ABSTRACT

Let the number of defectives found and the number of items inspected be recorded for each lot. The sample data then consists of the paired values (z1, y 1), (z2 , y2), 0 ° 0 , (za, Ya), (k, Ya+ 1), (k, Ya+2), 0 0 0 '(k, Ya+n), where Z;(i = 1, 2, 0 0 0 , a) is the number of defectives found in the ith accepted lot (z; < k) and k is the number of defectives found in each rejected lot. This sample could be described more concisely as consisting of the paired values (z;, y;), i = 1, 2, . . . , m with z; < k in accepted lots and z; = k in rejected lots. It is assumed that no further inspections are made from a rejected lot after the decision to reject has been reached. It follows from (14.1.3) that the likelihood function of a sample as described is

(14.1.10)

We take logarithms of L, differentiate with respect top, equate to zero, and solve for p to obtain

where

rk = number defectives found in r rejected lots aK = number nondefectives found in a accepted lots

2: y; = total number inspections from m = a + r lots i=l

The asymptotic variance of p can be obtained as V(p) = - l!E(iP In L!ap2 ),

which after certain algebraic reduction can be written as

(14.1.11)

(14.1.12)

(14.1.13)

(14.1.14)

For a sufficiently large number of lots, the mean of the observed values of Y should provide a reasonable approximation to E(Y), and the preceding variance might thereby be approximated as

V( ') == pq p m . (14.1.15) LYi i= 1

In the notation employed here, ASN = E(Y); that is, the average sample number is merely the expected value of the random variable Y. From the probability function of (14.1.5), it follows that

(14.1.16)

where B[p, n, k] is the cumulative binomial function

and n = k + K-1. (14.1.18)

The expression given in (14.1.17) for E(Y) was given by Phatak and Bhatt (1967), and an equivalent result was given by Craig (l968b).