stat : control charts : xbar/r chart - rensselaer...
TRANSCRIPT
5-14
5-12 continued (c) MTB : Stat : Control Charts : Xbar/R Chart : Ex5-12cx1,x2,x3,x4,x5
10987654321Subgroup 0
160
150
140
130
120
110
100
Sam
ple
Mea
n
1
Mean=130.9
UCL=154.5
LCL=107.3
100
50
0
Sam
ple
Ran
ge
R=40.9
UCL=86.4
LCL=0
Xbar/R Chart for Critical Dimension
The adjustment overcompensated for the upward shift. The process average is now between x and the LCL, with a run of ten points below, and even one sample (#36) below the LCL. 5-13. (a) Original Data MTB : Stat : Control Charts : Xbar/R Chart : Ex5-13aSt
2010Subgroup 0
85
80
75Sam
ple
Mea
n
Mean=79.53
UCL=84.58
LCL=74.49
20
10
0
Sam
ple
Ran
ge
R=8.745
UCL=18.49
LCL=0
Xbar/R Chart for Compressive Strength
Yes, the process is in control—though we should watch for a possible cyclic pattern in the averages.
5-15
5-13 continued (b) New Data MTB : Stat : Control Charts : Xbar/R Chart : Ex5-13bSt
15105Subgroup 0
87
82
77
72
Sam
ple
Mea
n1
1 1
Mean=79.53
UCL=84.57
LCL=74.49
30
20
10
0
Sam
ple
Ran
ge 1 1
11
1
11
R=8.745
UCL=18.49
LCL=0
Xbar/R Chart for Compressive Strength
Test Results for Xbar Chart TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 4 11 14 Test Results for R Chart TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 5 6 7 11 13 14 15 A strongly cyclic pattern in the averages is now evident, but more importantly, there are several out-of-control points on the range chart. 5-14. (a) Original Data MTB : Stat : Control Charts : Xbar/S Chart: Ex5-13aSt
2010Subgroup 0
85
80
75Sam
ple
Mea
n
Mean=79.53
UCL=84.64
LCL=74.43
876543210
Sam
ple
StD
ev
S=3.575
UCL=7.468
LCL=0
Xbar/S Chart for Compressive Strength
5-16
5-14 continued New Data MTB : Stat : Control Charts : Xbar/S Chart: Ex5-13bSt
15105Subgroup 0
87
82
77
72
Sam
ple
Mea
n
1
1 1
Mean=79.53
UCL=84.57
LCL=74.49
10
5
0
Sam
ple
StD
ev
11 1
11 1
1 1
S=3.575
UCL=7.383
LCL=0
Xbar/S Chart for Compressive Strength
Test Results for Xbar Chart TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 4 11 14 Test Results for S Chart TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 2 5 6 7 11 13 14 15 (c) Yes, the S chart detects the change in process variability more quickly than the R chart did, at sample #22 versus sample #24. 5-15.
5; 34.00; 4.7n x R (a) for n = 3
2
2
4
3
UCL 34 1.023(4.7) 38.81
LCL 34 1.023(4.7) 29.19
UCL 2.574(4.7) 12.1
UCL 0(4.7) 0
x
x
R
R
x A R
x A R
D R
D R
(b) The x control limits for n = 5 are "tighter" (31.29, 36.72) than those for n = 3 (29.19, 38.81). This means a 2 shift in the mean would be detected more quickly with a sample size of n = 5. (c) for n = 8
2
2
4
3
UCL 34 0.373(4.7) 35.75
LCL 34 0.373(4.7) 32.25
UCL 1.864(4.7) 8.76
UCL 0.136(4.7) 0.64
x
x
R
R
x A R
x A R
D R
D R
5-22
5-26. 30 30
1 1
5; 607.8; 144; 30i i
i i
n x R m f
(a)
1 1
2
2
4
3
607.8 14420.26; 4.8
30 30UCL 20.26 0.577(4.8) 23.03
LCL 20.26 0.577(4.8) 17.49
UCL 2.115(4.8) 10.152
LCL 0(4.8) 0
m m
i ii i
x
x
R
R
x Rx R
m mx A R
x A R
D R
D R
(b)
2ˆ / 4.8 / 2.326 2.064
xR d
16 20.26
ˆ Pr{ LSL} ( 2.064) 0.01952.064
p x
5-27. (a) MTB : Stat : Control Charts : R Chart : Ex5-27Det
151050
20
10
0
Sample Number
Sam
ple
Ran
geR Chart for Detent
1
R=6.467
UCL=13.67
LCL=0
TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 12 Process is not in statistical control -- sample #12 exceeds the upper control limit on the Range chart.
