Question

On the production line the company finds that 90.2% of products are made correctly. You are responsible for quality control and take batches of 30 products from the line and test them. What number of the 30 being incorrectly made would cause you to shut down production?

Answer 1

Solution

At the very outset, it is not possible to derive a general rule. The rule has to be governed by probability. To elaborate:

There are two scenarios:

Case 1

The derived rule shuts down the production and the subsequent investigation reveals that production process is normal, i.e., 90.2% or more of products are made correctly.

Case 2

The derived rule allows the production to continue but production process is not normal, i.e., percent of products made correctly is less than 90.2%, i.e., percent of defectives is more than 9.8%.

In both cases, the rule leads to a wrong decision.

Since both cases are possible, the rule is derived such that probability of these two wrong decisions is limited to some specified values.

Let X = number of defectives found in a sample of 30 products and let the rule be that if X > c, the production would be shut down.

For demonstration purposes, suppose we fix the probability of shutting down the production when the percent defective, p is less than or equal to 9.8% to say 0.05 and the probability of not shutting down the production when the percent defective is more than 9.8%,say 11%, to say 0.10. Then, we should have c such that:

P(X > c/p = 0.098) = 0.05 and P(X < c/p = 0.11) = 0.1.

A mathematical solution for c is highly involved. Normally, by trial and error method using Binomial probability the value of c is evolved.

For example, the first probability is satisfied when c = 6 because at c = 5, the probability is 0.07 and at c =6, it is 0.02.

DONE