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 \bar{X} chart and an S chart, using samples of size n = 5. These show the process to be in control. The values of \overline{\overline{X}} and \bar{s} are \overline{\overline{X}} = 71.8 mm and \bar{s} = 0.20 mm. Compute the value of C_{pk}. Is the process capability acceptable?
We estimate \hat{\mu}=\overline{\overline{X}}=71.8. To compute \hat{\sigma}, we find from Table A.8 that c_4 = 0.9400 when the sample size is 5. Therefore \hat{\sigma}=\bar{s}/ c_4=0.20 / 0.9400=0.2128. The specification limits are LSL = 71 .4 mm and USL = 72.8 mm. The value \hat{\mu} is closer to the LSL than to the USL. Therefore
\begin{aligned}C_{p k}&=\frac{\hat{\mu}- LSL}{3 \hat{\sigma}}=\frac{71.8-71.4}{(3)(0.2128)}\\&=0.6266\end{aligned}
Since C_{p k}<1, the process capability is not acceptable.