Formulas for the Group Sequential Sampling of Attributes

Abstract

When an infinite lot consisting of defective and nondefective items is investigated by means of a group sequential sampling plan, the use of matrices and vectors is helpful in determining the probabilities of various combinations of the two classes of items and in computing unbiased estimates of the lot fraction defective. For a sequential plan of the Bartky [1] type, the infinite summation of such vectors leads to an exact, explicit formula for the average number of items inspected, \begin{equation*}\tag{1.1}\bar n_p = p^{-1}\{L_p\lbrack G(h_1 + h_2 - 1) - (h_1 + h_2)\rbrack - G(h_2 - 1) + h_1 + h_2 - \lbrack h_1\rbrack\}\end{equation*}, where $p$ is the fraction defective in the lot, $L_p$ is the probability of arriving at a decision to accept the lot, $h_1$ and $h_2$ are parameters of the plan as defined by the Statistical Research Group [3], $G(i)$ is defined by Bartky's equation (36), and $\lbrack h_1\rbrack$ is the largest integer equal to or less than $h_1$. In approximating $L_p$, or in finding the parameters of a sequential plan with specified risks, the formulas proposed by Wald [2] and the Statistical Research Group can be improved by adding an adjustment, \begin{equation*}\tag{1.2}a = \frac{1}{3}(1 - 2s)\end{equation*}, to the value of $h_2$ wherever it occurs. Their formula for approximating $\bar n_p$ can be improved by adding the adjustment \begin{equation*}\tag{1.3} cq = aq/(1 - s)\end{equation*} wherever $h_2$ occurs, provided that the value of $L_p$ which appears in this formula is arrived at by employing adjustment (1.2).