Restaurants: Reservations The Orchard Caf has found that

Statistical Literacy What are the requirements for a probability distribution?

Statistical Literacy List the criteria for a binomial experiment. What does the random variable of a binomial experiment measure?

Critical Thinking For a binomial probability distribution, it is unusual for the number of successes to be less than \(\mu - 2.5 \sigma\) or greater than \(\mu + 2.5 \sigma\). (a) For a binomial experiment with 10 trials for which the probability of success on a single trial is 0.2, is it unusual to have more than five successes? Explain. (b) If you were simply guessing on a multiple-choice exam consisting of 10 questions with 5 possible responses for each question, would you be likely to get more than half of the questions correct? Explain.

Critical Thinking Consider a binomial experiment. If the number of trials is increased, what happens to the expected value? to the standard deviation? Explain.

Probability Distribution: Auto Leases Consumer Banker Association released a report showing the lengths of automobile leases for new automobiles. The results are as follows. Lease Length in Months Percent of Leases 1324 12.7% 2536 37.1% 3748 28.5% 4960 21.5% More than 60 0.2% (a) Use the midpoint of each class, and call the midpoint of the last class 66.5 months, for purposes of computing the expected lease term. Also find the standard deviation of the distribution. (b) Sketch a graph of the probability distribution for the duration of new auto leases.

Ecology: Predator and Prey Isle Royale, an island in Lake Superior, has provided an important study site of wolves and their prey. In the National Park Service Scientific Monograph Series 11, Wolf Ecology and Prey Relationships on Isle Royale, Peterson gives results of many wolfmoose studies. Of special interest is the study of the number of moose killed by wolves. In the period from 1958 to 1974, there were 296 moose deaths identified as wolf kills. The age distribution of the kills is as follows. Age of Moose in Years Number Killed by Wolves Calf (0.5 yr) 112 15 53 610 73 1115 56 1620 2 (a) For each age group, compute the probability that a moose in that age group is killed by a wolf. (b) Consider all ages in a class equal to the class midpoint. Find the expected age of a moose killed by a wolf and the standard deviation of the ages.

Insurance: Auto State Farm Insurance studies show that in Colorado, 55% of the auto insurance claims submitted for property damage were submitted by males under 25 years of age. Suppose 10 property damage claims involving automobiles are selected at random. (a) Let r be the number of claims made by males under age 25. Make a histogram for the r-distribution probabilities. (b) What is the probability that six or more claims are made by males under age 25? (c) What is the expected number of claims made by males under age 25? What is the standard deviation of the r-probability distribution?

Quality Control: Pens A stationery store has decided to accept a large shipment of ball-point pens if an inspection of 20 randomly selected pens yields no more than two defective pens. (a) Find the probability that this shipment is accepted if 5% of the total shipment is defective. (b) Find the probability that this shipment is not accepted if 15% of the total shipment is defective.

Criminal Justice: Inmates According to Harpers Index, 50% of all federal inmates are serving time for drug dealing. A random sample of 16 federal inmates is selected. (a) What is the probability that 12 or more are serving time for drug dealing? (b) What is the probability that 7 or fewer are serving time for drug dealing? (c) What is the expected number of inmates serving time for drug dealing?

Airlines: On-Time Arrivals Consumer Reports rated airlines and found that 80% of the flights involved in the study arrived on time (that is, within 15 minutes of scheduled arrival time). Assuming that the on-time arrival rate is representative of the entire commercial airline industry, consider a random sample of 200 flights. What is the expected number that will arrive on time? What is the standard deviation of this distribution?

Agriculture: Grapefruit It is estimated that 75% of a grapefruit crop is good; the other 25% have rotten centers that cannot be detected unless the grapefruit are cut open. The grapefruit are sold in sacks of 10. Let r be the number of good grapefruit in a sack. (a) Make a histogram of the probability distribution of r. (b) What is the probability of getting no more than one bad grapefruit in a sack? What is the probability of getting at least one good grapefruit in a sack? (c) What is the expected number of good grapefruit in a sack? (d) What is the standard deviation of the r probability distribution?

