Question

KALI, Inc., manufactures home appliances that are marketed under a variety of trade names. However, KALI does not manufacture every component used in its products. Several components are purchased directly from suppliers. For example, one of the components that KALI purchases for use in home air conditioners is an overload protector, a device that turns off the compressor if it overheats. The compressor can be seriously damaged if the overload protector does not function properly, and therefore KALI is concerned about the quality of the overload protectors. One way to ensure quality would be to test every component received through an approach known as 100% inspection. However, to determine proper functioning of an overload protector, the device must be subjected to time-consuming and expensive tests, and KALI cannot justify testing every overload protector it receives.

Instead, KALI uses an acceptance sampling plan to monitor the quality of the overload protectors. The acceptance sampling plan requires that KALI's quality control inspectors select and test a sample of overload protectors from each shipment. If very few defective units are found in the sample, the lot is probably of good quality and should be accepted. However, if a large number of defective units are found in the sample, the lot is probably of poor quality and should be rejected.

The quality control manager requested a producer's risk of 0.10 when

p₀

was 0.04 and a consumer's risk of 0.20 when

p₁

was 0.25. Consider the acceptance sampling plan based on a sample size of 15 and an acceptance number of 1. Answer the following questions.

(a)

What is the producer's risk for the

n = 15, c = 1

sampling plan? (Round your answer to four decimal places.)

(b)

What is the consumer's risk for the

n = 15, c = 1

sampling plan? (Round your answer to four decimal places.)

(c)

Does the

n = 15, c = 1

sampling plan satisfy the risks requested by the quality control manager? Discuss. (Select all that apply.)

The producer's risk is acceptable since it is less than 0.10.The producer's risk is not acceptable since it is greater than 0.10.The consumer's risk is acceptable since it is less than 0.20.The consumer's risk is not acceptable since it is greater than 0.20.

Answer 1

Define X=number of defectives in sample of size n=15. c=acceptance number=1

Accept the lot if X<=c

Assuming large lot, we assume X~Binomial(n,p), p=true proportion of defectives in the lot.

a) Producer's risk=P(rejecting a lot|p=.04)=1-P(X<=1|X~Binomial(15,.04))=0.1191096

b) Consumer's risk=P(accepting a lot|p=.25)=P(X<=1|X~Binomial(15,.25)=0.08018077

c) The producer's risk is not acceptable since it is greater than 0.10.

The consumer's risk is acceptable since it is less than 0.20