The quality control director of a small company that manufactures exercise equipment knows that 10% of their products are defective. Market research shows that only 30% of their customers will use the product in the first year after the sale. So, the manufacturer expects that 3% of the units will be returned for service under a one-year full warranty. This company expects to sell 825 units next year. (This problem is based on data from an actual company).

For this problem, assume that next year’s production can be considered a simple random sample of all the units produced.

a. Describe (mean, standard deviation and shape) the sampling distribution of p̂ , the proportion of the 825 units that will be returned for service.

b. What is the probability that more than 2.25% of the 825 units will be returned for service?

c. What is the probability that between 2% and 4% of the 825 units will be returned for service?

d. What is the probability that more than 4.5% of the 825 units will be returned for service?

e. Would it be unusual if in a random sample of 825 units, 4.5% were returned for service & why?

f. It has been found that the repair cost (to the company) for the products returned averages $375 per unit returned. An engineering change has been suggested that will reduce the defect rate from 10% to 4%, but this engineering change will add $6 to the cost of each unit produced. Due to competitive forces, there is no opportunity to raise prices to recover the additional cost.

If the decision to adopt the engineering change is driven by financial considerations only, should the company make the engineering change or not (justify your decision)?