a. List the experimental outcomes.
b. Define a random variable that represents the number of heads occurring on the two tosses.
c. Show what value the random variable would assume for each of the experimental outcomes.
d. Is this random variable discrete or continuous?
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5.2 Consider the experiment of a worker assembling a product.
a. Define a random variable that represents the time in minutes required to assemble the product.
b. What values may the random variable assume?
c. Is the random variable discrete or continuous?
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5.3 Three students scheduled interviews for summer employment at the Brookwood Institute. In each case the interview results in either an offer for a position or no offer. Experimental outcomes are defined in terms of the results of the three interviews.
a. List the experimental outcomes.
b. Define a random variable that represents the number of offers made. Is the random variable continuous?
c. Show the value of the random variable for each of the experimental outcomes.
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5.4
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5.5 To perform a certain type of blood analysis, lab technicians must perform two procedures. The first procedure requires either one or two separate steps, and the second procedure requires either one, two, or three steps.
a. List the experimental outcomes associated with performing the blood analysis.
b. If the random variable of interest is the total number of steps required to do the complete analysis (both procedures), show what value the random variable will assume for each of the experimental outcomes.
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5.6 Listed is a series of experiments and associated random variables. In each case, identify the values that the random variable can assume and state whether the random variable is discrete or continuous....
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5.7 The probability distribution for the random variable x follows....
a. Is this probability distribution valid? Explain.
b. What is the probability that x = 30?
c. What is the probability that x is less than or equal to 25?
d. What is the probability that x is greater than 30?
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5.8 The following data were collected by counting the number of operating rooms in use at Tampa General Hospital over a 20-day period: On three of the days only one operating room was used, on five of the days two were used, on eight of the days three were used, and on four days all four of the hospital’s operating rooms were used.
a. Use the relative frequency approach to construct a probability distribution for the number of operating rooms in use on any given day.
b. Draw a graph of the probability distribution.
c. Show that your probability distribution satisfies the required conditions for a valid discrete probability distribution.
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5.9 Nationally, 38% of fourth-graders cannot read an age-appropriate book. The following data show the number of children, by age, identified as learning disabled under special education. Most of these children have reading problems that should be identified and corrected before third grade. Current federal law prohibits most children from receiving extra help from special education programs until they fall behind by approximately two years’ worth of learning, and that typically means third grade or later (USA Today, September 6, 2001)....Suppose that we want to select a sample of children identified as learning disabled under special education for a program designed to improve reading ability. Let x be a random variable indicating the age of one randomly selected child.
a. Use the data to develop a probability distribution for x. Specify the values for the random variable and the corresponding values for the probability function f (x).
b. Draw a graph of the probability distribution.
c. Show that the probability distribution satisfies equations (5.1) and (5.2).
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5.10 The percent frequency distributions of job satisfaction scores for a sample of information systems (IS) senior executives and middle managers are as follows. The scores range from a low of 1 (very dissatisfied) to a high of 5 (very satisfied)....
a. Develop a probability distribution for the job satisfaction score of a senior executive.
b. Develop a probability distribution for the job satisfaction score of a middle manager.
c. What is the probability a senior executive will report a job satisfaction score of 4 or 5?
d. What is the probability a middle manager is very satisfied?
e. Compare the overall job satisfaction of senior executives and middle managers.
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5.11 A technician services mailing machines at companies in the Phoenix area. Depending on the type of malfunction, the service call can take 1, 2, 3, or 4 hours. The different types of malfunctions occur at about the same frequency.
a. Develop a probability distribution for the duration of a service call.
b. Draw a graph of the probability distribution.
c. Show that your probability distribution satisfies the conditions required for a discrete probability function.
d. What is the probability a service call will take three hours?
e. A service call has just come in, but the type of malfunction is unknown. It is 3:00 p.m. and service technicians usually get off at 5:00 p.m. What is the probability the service technician will have to work overtime to fix the machine today?
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5.12
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5.13 A psychologist determined that the number of sessions required to obtain the trust of a new patient is either 1, 2, or 3. Let x be a random variable indicating the number of sessions required to gain the patient’s trust. The following probability function has been proposed....
a. Is this probability function valid? Explain.
b. What is the probability that it takes exactly 2 sessions to gain the patient’s trust?
c. What is the probability that it takes at least 2 sessions to gain the patient’s trust?
