Category Introduction to the Mathematical and Statistical Foundations of Econometrics

Quality Control

1.2.1. Sampling without Replacement

As a second example, consider the following case. Suppose you are in charge of quality control in a light bulb factory. Each day N light bulbs are produced. But before they are shipped out to the retailers, the bulbs need to meet a minimum quality standard such as not allowing more than R out of N bulbs to be defective. The only way to verify this exactly is to try all the N bulbs out, but that will be too costly. Therefore, the way quality control is conducted in practice is to randomly draw n bulbs without replacement and to check how many bulbs in this sample are defective.


As in the Texas lotto case, the number M of different samples Sj of size n you can draw out of a set of N elements without replacement is

Each sample Sj is characterized by ...

Read More

Bayes’ Rule

Let A and B be sets in &. Because the sets A and A form a partition of the sample space ^, we have B = (B П A) U (B П A); hence,

P(B) = P(B П A) + P(B П A) = P(B|A)P(A) + P(B |A)P(A).


P(AB)= P(A П B) P(B|A)P(A)

( 1 ) P (B) P (B) ■

Combining these two results now yields Bayes’ rule:

Подпись: P (A| B)P (B | A) P (A)

P(B | A)P(A) + P(B| A)P(A) ■

Thus, Bayes’ rule enables us to compute the conditional probability P(A |B) if P(A) and the conditional probabilities P(B | A) and P(B | A) are given.

More generally, if Aj, j = 1, 2,…,n (< to) is a partition of the sample space ^ (i. e., the Aj’s are disjoint sets in Ж such that ^ = Un=j Aj), then

Подпись: P(Ai |B)P (B | Ai) P (Ai)

TTj=1 P (B| Aj) P (Aj) •

Bayes’ rule plays an important role in a special branch of statistics (and econometrics) called Bayesian ...

Read More

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


P(A(r)) = £ P({k}) = pr(n, K), (1.12)


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...

Read More

Sampling with Replacement

As a third example, consider the quality control example in the previous section except that now the light bulbs are sampled with replacement: After a bulb is tested, it is put back in the stock of N bulbs even if the bulb involved proves to be defective. The rationale for this behavior may be that the customers will at most accept a fraction R/N of defective bulbs and thus will not complain as long as the actual fraction K/N of defective bulbs does not exceed R /N. In other words, why not sell defective light bulbs if doing so is acceptable to the customers?

The sample space ^ and the a-algebra & are the same as in the case of sampling without replacement, but the probability measure P is different. Con­sider again a sample Sj of size n containing k defective light bulbs...

Read More