5-23
5-27 continued (b) Excluding sample #12: MTB : Stat : Control Charts : R Chart : Ex5-27Det
151050
20
10
0
Sample Number
Sam
ple
Ran
geR Chart for Detent less Sample 12
1
R=5.643
UCL=11.93
LCL=0
TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 12 (c) Without sample #12,
2ˆ / 5.64286 / 2.326 2.426x
R d . (d)
Without sample #12, USL LSL 0.3220 0.3200ˆPCR 1.374
ˆ6 6(2.426 0.0001)x
5-28. MTB : Stat : Control Charts : Xbar/R Chart : Ex5-27Det
15105Subgroup 0
5
0
-5
Sam
ple
Mea
n
1
1
1
Mean=-0.06667
UCL=3.663
LCL=-3.797
20
10
0
Sam
ple
Ran
ge
1
R=6.467
UCL=13.67
LCL=0
Xbar/R Chart for Detent
Test Results for Xbar Chart TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 1 2 12
Exercise 5-27 is simply an exercise in the estimation of σ and setting up a standard givenR chart. Exercise 5-28, since it is called a continuation of 5-27, is confusing. It is only acontinuation in the sense that it uses the same data. Treat it as an entirely separateproblem and it makes more sense.
5-23
5-27 continued (b) Excluding sample #12: MTB : Stat : Control Charts : R Chart : Ex5-27Det
151050
20
10
0
Sample Number
Sam
ple
Ran
ge
R Chart for Detent less Sample 12
1
R=5.643
UCL=11.93
LCL=0
TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 12 (c) Without sample #12,
2ˆ / 5.64286 / 2.326 2.426x
R d . (d)
Without sample #12, USL LSL 0.3220 0.3200ˆPCR 1.374
ˆ6 6(2.426 0.0001)x
5-28. MTB : Stat : Control Charts : Xbar/R Chart : Ex5-27Det
15105Subgroup 0
5
0
-5
Sam
ple
Mea
n
1
1
1
Mean=-0.06667
UCL=3.663
LCL=-3.797
20
10
0
Sam
ple
Ran
ge
1
R=6.467
UCL=13.67
LCL=0
Xbar/R Chart for Detent
Test Results for Xbar Chart TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 1 2 12
5-24
5-28 continued Test Results for R Chart TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 12 Need to remove samples 1, 2 and 12 from control limit calculations:
15105Subgroup 0
5
0
-5
Sam
ple
Mea
n
1
11
1
Mean=-0.6
UCL=2.380
LCL=-3.580
20
10
0
Sam
ple
Ran
ge
1
R=5.167
UCL=10.92
LCL=0
Xbar/R Chart for Detent, less samples 1, 2, 12
Test Results for Xbar Chart TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 1 2 3 12 Test Results for R Chart TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 12 Need to also remove sample 3 from control limit calculations:
15105Subgroup 0
5
0
-5
Sam
ple
Mea
n
1
11
1
Mean=-0.9091
UCL=1.870
LCL=-3.688
20
10
0
Sam
ple
Ran
ge
1
R=4.818
UCL=10.19
LCL=0
Xbar/R Chart for Detent, less samples 1, 2, 3, 12
Test Results for Xbar Chart TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 1 2 3 12 Test Results for R Chart TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 12
5-25
5-28 continued (a) MTB : Stat : Control Charts : Xbar/R Chart : Ex5-28All
252015105Subgroup 0
5
0
-5
Sam
ple
Mea
n
1
11
1
11
1 11
Mean=-0.9091
UCL=1.870
LCL=-3.688
20
10
0
Sam
ple
Ran
ge
1
1 1
R=4.818
UCL=10.19
LCL=0
Xbar/R Chart for Detent
Test Results for Xbar Chart TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 1 2 3 12 16 17 18 20 23 Test Results for R Chart TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 12 21 22 Given the large number of points beyond both the x and R control limits, the process appears to be unstable. (b) MTB : Stat : Control Charts : Xbar/R Chart : Ex5-28All
252015105Subgroup 0
5
0
-5
Sam
ple
Mea
n
1
1 1
1 1
Mean=1.232
UCL=5.546
LCL=-3.082
20
10
0
Sam
ple
Ran
ge
1
R=7.48
UCL=15.82
LCL=0
Xbar/R Chart for Detent
Test Results for Xbar Chart TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 1 12 13 16 17
5-26
5-28 (b) continued Test Results for R Chart TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 12 Remove out-of-control samples 1, 12, 13, 16, 17 and recalculate control limits.
