Final answer:
The process capability index (Cp) for the parts is 1.40, indicating the process is capable relative to its tolerance. The process capability performance index (Cpk) is 0.44, indicating the process is not centered well within the specification limits.
Step-by-step explanation:
The goal is to calculate the process capability index (Cp) and the process capability performance index (Cpk) for an injection-molded part whose diameter specifications are 22 mm ± 0.8 mm. The actual average diameter observed from 250 parts is 22.55 mm, and the standard deviation is 0.19 mm. First, we calculate the Cp, which is a measure of the potential capability of a process, disregarding the process mean. The formula for Cp is (USL - LSL) / (6×standard deviation), where USL is the upper specification limit and LSL is the lower specification limit. Given that the specification is ±0.8 mm around the target of 22 mm, USL = 22.8 mm and LSL = 21.2 mm. Therefore, Cp can be calculated as:
Cp = (22.8 - 21.2) / (6×0.19) = 1.6 / 1.14 = 1.40 (rounded to two decimal places).
Next, the Cpk, which takes into account the actual position of the mean relative to the specification limits, is calculated using the formula Cpk = min(USL - μ, μ - LSL) / (3×standard deviation), where μ is the actual process mean. The process mean is 22.55 mm, so:
Cpk = min(22.8 - 22.55, 22.55 - 21.2) / (3×0.19) = min(0.25, 1.35) / 0.57 = 0.25 / 0.57 = 0.44 (rounded to two decimal places).
The calculated Cp index indicates the injection molding process is capable relative to the specified tolerance if the process mean is centered. However, the Cpk value shows that although the process is capable, it is not centered well within the specification limits, which could lead to a higher level of parts not meeting the specification.