | 10.001 | 0.014 |
12 | 10.006 | 0.009 |
Twelve samples that contain five parts each were taken.
Determine why the process is not in control:
Further investigations are necessary to check whether the mean value of sample #3 is a freak incident outside the three sigma limits (which has a 0.27% probability). Perhaps 12 more samples can be drawn and examined whether such an incident occurs again. In case it happens again, the process needs to be examined in detail.
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Chapter 6 Solutions
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