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5.14 The following table is a partial probability distribution for the MRA Company’s projected profits (x = profit in $1000s) for the first year of operation (the negative value denotes a loss)....
a. What is the proper value for f (200)? What is your interpretation of this value?
b. What is the probability that MRA will be profitable?
c. What is the probability that MRA will make at least $100,000?
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5.15 The following table provides a probability distribution for the random variable x....
a. Compute E(x), the expected value of x.
b. Compute σ2, the variance of x.
c. Compute σ, the standard deviation of x.
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5.16 The following table provides a probability distribution for the random variable y....
a. Compute E(y).
b. Compute Var(y) and σ.
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5.17
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5.18 The American Housing Survey reported the following data on the number of bedrooms in owner occupied and renter-occupied houses in central cities (http://www.census.gov, March 31, 2003)....
a. Define a random variable x = number of bedrooms in renter-occupied houses and develop a probability distribution for the random variable. (Let x = 4 represent 4 or more bedrooms.)
b. Compute the expected value and variance for the number of bedrooms in renter occupied houses.
c. Define a random variable y = number of bedrooms in owner-occupied houses and develop a probability distribution for the random variable. (Let y = 4 represent 4 or more bedrooms.)
d. Compute the expected value and variance for the number of bedrooms in owner occupied houses.
e. What observations can you make from a comparison of the number of bedrooms in renter-occupied versus owner-occupied homes?
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5.19 The National Basketball Association (NBA) records a variety of statistics for each team. Two of these statistics are the percentage of field goals made by the team and the percentage of three-point shots made by the team. For a portion of the 2004 season, the shooting records of the 29 teams in the NBA showed the probability of scoring two points by making a field goal was .44, and the probability of scoring three points by making a three-point shot was .34 (http://www.nba.com, January 3, 2004).
a. What is the expected value of a two-point shot for these teams?
b. What is the expected value of a three-point shot for these teams?
c. If the probability of making a two-point shot is greater than the probability of making a three-point shot, why do coaches allow some players to shoot the three-point shot if they have the opportunity? Use expected value to explain your answer.
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5.20 The probability distribution for damage claims paid by the Newton Automobile Insurance Company on collision insurance follows....
a. Use the expected collision payment to determine the collision insurance premium that would enable the company to break even.
b. The insurance company charges an annual rate of $520 for the collision coverage. What is the expected value of the collision policy for a policyholder? (Hint: It is the expected payments from the company minus the cost of coverage.) Why does the policyholder purchase a collision policy with this expected value?
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5.21 The following probability distributions of job satisfaction scores for a sample of information systems (IS) senior executives and middle managers range from a low of 1 (very dissatisfied) to a high of 5 (very satisfied)....
a. What is the expected value of the job satisfaction score for senior executives?
b. What is the expected value of the job satisfaction score for middle managers?
c. Compute the variance of job satisfaction scores for executives and middle managers.
d. Compute the standard deviation of job satisfaction scores for both probability distributions.
e. Compare the overall job satisfaction of senior executives and middle managers.
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5.22 The demand for a product of Carolina Industries varies greatly from month to month. The probability distribution in the following table, based on the past two years of data, shows the company’s monthly demand....
a. If the company bases monthly orders on the expected value of the monthly demand, what should Carolina’s monthly order quantity be for this product?
b. Assume that each unit demanded generates $70 in revenue and that each unit ordered costs $50. How much will the company gain or lose in a month if it places an order based on your answer to part (a) and the actual demand for the item is 300 units?
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5.23
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5.24 The J. R. Ryland Computer Company is considering a plant expansion to enable the company to begin production of a new computer product. The company’s president must determine whether to make the expansion a medium- or large-scale project. Demand for the new product is uncertain, which for planning purposes may be low demand, medium demand, or high demand. The probability estimates for demand are .20, .50, and .30, respectively. Letting x and y indicate the annual profit in thousands of dollars, the firm’s planners developed the following profit forecasts for the medium- and large-scale expansion projects....
a. Compute the expected value for the profit associated with the two expansion alternatives. Which decision is preferred for the objective of maximizing the expected profit?
b. Compute the variance for the profit associated with the two expansion alternatives. Which decision is preferred for the objective of minimizing the risk or uncertainty?
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5.25 Consider a binomial experiment with two trials and p = .4.
a. Draw a tree diagram for this experiment (see Figure 5.3).
b. Compute the probability of one success, f (1).
c. Compute f (0).
d. Compute f (2).
e. Compute the probability of at least one success.
f. Compute the expected value, variance, and standard deviation.
