Chapter 6

Which of these variables are discrete and which are continuous random variables? a. The number of new accounts established by a salesperson in a year. b. The time between customer arrivals to a bank ATM. c. The number of customers in Big Nick's barber shop. d. The amount of fuel in your car's gas tank. e. The number of minorities on a jury. f. The outside temperature today.

In a binomial situation \(n=5\) and \(\pi=.40 .\) Determine the probabilities of the following events using the binomial formula. a. \(x=1\) b. \(x=2\)

An American Society of Investors survey found 30 percent of individual investors use a discount broker. In a random sample of nine individuals, what is the probability: a. Exactly two of the sampled individuals have used a discount broker? b. Exactly four of them have used a discount broker? c. None of them have used a discount broker?

The United States Postal Service reports 95 percent of first class mail within the same city is delivered within two days of the time of mailing. Six letters are randomly sent to different locations. a. What is the probability that all six arrive within two days? b. What is the probability that exactly five arrive within two days? c. Find the mean number of letters that will arrive within two days. d. Compute the variance and standard deviation of the number that will arrive within two days.

Industry standards suggest that 10 percent of new vehicles require warranty service within the first year. Jones Nissan in Sumter, South Carolina, sold 12 Nissans yesterday. a. What is the probability that none of these vehicles requires warranty service? b. What is the probability exactly one of these vehicles requires warranty service? c. Determine the probability that exactly two of these vehicles require warranty service. d. Compute the mean and standard deviation of this probability distribution.

A telemarketer makes six phone calls per hour and is able to make a sale on 30 percent of these contacts. During the next two hours, find: a. The probability of making exactly four sales. b. The probability of making no sales. c. The probability of making exactly two sales. d. The mean number of sales in the two-hour period.

A recent survey by the American Accounting Association revealed 23 percent of students graduating with a major in accounting select public accounting. Suppose we select a sample of 15 recent graduates. What is the probability two select public accounting? A telemarketer makes six phone calls per hour and is able to make a sale on 30 percent of these contacts. During the next two hours, find: a. The probability of making exactly four sales. b. The probability of making no sales. c. The probability of making exactly two sales. d. The mean number of sales in the two-hour period.

Can you tell the difference between Coke and Pepsi in a blind taste test? Most people say they can and have a preference for one brand or the other. However, research suggests that people can correctly identify a sample of one of these products only about 60 percent of the time. Suppose we decide to investigate this question and select a sample of 15 college students. a. How many of the 15 students would you expect to correctly identify Coke or Pepsi? b. What is the probability exactly 10 of the students surveyed will correctly identify Coke or Pepsi? C. What is the probability at least 10 of the students will correctly identify Coke or Pepsi?

In a binomial distribution \(n=8\) and \(\pi=.30\). Find the probabilities of the following events. a. \(x=2\) b. \(x \leq 2\) (the probability that \(x\) is equal to or less than 2). c. \(x \geq 3\) (the probability that \(x\) is equal to or greater than 3 )

In a binomial distribution \(n=12\) and \(\pi=.60 .\) Find the following probabilities. a. \(x=5\) b. \(x \leq 5\) c. \(x \geq 6\)

In a binomial distribution \(n=12\) and \(\pi=.60 .\) Find the following probabilities. a. \(x=5\) b. \(x \leq 5\). c. \(x \geq 6\)

In a binomial distribution \(n=12\) and \(\pi=.60 .\) Find the following probabilities. a. \(x=5\) b. \(x \leq 5\) c. \(x \geq 6\)

The speed with which utility companies can resolve problems is very important. GTC, the Georgetown Telephone Company, reports it can resolve customer problems the same day they are reported in 70 percent of the cases. Suppose the 15 cases reported today are representative of all complaints. a. How many of the problems would you expect to be resolved today? What is the standard deviation? b. What is the probability 10 of the problems can be resolved today? c. What is the probability 10 or 11 of the problems can be resolved today? d. What is the probability more than 10 of the problems can be resolved today?

Backyard Retreats, Inc., sells an exclusive line of pools, hot tubs, and spas. It is located just off the Bee Line Expressway in Orlando, Florida. The owner reports 20 percent of the customers entering the store will make a purchase of at least \(\$ 50 .\) Suppose 15 customers enter the store before 10 a.m. on a particular Saturday. a. How many of these customers would you expect to make a purchase of at least \(\$ 50 ?\) b. What is the probability exactly five of these customers make a purchase of at least \(\$ 50 ?\) c. What is the probability at least five of these customers make a purchase of at least \(\$ 50 ?\) d. What is the probability at least one customer makes a purchase of at least \(\$ 50 ?\)

In a Poisson distribution \(\mu=0.4\). a. What is the probability that \(x=0 ?\) b. What is the probability that \(x>0 ?\)

In a Poisson distribution \(\mu=4\). a. What is the probability that \(x=2 ?\) b. What is the probability that \(x \leq 2 ?\) c. What is the probability that \(x>2 ?\)

Ms. Bergen is a loan officer at Coast Bank and Trust. From her years of experience, she estimates that the probability is .025 that an applicant will not be able to repay his or her installment loan. Last month she made 40 loans. a. What is the probability that 3 loans will be defaulted? b. What is the probability that at least 3 loans will be defaulted?

