1. Nova Ceramics manufacture specialty ceramic parts for non-metallic applications. Parts must be precision drilled such that when assembled with other parts they are a perfect fit. Inspecting parts is time consuming so the practice is for customers to inspect a sample of items from a shipment and if no more than 1 item is defective, the shipment is accepted.
a) Suppose that the on average 0.1% of units are defective. The customer makes the decision on acceptance based upon a sample of only 10 items. What is the probability that the shipment will be accepted?
b) Suppose the defect rate is really 8%. What is the probability that the customer will reject the shipment?
c) To better discriminate between good and bad quality, the customer has several options. She could only accept shipment if all units were perfect (no defects). What is the probability of accepting a shipment with 0.1% defective? What is the probability of accepting one with 8% defective?
d) Another options is to still allow for one defect in the sample, but draw a larger sample. Suppose she inspects 50 units. What is the probability of accepting a shipment with 0.1% defective? What is the probability of accepting one with 8% defective?
e) The customer is very concerned about the possibility of accepting a shipment with 8% defective. Even when there are no defects in a sample of 10 units, the risk is too high. She prefers to accept shipments only if the sample has no defects. How large a sample would she need to inspect if she would tolerate at most a 1% chance of accepting a shipment with 8% defectives.
2. Young drivers who have had accidents believe that they are being unfairly treated by Citadel Insurance. Last year, Citadel refused to renew the auto insurance policies of 15% of policyholders of all ages who had had accidents that year.
a) In a random sample of 8 young drivers who had accidents, Citadel had refused to renew 3 policies. What is the probability that exactly 3 of 8 would not be renewed if Citadel treated young drivers the same as any others?
b) What is the probability that at least 4 of 8 policies would not be renewed?
c) If 4 of 8 in a random sample of young drivers were refused a renewal, is Citadel treating young drivers more harshly? Use the above probabilities to support your argument.