Q1 I am working as a consultant for a manufacturing facility that makes ... well, for confidentiality reasons, I'll keep the product details to myself.Suffices to say that when the production process is working properly, they have a unit (marginal) cost of $120 to make with a trivial amount of fixed cost and sell for $150 at factory door.The can produce about 500 units per day and sell all of them - to a total of 120,000 per year, which is our annual production capacity.

The reason I was called in is that the production process create defective units and he wants to understand more about them. Defects have always been a reality, with about 2% of units being defective.A defective unit costs $90 to repair otherwise it can be scrapped. There would not be a disposal fee, but throwing a defective unit out kind of feels like a waste.

He is thinking about replace one of the machines with a rental unit. The new unit would increase annual fixed costs by $100,000. The maker of the new machine claims its defect rate is below 1.6 % and will let him borrow it for a production run.

My client has asked me the following questions:

a. He thinks that his new employee 'Bob' produces a failure rate about twice that of 'Alice', 'Chris' and someone else whose name starts with the letter 'D' - his other employees.

1. How large a sample does he need to estimate Bob's failure rate within 0.25% with 95% confidence? Explain any important decisions in this calculation.
2. How should he construct the sample? Be specific in supporting your choice and use appropriate language.

b. Bob was fired and is no longer part of the problem.

A sample of 500 units was produced using the new machine, of which 40 were defective. Using the process from class, establish the test to help my client decide if he should replace the machine. For reasons he does not wish to explain, your client insists on an alpha of 1% for this test.