5.73 Benford’s law.

We learned in Chapter 4 that there is a striking fact that the first digits of numbers in legitimate records often follow a distribution known as Benford’s law. Here it is:

5.74 Wi-fi interruptions.

Refer to Example 5.16 (page 268) in which we were told that the mean number of wi-fi interruptions per day is 0.9. We also found in Example 5.16 that the probability of no interruptions on a given day is 0.4066.

5.75 Benford’s law, continued

Benford’s law suggests that the proportion of legitimate invoices with a first digit of 1, 2, or 3 is much greater than if the digits were distributed as equally likely outcomes. As a fraud investigator, you would be suspicious of some potential wrongdoing if the count of invoices with a first digit of 1, 2, or 3 is too low. You decide if the count is in the lower 5% of counts expected by Benford’s law, then you will call for a detailed investigation for fraud.

5.76 Environmental credits.

An opinion poll asks an SRS of 500 adults whether they favor tax credits for companies that demonstrate a commitment to preserving the environment. Suppose that, in fact, 45% of the population favor this idea. What is the probability that more than half of the sample are in favor?

5.77 Leaking gas tanks.

Leakage from underground gasoline tanks at service stations can damage the environment. It is estimated that 25% of these tanks leak. You examine 15 tanks chosen at random, independently of each other.

5.78 Is this coin balanced?

While he was a prisoner of the Germans during World War II, John Kerrich tossed a coin 10,000 times. He got 5067 heads. Take Kerrich’s tosses to be an SRS from the population of all possible tosses of his coin. If the coin is perfectly balanced, . Is there reason to think that Kerrich’s coin gave too many heads to be balanced? To answer this question, find the probability that a balanced coin would give 5067 or more heads in 10,000 tosses. What do you conclude?

5.79 Six Sigma.

Six Sigma is a quality improvement strategy that strives to identify and remove the causes of defects. Processes that operate with Six-Sigma quality produce defects at a level of 3.4 defects per million. Suppose 10,000 independent items are produced from a Six-Sigma process. What is the probability that there will be at least one defect produced?

5.80 Binomial distribution?

Suppose a manufacturing colleague tells you that 1% of items produced in first shift are defective, while 1.5% in second shift are defective and 2% in third shift are defective. He notes that the number of items produced is approximately the same from shift to shift, which implies an average defective rate of 1.5%. He further states that because the items produced are independent of each other, the binomial distribution with of 0.015 will represent the number of defective items in an SRS of items taken in any given day. What is your reaction?

5.81 Poisson distribution?

Suppose you find in your spam folder an average of two spam emails every 10 minutes. Furthermore, you find that the rate of spam mail from midnight to 6 A.M. is twice the rate during other parts of the day. Explain whether or not the Poisson distribution is an appropriate model for the spam process.

5.82 Airline overbooking.

Airlines regularly overbook flights to compensate for no-show passengers. In doing so, airlines are balancing the risk of having to compensate bumped passengers against lost revenue associated with empty seats. Historically, no-show rates in the airline industry range from 10 to 15 percent. Assuming a no-show rate of 12.5%, what is the probability that no passenger will be bumped if an airline books 215 passengers on a 200-seat plane?

5.83 Inventory control.

OfficeShop experiences a one-week order time to restock its HP printer cartridges. During this reorder time, also known as lead time, OfficeShop wants to ensure a high level of customer service by not running out of cartridges. Suppose the average lead time demand for a particular HP cartridge is 15 cartridges. OfficeShop makes a restocking order when there are 18 cartridges on the shelf. Assuming the Poisson distribution models the lead time demand process, what is the probability that OfficeShop will be short of cartridges during the lead time?

5.84 More about inventory control.

Refer to the previous exercise. In practice, the amount of inventory held on the shelf during the lead time is known as the reorder point. Firms use the term service level to indicate the percentage of the time that the amount of inventory is sufficient to meet demand during the reorder period. Use software and the Poisson distribution to determine the reorder points so that the service level is minimally