Automobiles arrive at the Elkhart exit of the Indiana Toll Road at the rate of two per minute. The distribution of arrivals approximates a Poisson distribution. a. What is the probability that no automobiles arrive in a particular minute? b. What is the probability that at least one automobile arrives during a particular minute?

It is estimated that 0.5 percent of the callers to the Customer Service department of Dell, Inc., will receive a busy signal. What is the probability that of today's 1,200 callers at least 5 received a busy signal?

Textbook authors and publishers work very hard to minimize the number of errors in a text. However, some errors are unavoidable. Mr. J. A. Carmen, statistics editor, reports that the mean number of errors per chapter is 0.8 . What is the probability that there are less than 2 errors in a particular chapter?

What is the difference between a random variable and a probability distribution?

For each of the following indicate whether the random variable is discrete or continuous. a. The length of time to get a haircut. b. The number of cars a jogger passes each morning while running. c. The number of hits for a team in a high school girls' softball game. d. The number of patients treated at the South Strand Medical Center between 6 and 10 p.m. each night. e. The distance your car traveled on the last fill-up. f. The number of customers at the Oak Street Wendy's who used the drive- through facility. g. The distance between Gainesville, Florida, and all Florida cities with a population of at least 50,000 .

What are the requirements for the binomial distribution?

Under what conditions will the binomial and the Poisson distributions give roughly the same results?

An investment will be worth \(\$ 1,000, \$ 2,000,\) or \(\$ 5,000\) at the end of the year. The probabilities of these values are \(.25, .60,\) and \(.15,\) respectively. Determine the mean and variance of the worth of the investment.

A Tamiami shearing machine is producing 10 percent defective pieces, which is abnormally high. The quality control engineer has been checking the output by almost continuous sampling since the abnormal condition began. What is the probability that in a sample of 10 pieces: a. Exactly 5 will be defective? 5 or more will be defective?

Thirty percent of the population in a southwestern community are Spanish- speaking Americans. A Spanish speaking person is accused of killing a non- Spanish speaking American and goes to trial. Of the first 12 potential jurors, only 2 are Spanish-speaking Americans, and 10 are not. The defendant's lawyer challenges the jury selection, claiming bias against her client. The government lawyer disagrees, saying that the probability of this particular jury composition is common. Compute the probability and discuss the assumptions.

An auditor for Health Maintenance Services of Georgia reports 40 percent of policyholders 55 years or older submit a claim during the year. Fifteen policyholders are randomly selected for company records. a. How many of the policyholders would you expect to have filed a claim within the last year? b. What is the probability that 10 of the selected policyholders submitted a claim last year? c. What is the probability that 10 or more of the selected policyholders submitted a claim last year? d. What is the probability that more than 10 of the selected policyholders submitted a claim last year?

Tire and Auto Supply is considering a 2 -for- 1 stock split. Before the transaction is finalized, at least two thirds of the 1,200 company stockholders must approve the proposal. To evaluate the likelihood the proposal will be approved, the CFO selected a sample of 18 stockholders. He contacted each and found 14 approved of the proposed split. What is the likelihood of this event, assuming two-thirds of the stockholders approve?

A federal study reported that 7.5 percent of the U.S. workforce has a drug problem. A drug enforcement official for the State of Indiana wished to investigate this statement. In her sample of 20 employed workers: a. How many would you expect to have a drug problem? What is the standard deviation? b. What is the likelihood that none of the workers sampled has a drug problem?

The Bank of Hawaii reports that 7 percent of its credit card holders will default at some time in their life. The Hilo branch just mailed out 12 new cards today. a. How many of these new cardholders would you expect to default? What is the standard deviation? b. What is the likelihood that none of the cardholders will default? c. What is the likelihood at least one will default?

Recent statistics suggest that 15 percent of those who visit Blair, Inc., an online retail clothier, make a purchase. Blair's planning department wishes to verify this claim. To do so, they selected a sample of 16 "hits" to the site and found that 4 actually made a purchase. a. What is the likelihood of exactly four purchases? b. How many purchases should they expect? c. What is the likelihood that four or more "hits" result in a purchase?

Dr. Richmond, a psychologist, is studying the daytime television viewing habits of college students. She believes 45 percent of college students watch soap operas during the afternoon. To further investigate, she selects a sample of \(10 .\) a. Develop a probability distribution for the number of students in the sample who watch soap operas. b. Find the mean and the standard deviation of this distribution. c. What is the probability of finding exactly four watch soap operas? d. What is the probability less than half of the students selected watch soap operas?

