Testing process performance based on capability index Cpk with critical values

Process capability index Cpk has been widely used in the manufacturing industry as a process performance measure. In this paper, we investigate the natural estimator of the index Cpk, and show that under the assumption of normality its distribution can be expressed as a mixture of the chi-square and the normal distributions. We also implement the theory of hypothesis testing using the natural estimator of Cpk, and provide efficient Maple programs to calculate the p-values as well as the critical values for various values of α-risk, capability requirements, and sample sizes. The behavior of the p-values and critical values as functions of the distribution parameters are investigated to obtain tight critical values for reliable testing. Based on the test, we develop a simple and practical procedure for in-plant applications. The practitioners can use the proposed procedure to determine whether their process meets the preset capability requirement, and make reliable decisions.