The following table contains the measurements of the key length dimension from a
ID: 378532 • Letter: T
Question
The following table contains the measurements of the key length dimension from a fuel injector. These samples of size five were taken at one-hour intervals. Use three-sigma control limits. Use Exhibit 10.13.
OBSERVATIONS
a. Calculate the mean and range for the above samples. (Do not round intermediate calculations. Round your answers to 3 decimal places.)
b. Determine X=X= and RR . (Do not round intermediate calculations. Round your answers to 3 decimal places.)
c. Determine the UCL and LCL for a %media:formula252.mml%-chart. (Do not round intermediate calculations. Round your answers to 3 decimal places.)
d. Determine the UCL and LCL for R-chart. (Leave no cells blank - be certain to enter "0" wherever required. Do not round intermediate calculations. Round your answers to 3 decimal places.)
e. What comments can you make about the process?
The process is out of statistical control.
The process is in statistical control.
OBSERVATIONS
SAMPLE NUMBER 1 2 3 4 5 1 0.484 0.483 0.485 0.506 0.487 2 0.497 0.498 0.515 0.493 0.524 3 0.494 0.482 0.511 0.482 0.523 4 0.493 0.498 0.485 0.483 0.485 5 0.473 0.509 0.526 0.462 0.486 6 0.471 0.483 0.503 0.484 0.506 7 0.495 0.512 0.483 0.472 0.505 8 0.523 0.499 0.491 0.473 0.481 9 0.489 0.498 0.518 0.505 0.513 10 0.493 0.503 0.515 0.506 0.494 11 0.494 0.485 0.463 0.483 0.489 12 0.478 0.452 0.514 0.425 0.516 13 0.522 0.512 0.484 0.525 0.512 14 0.483 0.525 0.503 0.505 0.499 15 0.494 0.515 0.492 0.515 0.513 16 0.471 0.498 0.473 0.484 0.522 17 0.463 0.484 0.508 0.481 0.493 18 0.519 0.485 0.484 0.487 0.473 19 0.498 0.528 0.483 0.494 0.483 20 0.508 0.483 0.474 0.505 0.512Explanation / Answer
Please find below table which presents mean value ( Xbar) and Range of each sample.
Following may be noted :
Xbar for each sample = Sum of 5 observations / 5
Range for each observation = Maximum value – Minimum value
SAMPLE NO
1
2
3
4
5
Range
Xbar
1
0.484
0.483
0.485
0.506
0.487
0.023
0.49
2
0.497
0.498
0.515
0.493
0.524
0.031
0.51
3
0.494
0.482
0.511
0.482
0.523
0.041
0.50
4
0.493
0.498
0.485
0.483
0.485
0.015
0.49
5
0.473
0.509
0.526
0.462
0.486
0.064
0.49
6
0.471
0.483
0.503
0.484
0.506
0.035
0.49
7
0.495
0.512
0.483
0.472
0.505
0.04
0.49
8
0.523
0.499
0.491
0.473
0.481
0.05
0.49
9
0.489
0.498
0.518
0.505
0.513
0.029
0.50
10
0.493
0.503
0.515
0.506
0.494
0.022
0.50
11
0.494
0.485
0.463
0.483
0.489
0.031
0.48
12
0.478
0.452
0.514
0.425
0.516
0.091
0.48
13
0.522
0.512
0.484
0.525
0.512
0.041
0.51
14
0.483
0.525
0.503
0.505
0.499
0.042
0.50
15
0.494
0.515
0.492
0.515
0.513
0.023
0.51
16
0.471
0.498
0.473
0.484
0.522
0.051
0.49
17
0.463
0.484
0.508
0.481
0.493
0.045
0.49
18
0.519
0.485
0.484
0.487
0.473
0.046
0.49
19
0.498
0.528
0.483
0.494
0.483
0.045
0.50
20
0.508
0.483
0.474
0.505
0.512
0.038
0.50
TOTAL =
0.803
9.89
Thus :
Xbar-bar = Mean of Xbar values = 9.894 / 20 = 0.4947
Rbar = Mean of Range values = 0.803 / 20 = 0.04015
Given is : Sample size = 5
Corresponding values of constants for n = 5 as derived from standard table for Xbar chart and Range charts as follows :
A2 = 0.577
D4 = 2.114
D3 = 0
Control charts are accordingly as follows :
Control limits for Xbar chart :
Upper Control Limit = UCL = Xbar-bar + A2.Rbar = 0.4947 + 0.577x 0.04015 = 0.4947 + 0.0231 = 0.5178
Lower Control Limit = LCL = Xbar – A2.Rbar = 0.4947 – 0.577 x 0.04015 = 0.4947 – 0.0231 = 0.4716
Control limits for Range chart :
Upper Control Limit = UCL = D4XRbar = 2.114 x 0.04015 = 0.0848
Lower Control Limit = D3x Rbar = 0
For a process to be in control , all data of Xbar and Range ( R) are to be within control limits. That means ,
Xbar to be in the range of = 0.4716 – 0.5178
Range to be in the range of = 0 – 0.0848
All data pertaining to Xbar in above table are within control Limits
However one of the Range data i..e 0.091 ( sample 12 ) is outside Upper Control Limit Range chart and hence process is not in control
SAMPLE NO
1
2
3
4
5
Range
Xbar
1
0.484
0.483
0.485
0.506
0.487
0.023
0.49
2
0.497
0.498
0.515
0.493
0.524
0.031
0.51
3
0.494
0.482
0.511
0.482
0.523
0.041
0.50
4
0.493
0.498
0.485
0.483
0.485
0.015
0.49
5
0.473
0.509
0.526
0.462
0.486
0.064
0.49
6
0.471
0.483
0.503
0.484
0.506
0.035
0.49
7
0.495
0.512
0.483
0.472
0.505
0.04
0.49
8
0.523
0.499
0.491
0.473
0.481
0.05
0.49
9
0.489
0.498
0.518
0.505
0.513
0.029
0.50
10
0.493
0.503
0.515
0.506
0.494
0.022
0.50
11
0.494
0.485
0.463
0.483
0.489
0.031
0.48
12
0.478
0.452
0.514
0.425
0.516
0.091
0.48
13
0.522
0.512
0.484
0.525
0.512
0.041
0.51
14
0.483
0.525
0.503
0.505
0.499
0.042
0.50
15
0.494
0.515
0.492
0.515
0.513
0.023
0.51
16
0.471
0.498
0.473
0.484
0.522
0.051
0.49
17
0.463
0.484
0.508
0.481
0.493
0.045
0.49
18
0.519
0.485
0.484
0.487
0.473
0.046
0.49
19
0.498
0.528
0.483
0.494
0.483
0.045
0.50
20
0.508
0.483
0.474
0.505
0.512
0.038
0.50
TOTAL =
0.803
9.89
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