ch08 - Chapter 8 Note to the Instructor: Some of the...

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Chapter 8 Note to the Instructor: Some of the revised control charts provided in the solutions retain the removed points only as place holders. Other revised control charts have been created by removing the out-of-control point from the worksheet and constructing the control charts again. Section 8-4 8-1. a) 48 . 28 r A CL LCL 95 . 33 CL 42 . 39 r A CL UCL chart x 2 2 = = = = + = 0 ) 35 . 5 ( 0 r D LCL 35 . 5 CL 77 . 13 r D UCL chart R 3 4 = = = = = = 20 10 Subgroup 0 40 35 30 Sample Mean X=33. 95 3.0SL=39.42 -3.0SL=28.48 15 10 5 0 Sample Range R=5.350 3.0SL=13.77 -3.0SL=0.00E+00 Xbar/R Chart for x1-x3 1 b) 22 . 29 LCL 28 . 34 CL 34 . 39 UCL chart x = = = 0 ) 947 . 4 ( 0 r D LCL 947 . 4 CL 74 . 12 UCL chart R 3 = = = = = 1
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20 10 Subgroup 0 40 35 30 Sample Mean X=34. 28 3.0SL=39.34 -3.0SL=29.22 15 10 5 0 Sample Range R=4.947 3.0SL=12.74 -3.0SL=0.00E+00 Xbar/R Chart for x1-x3 8-2. a) 39 . 0 20 80 . 7 r 925 . 18 20 50 . 378 x = = = = 64 . 18 ) 39 . 0 ( 729 . 0 925 . 18 r A CL LCL 925 . 18 CL 21 . 19 ) 39 . 0 ( 729 . 0 925 . 18 r A CL UCL chart x 2 2 = = = = = + = + = 0 ) 39 . 0 ( 0 r D LCL 39 . 0 CL 89 . 0 ) 39 . 0 ( 282 . 2 r D UCL chart R 3 4 = = = = = = = b) 2 ˆ 18.925 0.39 ˆ 0.1894 2.059 x r d µ σ == = 8-3. a) The trial control limits are as follows. 08 . 12 LCL 14 . 15 CL 20 . 18 UCL chart x = = = 0 LCL 2 . 4 CL 581 . 9 UCL chart R = = = 2
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20 10 Subgroup 0 19 18 17 16 15 14 13 12 Sample Mean X=15. 14 3.0SL=18.20 -3.0SL=12.08 10 5 0 Sample Range R=4.200 3.0SL=9.581 -3.0SL=0.00E+00 Xbar/R Chart for Problem 8-3 1 Based on the control charts, there is a single observation beyond the control limits. Observation 14 is above the upper control limit on the R chart. b) With Observation 14 removed, the control limits and charts are as follows. 31 . 12 LCL 14 . 15 CL 98 . 17 UCL chart x = = = 0 LCL 895 . 3 CL 885 . 8 UCL chart R = = = 20 10 Subgroup 0 18 17 16 15 14 13 12 X=15. 14 3.0SL=17.98 -3.0SL=12.31 10 5 0 R=3.895 3.0SL=8.885 -3.0SL=0.00E+00 Previous Xbar/R Chart for 8-3 All points are within the control limits. The process is said to be in statistical control. 8-4. 5476 . 3 ) 534 . 2 ( 4 . 1 r giving 534 . 2 d 4 . 1 d r 0 . 20 x 2 2 = = = = = 29 . 18 ) 5476 . 3 ( 483 . 0 0 . 20 r A CL LCL 0 . 20 CL 71 . 21 ) 5476 . 3 ( 483 . 0 0 . 20 r A CL UCL chart x 2 2 = = = = = + = + = 0 ) 5476 . 3 ( 0 r D LCL 5476 . 3 CL 11 . 7 ) 53476 . 3 ( 004 . 2 r D UCL chart R 3 4 = = = = = = = 3
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8-5. a) 2 . 47 25 1180 r 28 . 306 25 = = = 7657 x = 87 . 271 ) 2 . 47 ( 729 . 0 28 . 306 r A CL LCL 223 CL 69 . 340 ) 2 . 47 ( 729 . 0 28 . 306 r A CL UCL chart x 2 2 = = = = = + = + = 0 ) 2 . 47 ( 0 r D LCL 2 . 47 CL 71 . 107 ) 2 . 47 ( 282 . 2 r D UCL chart R 3 4 = = = = = = = b) 92 . 22 059 . 2 2 . 47 d r ˆ 28 . 306 x ˆ 2 = = = = = σ µ 8-6. a) 06237 . 0 LCL 06294 . 0 CL 06352 . 0 UCL chart x = = = 0 LCL 000996 . 0 CL 002106 . 0 UCL chart R = = = 25 20 15 10 5 Subgroup 0 0.064 0.063 0.062 Sample Mean X=0.06294 3.0SL=0.06352 -3.0SL=0.06237 0.003 0.002 0.001 0.000 Sample Range R=9.96E-04 3.0SL=0.002106 -3.0SL=0.00E+00 Xbar/R Chart for x1-x5 1 1 1 1 1 There are several points out of control. The control limits need to be revised. The points are 1, 14, 21, 22; or outside the control limits of the R chart: 15 b) Second revision, observations 5 and 6 are removed. Observation 12 is removed after the third revision. Observations 17 and 20 are removed after the fourth revision. Observations 8 and 11 are removed after the fifth revision. Observation 7 is removed after the sixth revision. After several revisions, the control chart is finally in control. 06263 . 0 LCL 06298 . 0 CL 06332 . 0 UCL chart x = = = 0 LCL 0006 . 0 CL 001269 . 0 UCL chart R = = = 4
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10 5 Subgroup 0 0.0634 0.0633 0.0632 0.0631 0.0630 0.0629 0.0628 0.0627 0.0626 Sample Mean X=0.06298 3.0SL=0.06332 -3.0SL=0.06263 0.0015 0.0010 0.0005 0.0000 Sample Range R=6.00E-04 3.0SL=0.001269 -3.0SL=0.00E+00 Xbar/R Chart for x1-x5 8-7. a) 1 1 1 137 . 5 LCL 324 . 6 CL 511 . 7 UCL chart x = = = 0 LCL 160 . 1 CL 986 . 2 UCL chart R = = = 25 20 15 10 5 Subgroup 0 8 7 6 5 X=6.324 3.0SL=7.511 -3.0SL=5.137 3 2 1 0 R=1.160 3.0SL=2.986 -3.0SL=0.00E+00 Xbar/R Chart for 8-7 b) 123 . 5 LCL 280 . 6 CL 437 . 7 UCL chart x = = = 0 LCL 131 . 1 CL 911 . 2 UCL chart R = = = 5
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25 20 15 10 5
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ch08 - Chapter 8 Note to the Instructor: Some of the...

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