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5.26 Consider a binomial experiment with n = 10 and p = .10.
a. Compute f (0).
b. Compute f (2).
c. Compute P(x ≤ 2).
d. Compute P(x ≥ 1).
e. Compute E(x).
f. Compute Var(x) and σ.
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5.27 Consider a binomial experiment with n = 20 and p = .70.
a. Compute f (12).
b. Compute f (16).
c. Compute P(x ≥ 16).
d. Compute P(x ≤ 15).
e. Compute E(x).
f. Compute Var(x) and σ.
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5.28 AHarris Interactive survey for InterContinental Hotels & Resorts asked respondents, “When traveling internationally, do you generally venture out on your own to experience culture, or stick with your tour group and itineraries?” The survey found that 23% of the respondents stick with their tour group (USA Today, January 21, 2004).
a. In a sample of 6 international travelers, what is the probability that 2 will stick with their tour group?
b. In a sample of 6 international travelers, what is the probability that at least 2 will stick with their tour group?
c. In a sample of 10 international travelers, what is the probability that none will stick with the tour group?
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5.29 In San Francisco, 30% of workers take public transportation daily (USA Today, December 21, 2005).
a. In a sample of 10 workers, what is the probability that exactly 3 workers take public transportation daily?
b. In a sample of 10 workers, what is the probability that at least 3 workers take public transportation daily?
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5.30 When a new machine is functioning properly, only 3% of the items produced are defective. Assume that we will randomly select two parts produced on the machine and that we are interested in the number of defective parts found.
a. Describe the conditions under which this situation would be a binomial experiment.
b. Draw a tree diagram similar to Figure 5.3 showing this problem as a two-trial experiment.
c. How many experimental outcomes result in exactly one defect being found?
d. Compute the probabilities associated with finding no defects, exactly one defect, and two defects.
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5.31 Nine percent of undergraduate students carry credit card balances greater than $7000 (Reader’s Digest, July 2002). Suppose 10 undergraduate students are selected randomly to be interviewed about credit card usage.
a. Is the selection of 10 students a binomial experiment? Explain.
b. What is the probability that 2 of the students will have a credit card balance greater than $7000?
c. What is the probability that none will have a credit card balance greater than $7000?
d. What is the probability that at least 3 will have a credit card balance greater than $7000?
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5.32 Military radar and missile detection systems are designed to warn a country of an enemy attack. A reliability question is whether a detection system will be able to identify an attack and issue a warning. Assume that a particular detection system has a .90 probability of detecting a missile attack. Use the binomial probability distribution to answer the following questions.
a. What is the probability that a single detection system will detect an attack?
b. If two detection systems are installed in the same area and operate independently, what is the probability that at least one of the systems will detect the attack?
c. If three systems are installed, what is the probability that at least one of the systems will detect the attack?
d. Would you recommend that multiple detection systems be used? Explain.
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5.33 Fifty percent of Americans believed the country was in a recession, even though technically the economy had not shown two straight quarters of negative growth (BusinessWeek, July 30, 2001). For a sample of 20 Americans, make the following calculations.
a. Compute the probability that exactly 12 people believed the country was in a recession.
b. Compute the probability that no more than 5 people believed the country was in a recession.
c. How many people would you expect to say the country was in a recession?
d. Compute the variance and standard deviation of the number of people who believed the country was in a recession.
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5.34 The Census Bureau’s Current Population Survey shows 28% of individuals, ages 25 and older, have completed four years of college (The New York Times Almanac, 2006). For a sample of 15 individuals, ages 25 and older, answer the following questions:
a. What is the probability 4 will have completed four years of college?
b. What is the probability 3 or more will have completed four years of college?
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5.35 A university found that 20% of its students withdraw without completing the introductory statistics course. Assume that 20 students registered for the course.
a. Compute the probability that two or fewer will withdraw.
b. Compute the probability that exactly four will withdraw.
c. Compute the probability that more than three will withdraw.
d. Compute the expected number of withdrawals.
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5.36
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5.37 Twenty-three percent of automobiles are not covered by insurance (CNN, February 23, 2006). On a particular weekend, 35 automobiles are involved in traffic accidents.
a. What is the expected number of these automobiles that are not covered by insurance?
b. What are the variance and standard deviation?
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5.38 Consider a Poisson distribution with μ = 3.
a. Write the appropriate Poisson probability function.
b. Compute f (2).
c. Compute f (1).
d. Compute P(x ≥ 2).
