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# Chap17 - 732 Chapter 18 Statistical Applications in Quality...

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220 Chapter 18: Statistical Applications in Quality and Productivity Management CHAPTER 17 17.1 (a) Proportion of nonconformances largest on Day 5, smallest on Day 3. 0 0.1 0.2 0.3 0 2 4 6 8 10 Day Proportion (b) n = 100, p = 1.48/10 = 0.148, 04147 . 0 100 ) 148 . 0 1 ( 148 . 0 3 148 . 0 ) 1 ( 3 = - - = - - = n p p p LCL , 25453 . 0 100 ) 148 . 0 1 ( 148 . 0 3 148 . 0 ) 1 ( 3 = - + = - + = n p p p UCL (c) Proportions are within control limits, so there does not appear to be any special causes of variation. 17.2 (a) Proportion of nonconformances largest on Day 4, smallest on Day 3. 0 0.1 0.2 0.3 0 2 4 6 8 10 Day Proportion (b) n = 1036/10 = 103.6, p = 148/1036 = 0.142857, (1 ) 0.142857(1 0.142857) 3 0.142857 3 0.039719 103.6 p p LCL p n - - = - = - = (1 ) 0.142857(1 0.142857) 3 0.142857 3 0.245995 103.6 p p UCL p n - - = + = + = (c) Proportions are within control limits, so there do not appear to be any special causes of variation.

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221 Chapter 18: Statistical Applications in Quality and Productivity Management 17.3 (a) n = 235, p = 538/4700 = 0.1145, 0522 . 0 235 ) 1145 . 0 1 ( 1145 . 0 3 1145 . 0 ) 1 ( 3 = - - = - - = n p p p LCL , 1768 . 0 235 ) 1145 . 0 1 ( 1145 . 0 3 1145 . 0 ) 1 ( 3 = - + = - + = n p p p UCL 0 0.05 0.1 0.15 0.2 0.25 0.3 0 5 10 15 20 Day Proportion The proportion of late arrivals on Day 13 is substantially out of control. Possible causes of this value should be investigated. In addition, the next four highest points all occur on a Friday. (b) The snowstorm would explain why the proportion of late arrivals was so high on Day 13. (c) The results would have been the same. The proportion of late arrivals on Day 13 would still have been above the UCL . 17.4 (a) (b) Although none of the points are outside the control limits, there is evidence of a pattern over time, since the first eight points are all below the center line and most of the later points are above the center line. Thus, the special causes that might be contributing to this pattern should be investigated before any change in the system of operation is contemplated.
Solutions to End-of-Section and Chapter Review Problems 222 17.5 (a) n = 102.5667, p = 0.308742, LCL = 0.171895, UCL = 0.44559 Note: The UCL and LCL are represented by jagged lines which are the control limits for each day depending on the sample size of that day. (b) Yes, the process gives an out of control signal because the proportions fall outside of the control limits on four of the 30 days. (c) n = 103.1923, p = 0.297428, LCL = 0.162428, UCL = 0.432428 Control Chart: Records not Processed Sigma level: 3 29 27 25 23 21 19 17 15 13 11 9 7 5 3 1 Proportion Nonconforming .6 .5 .4 .3 .2 .1 0.0 Records not Processe d UCL Center = .31 LCL

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223 Chapter 18: Statistical Applications in Quality and Productivity Management 17.6 (a) n = 113345/22 = 5152.0455, p = 1460/113345 = 0.01288, 00817 . 0 0455 . 5152 ) 0.01288 1 ( 0.01288 3 0.01288 ) 1 ( 3 = - - = - - = n p p p LCL 01759 . 0 5152.0455 ) 0.01288 1 ( 0.01288 3 0.01288 ) 1 ( 3 = - - = - + = n p p p UCL Note: The UCL and LCL are represented by jagged lines which are the control limits for each day depending on the sample size of that day. The proportion of unacceptable cans is below the LCL on Day 4. There is evidence of a pattern over time, since the last eight points are all above the mean and most of the earlier points are below the mean. Thus, the special causes that might be contributing to this pattern should be investigated before any change in the system of operation is contemplated.
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