You have an assembly process that yields a final dimension that needs to be controlled. The target value for this dimension is 75 mm. The design specifications are not symmetric and are given by:

\[ x^* = \frac{75 \times 2}{3} \]

Thus, USL = 77 and LSL = 72.

a) Process measurements show that for every 20,000 parts produced, you are seeing 17 parts out of specification, with a nearly even split between parts being above the USL and below the LSL. The standard deviation is 0.75. What would be the corresponding values for \(C_p\) and \(C_{pk}\)?

The corresponding values for process capability (CP) and process capability index (CPK) given the design specifications, standard deviation, and measurements would be approximately 1.11 and 0.89 respectively.

To calculate the process capability indices CP and CPK, we need to know the process specifications, standard deviation, and the mean value of the measurements. The equations are as follow:

CP = (USL - LSL) / (6*standard deviation)
CPK = Min [ (USL - mean) / (3*standard deviation), (mean - LSL) / (3*standard deviation)].

Normally, we calculate mean from the data provided, but this problem does not supply one. So we will use the target value of 75 as an approximation of the mean.

Plug the values into the formula, we have:

CP = (77 - 72) / (6*0.75) = 1.11
CPK = Min [ (77 - 75) / (3*0.75), (75 - 72) / (3*0.75) ] = Min [0.89, 1.33] = 0.89

The Cp index measures the potential capability of the process assuming it was centered within the specification limits. The Cpk index shows the actual process capability and tells us how close we are to the target and how much is the variation. A higher value of CP and CPK signifies that process is more capable.

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