Tests of significance for the mean of a finite lot

Abstract

Let ξ₁, ξ₂, ξ₃, ... be independent identically distributed random variables each with normal distribution with mean μ and variance σ². Tests for the process mean μ are well-known elements of statistical analysis: the Gauß test under known process variance σ², Student’st-test under unknown process variance σ². Let the process be partitioned in lots (ξ₁, ..., ξ _N ), (ξ _N+1, ..., ξ_2N ), ... of sizeN. Consider (ξ₁, ..., ξ _N ) as a stochastic representative of this lot sequence and let the lot be characterized by the lot mean$\frac{1}{N}\sum\limits_{i = 1}^N {\xi _i }$. The lot mean can be considered as a parameter of the joint conditional distribution function of the lot variables under$\frac{1}{N}\sum\limits_{i = 1}^N {\xi _i } = z$. The present paper investigates the analogies of the Gauß test and Student’st-test for the lot situation, i.e. tests of significance for the lot meanz under known and unknown process variance σ². This approach is of special interest for the statistical control of product quality in situations where the quality of a lot of items 1, 2, ...,N with quality characteristics ξ₁, ξ₂, ..., ξ _N is identified with the lot average$\frac{1}{N}\sum\limits_{i = 1}^N {\xi _i } = z$.