(a) Using the estimate for the entire industry, what is the expected number of defectives among the 1200 racquets made of Darwood aluminum? (b) How unusual is it that 20 defectives were found? What is the probability that x ≥ 20?

At the other end of this morning’s telephone call is an irate customer whose aluminum tennis racquet has cracked in half after just 14 months. The caller is an avid player who gets out on the court about two or three times a week, and he claims the racquet has never been abused in any way. He not only wants Thorndike Sports Equipment to send him another racquet, he also demands that the company reimburse him the $30 he lost after his racquet broke and he was no longer competitive with his opponent.

On Mr. Thorndike’s return to the office, Ted mentions the interesting phone call, and the elder Thorndike is not a bit surprised. He says the firm has been getting a lot of complaints in recent months, mainly from longtime customers who claim their latest Thorndike alu- minum racquet didn’t hold up as well as the ones they had used in the past.

Speculating, Mr. Thorndike goes on to point out that the company has had two aluminum suppliers for many years, but added a third supplier just a year and a half ago. He suspects that the most recent supplier may be shipping an aluminum alloy that is more brittle and prone to failure.

All of the racquets that have been returned over the past 5 years or so are in a wooden crate in the basement. Mr. Thorndike isn’t sure why, but 6 years ago he mandated that each racquet produced have a serial number that identifies when it was made and who sup- plied the aluminum for its construction. If someone were to ferret through the wooden crate, maybe he could shed some light on this business of customers calling up with bro- ken racquets.

Arriving at the office a bit early on Monday, Ted finds Luke Thorndike in a fit of rage. Mr. Thorndike is sipping coffee and looking over Ted’s notes on the defective racquets that were constructed with aluminum purchased from each of the company’s three sup- pliers.

In rummaging through the basement crate, Ted has found a total of 30 defective racquets. Of these, 5 were made of aluminum supplied by the Snowmet Corporation and 5 were made of aluminum purchased from Barstow Aluminum, Inc. These were the two sup- pliers from whom Thorndike Sports Equipment had been buying aluminum for many years, and with good results.

The cause of Mr. Thorndike’s anger is the total of 20 defective racquets made from alu- minum purchased from the company’s newest and lowest-priced supplier, Darwood Dis- count Metals, Inc. Luke recently placed a big order with Darwood, and he is counting the minutes until 10 a.m. so that he can call the West Coast firm, cancel the order, and give Mr. Darwood a big piece of his mind. The arrival of the morning mail only serves to heighten Luke’s anger, as three more complaints are received from disgruntled customers demand- ing immediate replacements for their broken racquets.

Ten o’clock finally rolls around, and Luke places a person-to-person collect call to Mr. Darwood. After berating Mr. Darwood and his products, Luke demands that the order be canceled immediately. Mr. Darwood does not appreciate the elder Thorndike’s tirade, but he is quite cognizant of the fact that 10% of his company’s profits come from Thorndike Sports Equipment. Though irritated, he patiently tries to reason with Mr. Thorndike.

According to Mr. Darwood, independent metallurgists have conducted a lot of tests in which they found Darwood aluminum to be every bit as good as the product supplied by Snowmet and Barstow. He suggests that the unusually high number of defective racquets found by Luke is merely a fluke. Already operating on a short fuse, Mr. Thorndike re- sponds that he doesn’t much care for silly rhymes in the middle of the morning, and he warns Darwood not to load any more aluminum onto Thorndike-bound trucks until he and Ted have a chance to further examine the information Ted collected over the week- end. He promises to call Mr. Darwood with a final decision by 3 p.m., Pacific Time.

Ted spends a very busy morning and skips lunch, but by 2 p.m. he comes up with some data that might prove useful. Most important, he has uncovered a research study in which it was found that 0.8% of all aluminum racquets end up being returned as defective. The number of Snowmet and Barstow racquets in the “defectives” crate is about 1% of those produced. However, of the 1200 racquets made from Darwood aluminum, 20 (about 1.7%) are defective. Ted decides to consider the situation as a binomial probability distribution in which the probability of a defect on a given trial is 0.008, corresponding to the research finding that 0.8% of all the aluminum racquets produced are defective.

(a) Using the estimate for the entire industry, what is the expected number of defectives among the 1200 racquets made of Darwood aluminum?

(b) How unusual is it that 20 defectives were found? What is the probability that x ≥ 20?