The Farley Manufacturing Company prides itself on the quality of its products. T

Question

The Farley Manufacturing Company prides itself on the quality of its products. The company is engaged in competition for a very important project. A key element is a part that ultimately goes into precision testing equipment. The specifications are 8.000plus or minus±3.000 millimeters. Management is concerned about the capability of the process to produce that part. The accompanying data were randomly collected during test runs of the process.

What is the process capability index, Cpk, equal to? (Enter your response rounded to three decimal places.) Observation (millimeters)

   Observation (millimeters)                          
Sample   1   2   3   4   5   6   7   8
1   7.600   9.900   8.300   9.800   7.700   7.000   9.200   8.300
2   9.900   7.200   7.400   7.300   9.700   9.600   7.500   7.600
3   8.400   7.800   7.300   8.200   8.600   8.100   7.100   9.800
4   8.000   7.300   9.700   7.400   7.600   8.100   7.800   8.300
5   8.600   8.400   7.400   7.600   7.300   8.900   7.600   6.900

Explanation / Answer

CPK- The process capability index is defined as the capability index of the process which explains the capability of the process to deliver the product within the given tolerance as per sigma levels.

The process capabilities is given by the formula as

Cpk = (USL -LSL) / 6 * sigma

USL = Mean + 3Sigma

LSL = Mean - 3sigma

Let us calculate the sigma and mean values for the given data as below.

32.619

Mean = Sum of all Xi / N

Mean = 326.200 / 40

Mean = 8.155

Standard Deviation = Square root { (sum of all value of [square [x-u]) / (n-1) }
Standard Deviation = Square root {32.619) / (39)}
Standard Deviation = Square root {0.836}
Standard Deviation = 0.915

Mean = 8.155 & Standard Deviation = sigma = 0.915

USL = Mean + 3Sigma = 8.155 + 3* 0.915 =10.90

LSL = Mean - 3sigma = 8.155 - 3* 0.915 = 5.41

Process capability index = Cpk

Cpk = (USL -LSL) / 6 * sigma

Cpk = (10.90 - 5.41) / (6*0.915)

Cpk = 5.49 / 5.487

Cpk = 1.001

Thus the CPk for the given data for the given process is equal to Cpk=1.001

S.No. X X-Mean Square (X-Mean) 1 7.600 -0.555 0.308 2 9.900 1.745 3.045 3 8.300 0.145 0.021 4 9.800 1.645 2.706 5 7.700 -0.455 0.207 6 7.000 -1.155 1.334 7 9.200 1.045 1.092 8 8.300 0.145 0.021 9 9.900 1.745 3.045 10 7.200 -0.955 0.912 11 7.400 -0.755 0.570 12 7.300 -0.855 0.731 13 9.700 1.545 2.387 14 9.600 1.445 2.088 15 7.500 -0.655 0.429 16 7.600 -0.555 0.308 17 8.400 0.245 0.060 18 7.800 -0.355 0.126 19 7.300 -0.855 0.731 20 8.200 0.045 0.002 21 8.600 0.445 0.198 22 8.100 -0.055 0.003 23 7.100 -1.055 1.113 24 9.800 1.645 2.706 25 8.000 -0.155 0.024 26 7.300 -0.855 0.731 27 9.700 1.545 2.387 28 7.400 -0.755 0.570 29 7.600 -0.555 0.308 30 8.100 -0.055 0.003 31 7.800 -0.355 0.126 32 8.300 0.145 0.021 33 8.600 0.445 0.198 34 8.400 0.245 0.060 35 7.400 -0.755 0.570 36 7.600 -0.555 0.308 37 7.300 -0.855 0.731 38 8.900 0.745 0.555 39 7.600 -0.555 0.308 40 6.900 -1.255 1.575 Total 326.200

32.619

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