A recent study conducted by Penn, Shone, and Borland, on behalf of LastMinute.com, revealed that 52 percent of business travelers plan their trips less than two weeks before departure. The study is to be replicated in the tri-state area with a sample of 12 frequent business travelers. a. Develop a probability distribution for the number of travelers who plan their trips within two weeks of departure. b. Find the mean and the standard deviation of this distribution. c. What is the probability exactly 5 of the 12 selected business travelers plan their trips within two weeks of departure? d. What is the probability 5 or fewer of the 12 selected business travelers plan their trips within two weeks of departure?

Suppose 1.5 percent of the new Nokia cell phones are defective. For a random sample of 200 antennas, find the probability that: a. None of the antennas is defective. b. Three or more of the antennas are defective.

An internal study by the Technology Services department at Lahey Electronics revealed company employees receive an average of two emails per hour. Assume the arrival of these emails is approximated by the Poisson distribution. a. What is the probability Linda Lahey, company president, received exactly 1 email between 4 p.m. and 5 p.m. yesterday? b. What is the probability she received 5 or more email during the same period? c. What is the probability she did not receive any email during the period?

Recent crime reports indicate that 3.1 motor vehicle thefts occur each minute in the United States. Assume that the distribution of thefts per minute can be approximated by the Poisson probability distribution. a. Calculate the probability exactly four thefts occur in a minute. b. What is the probability there are no thefts in a minute? c. What is the probability there is at least one theft in a minute?

New Process, Inc., a large mail-order supplier of women's fashions, advertises same-day service on every order. Recently the movement of orders has not gone as planned, and there were a large number of complaints. Bud Owens, director of customer service, has completely redone the method of order handling. The goal is to have fewer than five unfilled orders on hand at the end of 95 percent of the working days. Frequent checks of the unfilled orders at the end of the day reveal that the distribution of the unfilled orders follows a Poisson distribution with a mean of two orders. a. Has New Process, Inc., lived up to its internal goal? Cite evidence b. Draw a histogram representing the Poisson probability distribution of unfilled orders.

The National Aeronautics and Space Administration (NASA) has experienced two disasters. The Challenger exploded over the Atlantic Ocean in 1986 and the Columbia exploded over East Texas in \(2003 .\) There have been a total of 113 space missions. Assume failures continue to occur at the same rate and consider the next 23 missions. What is the probability of exactly two failures? What is the probability of no failures?

According to the "January theory," if the stock market is up for the month of January, it will be up for the year. If it is down in January, it will be down for the year. According to an article in The Wall Street Journal, this theory held for 29 out of the last 34 years. Suppose there is no truth to this theory; that is, the probability it is either up or down is .50. What is the probability this could occur by chance? (You will probably need a software package such as Excel or MINITAB.)

During the second round of the 1989 U.S. Open golf tournament, four golfers scored a hole in one on the sixth hole. The odds of a professional golfer making a hole in one are estimated to be 3,708 to \(1,\) so the \(\begin{array}{lllll}\text { probability is } & 1 / 3,709 . & \text { There were } & 155 & \text { golfers }\end{array}\) participating in the second round that day. Estimate the probability that four golfers would score a hole in one on the sixth hole.

On September \(18,2003,\) hurricane Isabel struck the North Carolina Coast, causing extensive damage. For several days prior to reaching land the National Hurricane Center had been predicting the hurricane would come on shore between Cape Fear, North Carolina, and the North Carolina-Virginia border. It was estimated that the probability the hurricane would actually strike in this area was .95. In fact, the hurricane did come on shore almost exactly as forecast and was almost in the center of the strike area. Suppose the National Hurricane Center forecasts that hurricanes will hit the strike area with a .95 probability. Answer the following questions: a. What probability distribution does this follow? b. What is the probability that 10 hurricanes reach landfall in the strike area? c. What is the probability at least one of 10 hurricanes reaches land outside the strike area?

A recent CBS News survey reported that 67 percent of adults felt the U.S. Treasury should continue making pennies. Suppose we select a sample of 15 adults. a. How many of the 15 would we expect to indicate that the Treasury should continue making pennies? What is the standard deviation? b. What is the likelihood that exactly 8 adults would indicate the Treasury should continue making pennies? c. What is the likelihood at least 8 adults would indicate the Treasury should continue making pennies?

From the 2006 Baseball data, the New York Mets hit 200 home runs. So over the 162 -game season, they hit an average of 1.23 home runs per game. Assume the number of home runs hit per game follows the Poisson probability distribution. If you are not a Mets fan, pick your favorite team and answer the following questions. a. What is the interval or "continuum" in this instance? b. In what percentage of their games would you estimate that the Mets did not hit any home runs? c. In what percentage of the games did they hit at least one home run? d. Is there any chance the Mets could hit five home runs in a game?