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Task Conventional control charts are to be used on a process manufacturing small components with a specified length of 60 ± 1.5mm. Two identical machines are involved in making the components and process capability studies carried out on them reveal the following data : Sample size, n = 5 1.Calculate the control limits to be used on a mean and range chart for each machine and give the reasons for any differences between them. 2.Compare the results from each machine with the appropriate control chart limits and the specification tolerances.

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Mean Chart Machine 1 Action Lines at: X ± A 2 R = 60.0315+(0.58*2.385)=61.414 60.0315-(0.58*2.385)=58.648 Warning Lines at: X ± 2/3A 2 R = 60.0315+(0.39*2.385)=60.961 60.0315-(0.39*2.385)=59.101 Mean Chart Machine 2 Action Lines at: X ± A 2 R = 60.055+(0.58*0.395)=60.284 60.055-(0.58*0.395)=59.825 Warning Lines at: X ± 2/3A 2 R = 60.055+(0.39*0.395)=60.209 60.055-(0.39*0.395)=59.900 Range Chart Machine 1 Upper Action Line at: D’0.0001 R = 5.581 Upper warning Line at: D’0.025 R = 4.317 Lower Warning Line at: D’0.975 R = 0.882 Lower Action Line at: D’0.999 R = 0.382 Range Chart Machine 2 Upper Action Line at: D’0.0001 R = 0.924 Upper warning Line at: D’0.025 R = 0.715 Lower Warning Line at: D’0.975 R = 0.146 Lower Action Line at: D’0.999 R = 0.063 For n = 5, A 2 = 0.58 and A 2 * 0.66 = 0.39 X = 60.0315 R = 2.385 X = 60.055 R = 0.395

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Mean Chart M1 60.0315 60.96 61.41 59.10 58.64 60.1 59.92 60.37 59.91 60.01 60.18 59.67 60.57 59.68 59.55 59.98 60.22 60.54 60.68 59.24 59.48 60.2 60.27 59.57 60.49

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Mean Chart M2 60.055 60.20 60.28 59.90 59.82 60.86 59.1 60.32 60.05 58.95 59.12 58.8 59.68 60.14 60.96 61.05 60.84 61.01 60.82 59.14 59.0159.08 59.25 61.5 61.42

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Range Chart M1 2.38 4.31 5.58 0.88 0.38 2.5 2.2 3 2.4 2.7 1.7 3.4 1.7 1.5 2.3 2.7 3.3 3.6 0.9 1.4 2.7 2.8 1.5 3.2

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Range Chart M2 0.39 0.71 0.924 0.146 0.063 0.5 0.4 0.6 0.2 0.3 0.7 0.5 0.4 0.6 0.3 0.2 0.5 0.4 0.6 0.5 0.1 0.2 0.3 0.4

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Each machine is identical so the differences in stats for each machine can be explained by random causes of variation impacting on machine 2.

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Accuracy and Precision NO Mean or Range values which lie outside the Action Limits (zone 3) NO more than about 1 in 40 values between the Warning and Action Limits (zone 2) NO incidence of two consecutive Mean or Range values which lie outside the same Warning Limit on either the mean or the range chart (zone 2) NO run or trend of five or more which also infringes a warning or action limit (zone 2 or 3) NO runs of more than six sample Means which lie either above or below the Grand Mean (zone 1) NO trends of more than six values of the sample Means which are either rising or falling (zone 1). M1 is stable in both accuracy and precision. Special causes of variation are absent. M2 is unstable in its accuracy but stable in precision. If this range chart for M2 is in fact unstable (with multiple values in the warning zone, i.e. more than one in forty) then we need to assume that the mean chart for M2 is unstable since the Range is used to calculate the lines in the mean chart. Since M2 and M1 are identical we must assume that the instability of M2 is caused by random causes.

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Tolerances The VOC is for a specified length of 60mm ± 1.5mm. Therefore σ is 0.5mm and 6 σ is 3mm. We can disregard M2 because it is unstable M1 with a 60.03 mean and 3SE at 61.444 and -3SE at 59.825 will meet the VOC

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