Thirty samples of size 3 were taken from a machining process over a 15-hour peri

Question

Thirty samples of size 3 were taken from a machining process over a 15-hour period. Control charts (x-Bar & R-Chart) were built by the Director of Quality:

a) Calculate Cp and Cpk , assuming that the specifications are 3.75 ± 0.30.

b) Interpret the results for the manager of this process. Include comments on the above graphs as well as Cp and Cpk. What do you conclude about the process?

X-bar and R-Chart This spreadsheet is designed for up to 50 samples, each of a constant sample size from 2 to 10. Enter data only in yellow cells Charts are displayed below the calculations. Some resizing or rescaling of the charts may be required Average Standard deviation 3.5260 0.4166 Grand Average Average Range 3.53 0.67 1.02 0 2.57 1.69 X-bar Chart Lower control limit Upper control limit 4.5 3.5 2.5 2 1.5 0.5 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 Sample number R-Chart Ranges Lower control limit Upper control limit 1.0 1.4 12 0.8 1 3 5 7 9 1113 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 Sample number

Explanation / Answer

a)

LSL = 3.75-0.3 = 3.45

USL = 3.75+0.3 = 4.05

µ = 3.5260

= 0.4166

Cp = (USL-LSL)6 =(4.05-3.45)/(6*0.4166) = 0.24

Cpk = MIN[(µ-LSL)/3, (USL-µ)/3] = MIN[(3.5260-3.45)/(3*0.4166), (4.05-3.5260)/(3*0.4166)] = 0.0608

b) Looking at the Xbar and R charts, the observed values lie within the upper and lower control limits for both charts. Therefore, the process is considered to be in statistical control.

However, Cp and Cpk values are less than 1.33, therefore, the process is not capable of producing output within specification limits, as per 3 sigma standards.

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