# Examples of Quality Control and the Capability Index

${{C}_{pk}}=\min \left\{ \frac{4.1-4.02}{3\times 0.03},\,\,\frac{4.02-3.9}{3\times 0.03} \right\}=\min \left\{ 0.88889,\,\,1.33333 \right\}=0.88889$
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$X\tilde{\ }N\left( 13.001,\,\,{{0.0025}^{2}} \right)$ $L\left( X \right)=\left\{ \begin{array}{ll} & 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,12.997\le X\le 13.003 \\ & -20\,\,\,\,\,\,\,\,\,\,\,\,\,\text{else} \\ \end{array} \right.$ $E\left( L\left( X \right) \right)=-20\left[ 1-\Pr \left( 12.997\le X\le 13.003 \right) \right]$ $= -20\left[ 1-\Pr \left( \frac{12.997-13.001}{0.0025}\le \frac{X-13.001}{0.0025}\le \frac{13.003-13.001}{0.0025} \right) \right]=-20\left( 1-\Pr \left( -1.6\le Z\le 0.8 \right) \right)$ $=-20\left( 1-0.733345 \right)=-5.33309$ $40,000\times 5.33309=\213,324$ $E\left( L\left( X \right) \right)=-20\left[ 1-\Pr \left( 12.997\le X\le 13.003 \right) \right]$ $= -20\left[ 1-\Pr \left( \frac{12.997-13}{0.0025}\le \frac{X-13}{0.0025}\le \frac{13.003-13}{0.0025} \right) \right]=-20\left( 1-\Pr \left( -1.2\le Z\le 1.2 \right) \right)$ $x=-20\left( 1-0.769861 \right)=-4.60279$ $40,000\times 4.60279=\184,112$ ${{C}_{pk}}=\min \left\{ \frac{0.43-0.401}{3\times 0.002},\,\,\frac{0.401-0.37}{3\times 0.002} \right\}=\min \left\{ 4.83333,\,\,5.16667 \right\}=4.833$ .