Introduction to the Mathematical and Statistical Foundations of Econometrics

Quality Control in Practice

The problem in applying this result in quality control is that K is unknown. Therefore, in practice the following decision rule as to whether K < R or not is followed. Given a particular number r < n, to be determined at the end of this subsection, assume that the set of N bulbs meets the minimum quality requirement K < R if the number k of defective bulbs in the sample is less than or equal to r. Then the set A(r) = {0, 1,..., r} corresponds to the assumption that the set of N bulbs meets the minimum quality requirement K < R, hereafter indicated by “accept,” with probability

r

P(A(r)) = £ P({k}) = pr(n, K), (1.12)

k=0

say, whereas its complement A(r) = {r + 1n} corresponds to the assump­tion that this set of N bulbs does not meet this quality requirement, hereafter indicated by “reject,” with corresponding probability

P(A(r)) = 1 - Pr(n, K).

Given r, this decision rule yields two types of errors: a Type I error with prob­ability 1 - pr (n, K) if you reject, whereas in reality K < R, and a Type II error with probability pr(K, n) if you accept, whereas in reality K > R. The probability of a Type I error has upper bound

pi(r, n) = 1 - minpr(n, K), (1.13)

K < R

and the probability of a Type II error upper bound

p2(r, n) = maxpr(n, K). (1.14)

K >R

To be able to choose r, one has to restrict either p1(r, n) or p2(r, n), or both. Usually it is the former option that is restricted because a Type I error may cause the whole stock of N bulbs to be trashed. Thus, allow the probability of a Type I error to be a maximal a such as a = 0.05. Then r should be chosen such that p1(r, n) < a. Because p1(r, n) is decreasing in r, due to the fact that (1.12) is increasing in r, we could in principle choose r arbitrarily large. But because p2(r, n) is increasing in r, we should not choose r unnecessarily large. Therefore, choose r = r(n|a), where r(n|a) is the minimum value of r for which p1(r, n) < a. Moreover, if we allow the Type II error to be maximal в, we have to choose the sample size n such that p2(r(n |a), n) < в.

As we will see in Chapters 5 and 6, this decision rule is an example of a statistical test, where H0 : K < R is called the null hypothesis to be tested at

This section may be skipped.

the a x 100% significance level against the alternative hypothesis H1 : K > R. The number r (n |a) is called the critical value of the test, and the number k of defective bulbs in the sample is called the test statistic.