Question 10.11: The design specifications for a piston rod used in an automa......
The design specifications for a piston rod used in an automatic transmission call for the rod length to be between 71.4 and 72.8 mm. The process is monitored with an \overline{X} chart and an S chart, using samples of size n = 5. These show the process to be in control. The values of \overset{=}{X} and \overline{s} are \overset{=}{X} = 71.8 mm and \overline{s} = 0.20 mm. Compute the value of C_{pk}. Is the process capability acceptable?
We estimate 𝜇̂ = \overset{=}{X} = 71.8. To compute 𝜎̂, we find from Table A.8 that c_{4} = 0.9400 when the sample size is 5. Therefore 𝜎̂ = \overline{s} ∕ c_{4} = 0.20 ∕ 0.9400 = 0.2128. The specification limits are LSL = 71.4 mm and USL = 72.8 mm. The value 𝜇̂ is closer to the LSL than to the USL. Therefore,
C_{pk} = \frac{\hat{\mu}\ −\ LSL}{3\hat{\sigma}} = \frac{71.8\ −\ 71.4}{(3)(0.2128)}
= 0.6266
Since C_{pk} < 1, the process capability is not acceptable.