Z and t Distributions in Hypothesis Testing: Unequal Division of Type I Risk

PurposeWhile Six Sigma metrics have been studied by researchers in detail for normal distribution-based data, in this paper, we have attempted to study the Six Sigma metrics for two-parameter Weibull distribution that is useful in many life test data analyses.Design/methodology/approachIn the theory of Six Sigma, most of the processes are assumed normal and Six Sigma metrics are determined for such a process of interest. In reliability studies non-normal distributions are more appropriate for life tests. In this paper, a theoretical procedure is developed for determining Six Sigma metrics when the underlying process follows two-parameter Weibull distribution. Numerical evaluations are also considered to study the proposed method.FindingsIn this paper, by matching the probabilities under different normal process-based sigma quality levels (SQLs), we first determined the Six Sigma specification limits (Lower and Upper Six Sigma Limits- LSSL and USSL) for the two-parameter Weibull distribution by setting different values for the shape parameter and the scaling parameter. Then, the lower SQL (LSQL) and upper SQL (USQL) values are obtained for the Weibull distribution with centered and shifted cases. We presented numerical results for Six Sigma metrics of Weibull distribution with different parameter settings. We also simulated a set of 1,000 values from this Weibull distribution for both centered and shifted cases to evaluate the Six Sigma performance metrics. It is found that the SQLs under two-parameter Weibull distribution are slightly lesser than those when the process is assumed normal.Originality/valueThe theoretical approach proposed for determining Six Sigma metrics for Weibull distribution is new to the Six Sigma Quality practitioners who commonly deal with normal process or normal approximation to non-normal processes. The procedure developed here is, in fact, used to first determine LSSL and USSL followed by which LSQL and USQL are obtained. This in turn has helped to compute the Six Sigma metrics such as defects per million opportunities (DPMOs) and the parts that are extremely good per million opportunities (EGPMOs) under two-parameter Weibull distribution for lower-the-better (LTB) and higher-the-better (HTB) quality characteristics. We believe that this approach is quite new to the practitioners, and it is not only useful to the practitioners but will also serve to motivate the researchers to do more work in this field of research.

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Six Sigma metrics based on lognormal distribution for life tests

Ravichandran

2019

IJQRM

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Purpose The purpose of this paper is to propose an approach for studying the Six Sigma metrics when the underlying distribution is lognormal. Design/methodology/approach The Six Sigma metrics are commonly available for normal processes that are run in the long run. However, there are situations in reliability studies where non-normal distributions are more appropriate for life tests. In this paper, Six Sigma metrics are obtained for lognormal distribution. Findings In this paper, unlike the normal process, for lognormal distribution, there are unequal tail probabilities. Hence, the sigma levels are not the same for left-tail and right-tail defects per million opportunities (DPMO). Also, in life tests, while left-tail probability is related to DPMO, the right tail is considered as extremely good PMO. This aspect is introduced and based on which the sigma levels are determined for different parameter settings and left- and right-tail probability combinations. Examples are given to illustrate the proposed approach. Originality/value Though Six Sigma metrics have been developed based on a normality assumption, there have been no studies for determining the Six Sigma metrics for non-normal processes, particularly for life test distributions in reliability studies. The Six Sigma metrics developed here for lognormal distribution is new to the practitioners, and this will motivate the researchers to do more work in this field of research.

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Z and t Distributions in Hypothesis Testing: Unequal Division of Type I Risk

Cited by 3 publications

References 10 publications

Performance evaluation of exponential distribution using Six Sigma-based tail probabilities

Performance evaluation of exponential distribution using Six Sigma-based tail probabilities

Six Sigma quality evaluation of life test data based on Weibull distribution

Six Sigma metrics based on lognormal distribution for life tests

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