# A normally distributed quality characteristic has specification limits at LSL = 50 and USL =...

## Question:

A normally distributed quality characteristic has specification limits at {eq}LSL = 50 {/eq} and {eq}USL = 60 {/eq}.

A random sample of size {eq}35 {/eq} results in {eq}\bar{x} = 55.5 {/eq} and {eq}s = 0.9 {/eq}.

a. Calculate a point estimate of {eq}C_{pk} {/eq}.

b. Find a {eq}95 {/eq}% confidence interval on {eq}C_{pk} {/eq}.

c. Is this a {eq}6 \alpha {/eq}-process? explain.

## Process capability Measurement:

Cpk is the process capability measurement tool that measures the capability of a process to produce the output limits (range) according to the specifications.

(a)

{eq}\begin{align} C_{pk}\displaystyle&=\frac{min(USL-\mu, \mu-LSL)}{3\hat{\sigma}}\\ \displaystyle&=\frac{min(60-55.5, 55.5-50)}{3\times 0.9}\\ \displaystyle&=\frac{min(4.5, 5.5)}{3\times 0.9}\\ \displaystyle&=\frac{4.5}{3\times 0.9}\\ &=1.67 \end{align} {/eq}

(b)

{eq}\hat{C}_{pk}\pm Z_{1-\frac{\alpha}{2}}\hat{\sigma}_{pk}\\ 1.67\pm Z_{1-\frac{0.05}{2}}\times 0.9\\ 1.67\pm Z_{0.975}\times 0.9\\ 1.67\pm 1.96\times 0.9\\ \left ( -0.094, 3.434 \right ) {/eq}

(c)

Since the limits of {eq}C_{pk} {/eq}

do not contain the {eq}(1-\alpha)\times 100\% {/eq} of the measurements of the process even when in control, this is not a {eq}6-\sigma {/eq} process.

This is because {eq}\bar{x}\neq \mu {/eq} and {eq}\bar{s}\neq \hat{\sigma} {/eq}. 