252015105Subgroup 0
5
0
-5
Sam
ple
Mea
n
1
1 1
1 1 1
Mean=0.99
UCL=4.941
LCL=-2.961
20
10
0
Sam
ple
Ran
ge
1
R=6.85
UCL=14.48
LCL=0
Xbar/R Chart for Detent
Test Results for Xbar Chart TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 1 12 13 16 17 20 Test Results for R Chart TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 12 Sample 20 is now out-of-control, try removing and recalculating limits.
252015105Subgroup 0
5
0
-5
Sam
ple
Mea
n
1
1 1
1 11 1
Mean=0.7789
UCL=4.665
LCL=-3.107
20
10
0
Sam
ple
Ran
ge
1
R=6.737
UCL=14.24
LCL=0
Xbar/R Chart for Detent
Test Results for Xbar Chart TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 1 12 13 16 17 18 20
5-27
5-28 (b) continued Test Results for R Chart TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 12 Sample 18 is now out-of-control, for a total 7 of the 25 samples, with runs of points both above and below the centerline. This suggests that the process is inherently unstable, and that the sources of variation need to be identified and removed. 5-29. (a)
20 10
, ,1 1
20
2 , 21
10
2 , 21
5; 20; 10; 18.608; 6.978
ˆ / / (18.086 / 20) / 2.326 0.400
ˆ / / (6.978 /10) / 2.326 0.300
x y x i y ii i
x x x i xi
y y y i yi
n m m R R
R d R m d
R d R m d
(b) Want Pr{( ) 0.09} 0.006x y .
Let z = x - y. Then 2 2 2 2ˆ ˆ ˆ 0.4 0.3 0.500z x y .
1
0.090.006
ˆ
0.09(0.006)
0.500
0.092.5121
0.500
2.5121(0.500) 0.09 1.346
z
z
z
z
z
5-30.
30 30
1 1
6; 12,870; 1350; 30i ii i
n x R m
(a)
1
4
3
135045.0
30UCL 2.004(45.0) 90.18
LCL 0(45.0) 0
m
ii
R
R
RR
mD R
D R
5-31
5-38. 30 30
1 1
7; 2700; 120; 30i i
i i
n x R m
(a)
1 1
2
2
4
3
2700 12090; 4
30 30UCL 90 0.419(4) 91.676
LCL 90 0.419(4) 88.324
UCL 1.924(4) 7.696
LCL 0.076(4) 0.304
m m
i ii i
x
x
R
R
x Rx R
m mx A R
x A R
D R
D R
(b)
2ˆ / 4 / 2.704 1.479
xR d
(c)
4ˆ 0.9594(1.479) 1.419
UCL 1.882(1.419) 2.671
LCL 0.118(1.419) 0.167
x
S
S
S c
5-39.
/ 2 0.01 / 2 0.005
9; 600; 12; =0.01
2.576
12UCL 600 2.576 610.3
9
12LCL 600 2.576 589.7
9
x
xx x
xx x
n
k Z Z Z
k kn
k kn
5-40.
2ˆ / 20.59 / 2.059 10
Pr{detect shift on 1st sample} Pr{ LCL} Pr{ UCL} Pr{ LCL} 1 Pr{ UCL}
LCL UCL 785 790 815 7901 1
10 4 10 4
( 1) 1 (5) 0.1587 1 1.0000
x
new new
x x
R d
x x x x
0.1587
5-41.
1
1 1 1 1ARL 6.30
1 1 Pr{not detect} Pr{detect} 0.1587
This is the addition to Problem 6:
2
20.59ˆ 10.000.
2.059r
dσ = = =
The shift in sigma units is 1, and so d = 2. Entering Table 3.2 with 2 gives an ARL of3.13.
5-38
5-50. (a) MTB : Stat: Control Charts : I-MR Chart : Ex5-50aTh
302010Subgroup 0
70
60
50
40
30
Indi
vidua
l Val
ue
Mean=49.85
UCL=65.14
LCL=34.55
20
10
0
Mov
ing
Ran
ge
R=5.752
UCL=18.79
LCL=0
I and MR Chart for Oxide Thickness
The process is in statistical control. MTB : Graph : Probability Plots : Ex5-50aTh
605040
99
95
90
80706050403020
10
5
1
Thickness
Perc
ent
Normal Probability Plot for Oxide Thickness
The normality assumption is reasonable.
