Auto pistons at Wemming Chung's plant in Shanghai are produced in a forging process, and the diameter is a critical factor that must be controlled. From sample sizes of 10 pistons produced each day, the mean and the range of this diameter have been as follows: Day Mean x (mm) Range R (mm) 1 156.9 4.4 2 155.2 4.4 3 155.6 4.3 4 153.5 4.8 5 156.6 4.7 Part 2 a) What is the value of x? x= _______ mm (round your response to two decimal places). b) What is the value of Upper R overbarR ? Upper R overbarRequals= _______ mm (round your response to two decimal places). c) What are the UCL Subscript x overbarUCLx and LCL Subscript x overbarLCLx using 3-sigma ? Upper Control Limit (UCL Subscript x overbarUCLx ) = _________ mm (round your response to two decimal places). Lower Control Limit (LCL Subscript x overbarLCLx ) = ________ mm (round your response to two decimal places). d) What are the UCL Subscript Upper RUCLR and LCL Subscript Upper RLCLR using 3-sigma ? Upper Control Limit (UCL Subscript Upper RUCLR ) = __________ mm (round your response to two decimal places). Lower Control Limit (LCL Subscript Upper RLCLR ) = __________ mm (round your response to two decimal places). e) If the true diameter mean should be 155 mm and you want this as your center (nominal) line, what are the new UCL Subscript x overbarUCLx and LCL Subscript x overbarLCLx ? Upper Control Limit (UCL Subscript x overbarUCLx ) = _________ mm (round your response to two decimal places). Lower Control Limit (LCL Subscript x overbarLCLx ) = ________ mm (round your response to two decimal places).
Auto pistons at Wemming Chung's plant in Shanghai are produced in a forging process, and the diameter is a critical factor that must be controlled. From sample sizes of 10 pistons produced each day, the mean and the range of this diameter have been as follows: Day Mean x (mm) Range R (mm) 1 156.9 4.4 2 155.2 4.4 3 155.6 4.3 4 153.5 4.8 5 156.6 4.7 Part 2 a) What is the value of x? x= _______ mm (round your response to two decimal places). b) What is the value of Upper R overbarR ? Upper R overbarRequals= _______ mm (round your response to two decimal places). c) What are the UCL Subscript x overbarUCLx and LCL Subscript x overbarLCLx using 3-sigma ? Upper Control Limit (UCL Subscript x overbarUCLx ) = _________ mm (round your response to two decimal places). Lower Control Limit (LCL Subscript x overbarLCLx ) = ________ mm (round your response to two decimal places). d) What are the UCL Subscript Upper RUCLR and LCL Subscript Upper RLCLR using 3-sigma ? Upper Control Limit (UCL Subscript Upper RUCLR ) = __________ mm (round your response to two decimal places). Lower Control Limit (LCL Subscript Upper RLCLR ) = __________ mm (round your response to two decimal places). e) If the true diameter mean should be 155 mm and you want this as your center (nominal) line, what are the new UCL Subscript x overbarUCLx and LCL Subscript x overbarLCLx ? Upper Control Limit (UCL Subscript x overbarUCLx ) = _________ mm (round your response to two decimal places). Lower Control Limit (LCL Subscript x overbarLCLx ) = ________ mm (round your response to two decimal places).
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter2: Introduction To Spreadsheet Modeling
Section: Chapter Questions
Problem 20P: Julie James is opening a lemonade stand. She believes the fixed cost per week of running the stand...
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Question
4.
Refer to
Table S6.1 - Factors for Computing Control Chart Limits (3 sigma)
for this problem.LOADING...
Auto pistons at Wemming Chung's plant in Shanghai are produced in a forging process, and the diameter is a critical factor that must be controlled. From sample sizes of
10
pistons produced each day, the mean and the range of this diameter have been as follows:|
Day
|
Mean x
(mm)
|
Range R
(mm)
|
|
1
|
156.9
|
4.4
|
|
2
|
155.2
|
4.4
|
|
3
|
155.6
|
4.3
|
|
4
|
153.5
|
4.8
|
|
5
|
156.6
|
4.7
|
Part 2
a) What is the value of
x?
x= _______ mm (round your response to two decimal places).
b) What is the value of
Upper R overbarR ?
Upper R overbarRequals= _______ mm (round your response to two decimal places).
c) What are the
UCL Subscript x overbarUCLx and
LCL Subscript x overbarLCLx using 3-sigma ?
Upper Control Limit
(UCL Subscript x overbarUCLx )
= _________ mm (round your response to two decimal places).
Lower Control Limit
(LCL Subscript x overbarLCLx )
= ________ mm (round your response to two decimal places).
d) What are the UCL Subscript Upper RUCLR and LCL Subscript Upper RLCLR using 3-sigma ?
Upper Control Limit
(UCL Subscript Upper RUCLR )
= __________ mm (round your response to two decimal places).
Lower Control Limit
(LCL Subscript Upper RLCLR )
= __________ mm (round your response to two decimal places).
e) If the true diameter mean should be 155 mm and you want this as your center (nominal) line, what are the new
UCL Subscript x overbarUCLx
and LCL Subscript x overbarLCLx ?
Upper Control Limit
(UCL Subscript x overbarUCLx )
= _________ mm (round your response to two decimal places).
Lower Control Limit
(LCL Subscript x overbarLCLx )
= ________ mm (round your response to two decimal places).
Transcribed Image Text:**Sample Size and Control Chart Constants**
This table provides important constants used for control charts in statistical process control. These constants are determined based on the sample size \( n \) and help in calculating control limits.
| Sample Size, \( n \) | Mean Factor, \( A_2 \) | Upper Range, \( D_4 \) | Lower Range, \( D_3 \) |
|----------------------|---------------------|-----------------|-----------------|
| 2 | 1.880 | 3.268 | 0 |
| 3 | 1.023 | 2.574 | 0 |
| 4 | 0.729 | 2.282 | 0 |
| 5 | 0.577 | 2.115 | 0 |
| 6 | 0.483 | 2.004 | 0 |
| 7 | 0.419 | 1.924 | 0.076 |
| 8 | 0.373 | 1.864 | 0.136 |
| 9 | 0.337 | 1.816 | 0.184 |
| 10 | 0.308 | 1.777 | 0.223 |
| 12 | 0.266 | 1.716 | 0.284 |
**Explanation:**
- **Sample Size, \( n \):** The number of observations in each sample.
- **Mean Factor, \( A_2 \):** A constant used to calculate control limits for the average chart.
- **Upper Range, \( D_4 \):** A constant that provides the upper control limit for the range chart.
- **Lower Range, \( D_3 \):** A constant that provides the lower control limit for the range chart; it becomes positive for sample sizes 7 and above.
These constants are essential for quality control processes as they help in constructing control charts to monitor process stability and capability.
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