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# Chapter 21 - Chapter 21 21.1Chance variation is caused by a...

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Chapter 21 21.1Chance variation is caused by a number of randomly occurring events that are part of the production process and that in general cannot be eliminated without changing the process. 21.3 Special variation is caused by specific events or factors that are frequently temporary and that can usually be identified and eliminated 21.4a Chance variation represents the variation in student achievement caused by differences in preparation, motivation, and ability. b Special variation represents variation due to specific event or factors that can be corrected. 21.5 0124 . ) 9938 . 1 ( 2 ) 5 . 2 | z (| P = - = = α 21.6 ARL = 0124 . 1 = 81 21.7 0456 . ) 9772 . 1 ( 2 ) 0 . 2 | z (| P = - = = α 21.8 ARL = 0456 . 1 = 22 21.9 ARL = 0026 . 1 = 385 Number of units = Production × ARL = 100(385) = 38,500 21.10a From Beta-mean spreadsheet, β = .6603 b Probability = .6603 8 = .0361 21.11 P = 1 − β = 1 − .6603 = .3397; ARL = 3397 . 1 P 1 = = 2.94 21.12 Number of units = Production × ARL = 50(385) = 19,250 21.13a From Beta-mean spreadsheet, β = .8133 b Probability = .8133 8 = .1914 193

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β = 1 − .8133 = .1867; ARL = 1867 . 1 P 1 = = 5.36 21.15 Sampling 3 units per hour means that on average we will produce 38,500 units before erroneously concluding that the process is out of control when it isn’t. Sampling 2 units per half hour reduces this figure by 50%. Sampling 4 units per hour means that when the process goes out of control, the probability of not detecting a shift of 1.5 standard deviations is .6603 and we will produce on average 2.94 × 100 = 294 units until the chart indicates a problem. Sampling 2 units per half hour increases the probability of not detecting the shift to .8133 and decreases the average number of units produced when the process is out of control to 50 × 5.36 = 268. 21.16 Number of units = Production × ARL = 2000(385) = 770,000 21.17a From Beta-mean spreadsheet, β = .7388 b Probability = .7388 4 = .2979 21.18 P = 1 − β = 1 − .7388 = .2612; ARL = 2612 . 1 P 1 = = 3.83 21.19 Number of units = Production × ARL = 4000(385) = 1,540,000 21.20a From Beta-mean spreadsheet, β = .3659 b Probability = .3659 4 = .0179 21.21 P = 1 − β = 1 − .3659 = .6341; ARL = 6341 . 1 P 1 = = 1.58 21.22 Sampling 10 units per half hour means that on average we will produce 770,000 units before erroneously concluding that the process is out of control when it isn’t. Sampling 20 units per hour doubles this figure. Sampling 10 units per half hour means that when the process goes out of control, the probability of not detecting a shift of .75 standard deviations is .7388 and we will produce on average 3.83 × 2000 = 7660 units until the chart indicates a problem. Sampling 20 units per hour decreases the probability of not detecting the shift to .3659 and decreases the average number of units produced when the process is out of control to 4000 × 1.58 = 6320. 21.23 Centerline =
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Chapter 21 - Chapter 21 21.1Chance variation is caused by a...

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