5-39
5-50 continued (b) MTB : Stat: Control Charts : I-MR Chart : Ex5-50bTh
40302010Subgroup 0
70
60
50
40
30
Indi
vidua
l Val
ue 5 6 2
12 2
Mean=49.85
UCL=65.15
LCL=34.55
20
10
0
Mov
ing
Ran
ge
R=5.752
UCL=18.79
LCL=0
I and MR Chart for Oxide Thickness
Test Results for I Chart TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 38 TEST 2. 8 points in a row on same side of center line. Test Failed at points: 37 38 39 40 TEST 5. 2 out of 3 points more than 2 sigmas from center line (on one side of CL). Test Failed at points: 34 39 40 TEST 6. 4 out of 5 points more than 1 sigma from center line (on one side of CL). Test Failed at points: 35 37 38 39 40 We have turned on some of the sensitizing rules in Minitab to illustrate their use. There is a run above the centerline, several 4 of 5 beyond 1 sigma, and several 2 of 3 beyond 2 sigma on the x chart. However, even without use of the sensitivity rules, it is clear that the process is out of control during this period of operation.
5-40
5-50 continued (c) MTB : Stat: Control Charts : I-MR Chart : Ex5-50cTh
605040302010Subgroup 0
70
60
50
40
30
Indi
vidua
l Val
ue 56 2
122
Mean=49.85
UCL=65.15
LCL=34.55
20
10
0
Mov
ing
Ran
ge
1
R=5.752
UCL=18.79
LCL=0
I and MR Chart for Oxide Thickness
The process has been returned to a state of statistical control. 5-51. (a) MTB : Graph : Probability Plot : Ex5-51C
1009080706050
99
95
90
80706050403020
10
5
1
Data
Perc
ent
Normal Probability Plot for Concentration
The normality assumption is a little bothersome for the concentration data, in particular due to the curve of the larger values and three distant values.
5-41
5-51 continued (b) MTB : Stat : Control Charts : I-MR Chart : Ex5-51C
302010Subgroup 0
110100908070605040
Indi
vidua
l Val
ue
5
Mean=73.73
UCL=104.9
LCL=42.59
40
30
20
10
0
Mov
ing
Ran
ge
1
R=11.71
UCL=38.26
LCL=0
I and MR Chart for Concentration
Test Results for I Chart TEST 5. 2 out of 3 points more than 2 sigmas from center line (on one side of CL). Test Failed at points: 11 Test Results for MR Chart TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 17 The process is not in control, with two Western Electric rule violations. (c) MTB : Graph : Probability Plot : Ex5-51lnC
4.654.554.454.354.254.154.053.95
99
95
90
80706050403020
10
5
1
Data
Perc
ent
Normal Probability Plot for ln(Concentration)
The normality assumption is still troubling for the natural log of concentration, again due to the curve of the larger values and three distant values.
5-42
5-51 continued (d) MTB : Stat : Control Charts : I-MR Chart : Ex5-51lnC
302010Subgroup 0
4.84.74.64.54.44.34.24.14.03.93.8
Indi
vidua
l Val
ue 5
Mean=4.288
UCL=4.708
LCL=3.869
0.60.50.40.30.20.10.0
Mov
ing
Ran
ge
1
R=0.1577
UCL=0.5154
LCL=0
I and MR Chart for ln(Concentration)
Test Results for I Chart TEST 5. 2 out of 3 points more than 2 sigmas from center line (on one side of CL). Test Failed at points: 11 Test Results for MR Chart TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 17 The process is still not in control, with the same to Western Electric Rules violations. There does not appear to be much difference between the two control charts (actual and natural log). 5-52. (a) MTB : Graph : Probability Plot : Ex5-52Pur
0.870.860.850.840.830.820.810.800.790.78
99
95
90
80706050403020
10
5
1
Purity
Perc
ent
Normal Probability Plot for Purity
Purity is not normally distributed.
9-1
CHAPTER 9 EXERCISE SOLUTIONS MINITAB Release 13.1 Worksheet “CHAP09.MTW” Microsoft Excel 2000 Workbook “CHAP09.XLS” 9-1. ˆ ˆ2.530, 15, 101.40
ˆ ˆ2.297, 9, 60.444
ˆ ˆ1.815, 18, 75.333
ˆ ˆ1.875, 18, 50.111
nA A AnB B BnC C CnD D D
Standard deviations are approximately the same, so the DNOM chart can be used. ˆ 2.245, 3.8R , n = 3 x chart: CL = 0.55, UCL = 4.44, LCL = 3.34 R chart: CL = 3.8, UCL = 4D R = 2.574 (3.8) = 9.78, LCL = 0 MTB : Stat : Control Charts : Xbar/R Chart : Ex9-1X
2010Subgroup 0
543210
-1-2-3-4
Sam
ple
Mea
n
Mean=0.55
UCL=4.438
LCL=-3.338
10
5
0
Sam
ple
Ran
ge
R=3.8
UCL=9.782
LCL=0
DNOM Xbar/R Chart for Measurements
Process is in control.