Using Normal Approximation on Testing and Determining Sample Size for C(subscript pk)

常態近似法檢定C(下标 pk)指標及樣本數需求之決定

林鴻欽(Hung-Chin Lin)

《工業工程學刊》 23卷1期 (2006/01) Pp. 1-11

https://doi.org/10.29977/JCIIE.200601.0001

摘要

製程能力指標(process capability indices)是一無單位的量化值，用以衡量製程生產出符合要求規格產品能力的重要依據。由於製程能力指標是製程平均數、製程標準差與設定的要求規格（規格上、下限和製程目標值）之函數，而且製程平均數與製程標準差大多是未知且必須經由樣本數據估計的情況下，因此，製程能力指標該視為滿足某種分配性質的隨機變數；而不應只是根據樣本數據以點估計的方式來評估製程的績效，不然將會忽略抽樣誤差所造成的效應。本文將針對其中一個重要製程能力指標C(下标 pk)，根據統計假設檢定的理論考慮抽樣誤差，發展較具有可靠性的方法，以評估製程是否能夠滿足要求的規格。應用Hamaker近似法則以常態逼近方式以避免原本複雜的非中t分配(non-central t distribution)，而過程中也將展現此近似方法的效果。再者，本文將進一步建構C(下标 pk)指標的評估計畫，即決定適當的判定臨界值以及需要達到最少的樣本總數。最後得到的程序與方便直接運算的簡易計算公式將可以提供企業界於實務應用上之參考。

關鍵字

製程能力指標 ； 製程平均數 ； 製程標準差 ； 非中t分配 ； Hamaker近似法

並列摘要

Process capability indices are widely used to measure the ability of the process to manufacture products meeting established specification limits. Simply reporting and then making decision from the calculated estimate of the capability index along is not reliable since sampling errors are ignored. However, a more appropriate estimate would be provided by a hypothesis testing. Using a Bayesian-like estimator of C(subscript pk) index, this paper presents a generalized approach for testing C(subscript pk) index and power computations determining whether a process meets the capability requirement Applying Hamaker's approximation, the testing procedure can be adequately simplified as a normal approximation and that the more complicated use of the non-central t distribution can be avoided. It allows for testing at any a-level and any sample size. Additionally, an analysis is provided to offer the practitioner the sampling requirements to obtain a fixed power.

並列關鍵字

process capability indices ； critical value ； power ； non-central t distribution ； Hamaker's approximation

參考文獻

Bissell, A. F.(1990).How reliable is your capability index.Applied Statistics.(Applied Statistics).

Chen, S. M.,W. L. Peam(1997).The asymptotic distribution of the estimated process capability index C (subscript pk).Communications in Statistics-Theory and Methods.(Communications in Statistics-Theory and Methods).

Cheng. S. W.(1994).Practical implementation of the process capability indices.Quality Engineering.(Quality Engineering).

Chou, Y. M.,A. Polansky(1993).Power of tests for some process capability indices.Communications in Statis tics-Simulation and Computation.(Communications in Statis tics-Simulation and Computation).

Hamaker, H. C.(1979).Acceptance sampling for percent defective by variables and by attributes.Journal of Quality Technology.11(2),139-148.