# Producer’s And Consumer’s Risk And Average Outgoing Quality

**Producer’s and consumer’s risk
**

With acceptance sampling, two parties are usually involved: the producer of the product and the consumer of the product. When specifying a sampling plan, each party wants to avoid costly mistakes in accepting or rejecting a lot. The producer wants to avoid the mistake of having a good lot rejected

It bears repeating that sampling always runs the danger of leading to an erroneous conclusion. Let us say in this example that the total population under scrutiny is a load of 1,000 computer chips, of which in reality only 30 (or 3%) are defective. This means that we would want to accept the shipment of chips, because 4% is the allowable defect rate. But if a random sample of n = 50 chips were drawn, we could conceivably end up with zero defects and accept that shipment (that is, it is OK) or we could find all 30 defects in the sample. If the latter happened, we could wrongly conclude that the whole population was 60% defective and reject them all.

(producer’s risk) because he or she usually must replace the rejected lot. Conversely, the customer or consumer wants to avoid the mistake of accepting a bad lot because defects found in a lot that has already been accepted are usually the responsibility of the customer (consumer’s risk). The OC curve shows the features of a particular sampling plan, including the risks of making a wrong decision.

To help you understand the theory underlying the use of sampling plans, we will illustrate how an OC curve is constructed statistically.In attribute sampling, where products are determined to be either good or bad, a binomial distribution is usually employed to build the OC curve. The binomial equation is

where n = number of items sampled (called trials)

p = probability that an x (defect) will occur on any one trial

P(x) = probability of exactly x results in n trials

When the sample size (n) is large and the percent defective (p) is small, however, the Poisson distribution can be used as an approximation of the binomial formula. This is convenient because binomial calculations can become quite complex, and because cumulative Poisson tables are readily available. Our Poisson table appears in Appendix II of the text.

In a Poisson approximation of the binomial distribution, the mean of the binomial, which is np, is used as the mean of the Poisson, which is λ; that is,

λ = np

**AVERAGE OUTGOING QUALITY**

In most sampling plans, when a lot is rejected, the entire lot is inspected and all of the defective items are replaced. Use of this replacement technique improves the average outgoing quality in terms of percent defective. In fact, given (1) any sampling plan that replaces all defective items encountered and (2) the true incoming percent defective for the lot, it is possible to determine the average outgoing quality (AOQ) in percent defective. The equation for AOQ is

where P_{d} = true percent defective of the lot

P_{a} = probability of accepting the lot

N= number of items in the lot

n = number of items in the sample