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 X chart and an S chart, using samples of size n = 5. These show the process to be in control. The values of X and s are X = 71.8 mm

Question

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 [latex]\bar{X}[/latex] chart and an S chart, using samples of size n = 5. These show the process to be in control. The values of [latex]\overline{\overline{X}}[/latex] and [latex]\bar{s}[/latex] are [latex]\overline{\overline{X}}[/latex] = 71.8 mm and [latex]\bar{s}[/latex] = 0.20 mm. Compute the value of C[latex]_{pk}[/latex]. Is the process capability acceptable?

Accepted Answer

We estimate [latex]\hat{\mu}=\overline{\overline{X}}=71.8[/latex]. To compute [latex]\hat{\sigma}[/latex], we find from Table A.8 that c[latex]_4[/latex] = 0.9400 when the sample size is 5. Therefore [latex]\hat{\sigma}=\bar{s}/ c_4=0.20 / 0.9400=0.2128[/latex]. The specification limits are LSL = 71 .4 mm and USL = 72.8 mm. The value [latex]\hat{\mu}[/latex] is closer to the LSL than to the USL. Therefore

[latex]\begin{aligned}C_{p k}&=\frac{\hat{\mu}- LSL}{3 \hat{\sigma}}=\frac{71.8-71.4}{(3)(0.2128)}\&=0.6266\end{aligned}[/latex]

Since [latex]C_{p k}<1[/latex], the process capability is not acceptable.

Question 10.11: The design specifications for a piston rod used in an automa...

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