Restaurants: Reservations The Orchard Caf has found that about 5% of the diners who make reservations dont show up. If 82 reservations have been made, how many diners can be expected to show up? Find the standard deviation of this distribution.

College Life: Student Government The student government claims that 85% of all students favor an increase in student fees to buy indoor potted plants for the classrooms. A random sample of 12 students produced 2 in favor of the project. What is the probability that 2 or fewer in the sample will favor the project, assuming the student governments claim is correct? Do the data support the student governments claim, or does it seem that the percentage favoring the increase in fees is less than 85%?

Quota Problem: Financial Suppose you are a (junk) bond broker who buys only bonds that have a 50% chance of default. You want a portfolio with at least five bonds that do not default. You can dispose of the other bonds in the portfolio with no great loss. How many such bonds should you buy if you want to be 94.1% sure that five or more will not default?

Theater: Coughs A person with a cough is a persona non grata on airplanes, elevators, or at the theater. In theaters especially, the irritation level rises with each muffled explosion. According to Dr. Brian Carlin, a Pittsburgh pulmonologist, in any large audience youll hear about 11 coughs per minute (Source: USA Today). (a) Let r  number of coughs in a given time interval. Explain why the Poisson distribution would be a good choice for the probability distribution of r. (b) Find the probability of three or fewer coughs (in a large auditorium) in a 1-minute period. (c) Find the probability of at least three coughs (in a large auditorium) in a 30-second period.

Accident Rate: Small Planes Flying over the western states with mountainous terrain in a small aircraft is 40% riskier than flying over similar distances in flatter portions of the nation, according to a General Accounting Office study completed in response to a congressional request. The accident rate for small aircraft in the 11 mountainous western states is 2.4 accidents per 100,000 flight operations (Source: Denver Post). (a) Let r  number of accidents for a given number of operations. Explain why the Poisson distribution would be a good choice for the probability distribution of r. (b) Find the probability of no accidents in 100,000 flight operations. (c) Find the probability of at least 4 accidents in 200,000 flight operations.

Banking: Loan Defaults Records over the past year show that 1 out of 350 loans made by Mammon Bank have defaulted. Find the probability that 2 or more out of 300 loans will default. Hint: Is it appropriate to use the Poisson approximation to the binomial distribution?

Car Theft: Hawaii In Hawaii, the rate of motor vehicle theft is 551 thefts per 100,000 vehicles (Reference: U.S. Department of Justice, Federal Bureau of Investigation). A large parking structure in Honolulu has issued 482 parking permits. (a) What is the probability that none of the vehicles with a permit will eventually be stolen? (b) What is the probability that at least one of the vehicles with a permit will eventually be stolen? (c) What is the probability that two or more of the vehicles with a permit will eventually be stolen? Note: The vehicles may or may not be stolen from the parking structure. Hint: Is it appropriate to use the Poisson approximation to the binomial? Explain.

General: Coin Flip An experiment consists of tossing a coin a specified number of times and recording the outcomes. (a) What is the probability that the first head will occur on the second trial? Does this probability change if we toss the coin three times? What if we toss the coin four times? What probability distribution model do we use to compute these probabilities? (b) What is the probability that the first head will occur on the fourth trial? after the fourth trial?

Testing: CPA Exam Cathy is planning to take the Certified Public Accountant Examination (CPA exam). Records kept by the college of business from which she graduated indicate that 83% of the students who graduated pass the CPA exam. Assume that the exam is changed each time it is given. Let n  1, 2, 3, . . . represent the number of times a person takes the CPA exam until the first pass. (Assume the trials are independent.) (a) What is the probability that Cathy passes the CPA exam on the first try? (b) What is the probability that Cathy passes the CPA exam on the second or third try?