Sample size is 6. Using subgroups 1 to 15 complete the following: a.Construct an
ID: 3291655 • Letter: S
Question
Sample size is 6. Using subgroups 1 to 15 complete the following:
a.Construct an Xbar & R chart.
b.Does the process appear to be in control? Why or why not?
c If the optimum response time is 4.5 ± 1.3 minutes, use the estimated standard deviation that you calculate from the control chart information and calculate Cp, and Cpk. Explain what the indices tell you about the process
Sample
1
2
3
4
5
6
7
8
3.81
4.08
4.19
3.88
4.15
4.95
4.01
4.85
3.93
3.82
4.30
4.16
4.93
4.12
4.89
4.05
3.89
3.98
4.17
4.12
4.86
4.08
4.87
4.08
3.91
4.21
4.23
4.24
4.98
4.23
4.98
3.98
4.08
4.00
4.12
4.17
4.58
4.12
4.89
4.07
4.01
4.06
4.15
4.17
4.97
4.13
4.88
3.95
9
10
11
12
13
14
15
4.24
4.99
4.07
4.71
3.91
4.60
4.57
4.87
4.16
4.88
3.98
4.64
3.94
4.06
5.33
4.16
4.89
4.04
4.68
4.00
3.95
4.83
4.18
4.86
3.94
4.57
3.85
3.96
4.93
4.29
4.93
3.94
4.57
3.86
3.98
4.83
4.37
4.97
3.90
4.52
3.88
3.95
Sample size is 6. Using subgroups 1 to 15 complete the following:
a.Construct an Xbar & R chart.
b.Does the process appear to be in control? Why or why not?
c If the optimum response time is 4.5 ± 1.3 minutes, use the estimated standard deviation that you calculate from the control chart information and calculate Cp, and Cpk. Explain what the indices tell you about the process
Sample
1
2
3
4
5
6
7
8
3.81
4.08
4.19
3.88
4.15
4.95
4.01
4.85
3.93
3.82
4.30
4.16
4.93
4.12
4.89
4.05
3.89
3.98
4.17
4.12
4.86
4.08
4.87
4.08
3.91
4.21
4.23
4.24
4.98
4.23
4.98
3.98
4.08
4.00
4.12
4.17
4.58
4.12
4.89
4.07
4.01
4.06
4.15
4.17
4.97
4.13
4.88
3.95
9
10
11
12
13
14
15
4.24
4.99
4.07
4.71
3.91
4.60
4.57
4.87
4.16
4.88
3.98
4.64
3.94
4.06
5.33
4.16
4.89
4.04
4.68
4.00
3.95
4.83
4.18
4.86
3.94
4.57
3.85
3.96
4.93
4.29
4.93
3.94
4.57
3.86
3.98
4.83
4.37
4.97
3.90
4.52
3.88
3.95
Explanation / Answer
Part (a): Construction of Xbar-R Charts
Let Xbari and Ri be respectively the sample average and range
Calculated values of Xbari and Ri are given below:
Xbari
3.938333
4.025
4.193333
4.123333
4.745
Ri
0.27
0.39
0.18
0.36
0.83
4.753333
4.163333
4.838333
4.358333
4.766667
0.97
0.9
1.09
0.83
0.9
4.085
4.481667
4.021667
4.078333
0.81
0.77
0.75
0.62
From these, the following are obtained:
Xdouble bar
4.322889
Rbar
0.702667
A2(6)
0.483
D3(6)
0
D4(6)
2.004
d2(6)
2.534
[A2, D3, D4 and d2 are obtained from standard Control Chart Constants Table.]
LCL Xbar
Xdouble bar - A2Rbar
3.983501
UCL Xbar
Xdouble bar + A2Rbar
4.662277
LCL R
D3Rbar
0
UCL R
D4Rbar
1.408144
So,
Xbar chart has 3.984 and 4.663 as the limits and RChart has 0 and 1.408 as limits. ANSWER
Part (b)
Visually examining the xbari’s and Ri’s vis-à-vis the respective limits we find that
Process is under control with respect to range, but not with respect to Xbar. ANSWER
[Sub-group 1 has a lower Xbar, subgroups 5, 7, 9 and 11 have higher Xbar.]
Part (c)
Estimate of standard deviation = Rbar/d2 = 0.2773. ANSWER
Cp = Specification band/6SD = (2 x 1.3)/(6 x 0.2772) = 1.563 ANSWER 2
[2 x 1.3 is obtained from the given Specification limit: 4.5 ± 1.3 min]
Cpk = [Min{(mean - LSL), (USL - mean)}]/3SD
= [min{(4.322 – 3.2), (5.8 – 4.322)}]/(3 x 0.2773)
= [{min(1.122, 1.478)}]/0.8319 = 1.122/0.8319 = 1.349 ANSWER 3
Cp indicates whether or not the process is capable of meeting the specification and Cpk tells whether or not the process is centered between the specification limits. ANSWER
Xbari
3.938333
4.025
4.193333
4.123333
4.745
Ri
0.27
0.39
0.18
0.36
0.83
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