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5.39 Consider a Poisson distribution with a mean of two occurrences per time period.
a. Write the appropriate Poisson probability function.
b. What is the expected number of occurrences in three time periods?
c. Write the appropriate Poisson probability function to determine the probability of x occurrences in three time periods.
d. Compute the probability of two occurrences in one time period.
e. Compute the probability of six occurrences in three time periods.
f. Compute the probability of five occurrences in two time periods.
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5.40 Phone calls arrive at the rate of 48 per hour at the reservation desk for Regional Airways.
a. Compute the probability of receiving three calls in a 5-minute interval of time.
b. Compute the probability of receiving exactly 10 calls in 15 minutes.
c. Suppose no calls are currently on hold. If the agent takes 5 minutes to complete the current call, how many callers do you expect to be waiting by that time? What is the probability that none will be waiting?
d. If no calls are currently being processed, what is the probability that the agent can take 3 minutes for personal time without being interrupted by a call?
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5.41 During the period of time that a local university takes phone-in registrations, calls come in at the rate of one every two minutes.
a. What is the expected number of calls in one hour?
b. What is the probability of three calls in five minutes?
c. What is the probability of no calls in a five-minute period?
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5.42 More than 50 million guests stay at bed and breakfasts (B&Bs) each year. The Web site for the Bed and Breakfast Inns of North America (http://www.bestinns.net), which averages approximately 7 visitors per minute, enables many B&Bs to attract guests (Time, September 2001).
a. Compute the probability of no Web site visitors in a 1-minute period.
b. Compute the probability of 2 or more Web site visitors in a 1-minute period.
c. Compute the probability of 1 or more Web site visitors in a 30-second period.
d. Compute the probability of 5 or more Web site visitors in a 1-minute period.
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5.43 Airline passengers arrive randomly and independently at the passenger-screening facility at a major international airport. The mean arrival rate is 10 passengers per minute.
a. Compute the probability of no arrivals in a one-minute period.
b. Compute the probability that three or fewer passengers arrive in a one-minute period.
c. Compute the probability of no arrivals in a 15-second period.
d. Compute the probability of at least one arrival in a 15-second period.
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5.44 An average of 15 aircraft accidents occur each year (The World Almanac and Book of Facts, 2004).
a. Compute the mean number of aircraft accidents per month.
b. Compute the probability of no accidents during a month.
c. Compute the probability of exactly 1 accident during a month.
d. Compute the probability of more than 1 accident during a month.
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5.45 The National Safety Council (NSC) estimates that off-the-job accidents cost U.S. businesses almost $200 billion annually in lost productivity (National Safety Council, March 2006). Based on NSC estimates, companies with 50 employees are expected to average three employee off-the-job accidents per year. Answer the following questions for companies with 50 employees.
a. What is the probability of no off-the-job accidents during a one-year period?
b. What is the probability of at least two off-the-job accidents during a one-year period?
c. What is the expected number of off-the-job accidents during six months?
d. What is the probability of no off-the-job accidents during the next six months?
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5.46
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5.47 Suppose N = 15 and r = 4. What is the probability of x = 3 for n = 10?
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5.48 In a survey conducted by the Gallup Organization, respondents were asked, “What is your favorite sport to watch?” Football and basketball ranked number one and two in terms of preference (http://www.gallup.com, January 3, 2004). Assume that in a group of 10 individuals, 7 preferred football and 3 preferred basketball. A random sample of 3 of these individuals is selected.
a. What is the probability that exactly 2 preferred football?
b. What is the probability that the majority (either 2 or 3) preferred football?
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5.49 Blackjack, or twenty-one as it is frequently called, is a popular gambling game played in Las Vegas casinos. A player is dealt two cards. Face cards (jacks, queens, and kings) and tens have a point value of 10. Aces have a point value of 1 or 11. A 52-card deck contains 16 cards with a point value of 10 (jacks, queens, kings, and tens) and four aces.
a. What is the probability that both cards dealt are aces or 10-point cards?
b. What is the probability that both of the cards are aces?
c. What is the probability that both of the cards have a point value of 10?
d. A blackjack is a 10-point card and an ace for a value of 21. Use your answers to parts (a), (b), and (c) to determine the probability that a player is dealt blackjack. (Hint: Part (d) is not a hypergeometric problem. Develop your own logical relationship as to how the hypergeometric probabilities from parts (a), (b), and (c) can be combined to answer this question.)
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5.50 Axline Computers manufactures personal computers at two plants, one in Texas and the other in Hawaii. The Texas plant has 40 employees; the Hawaii plant has 20. A random sample of 10 employees is to be asked to fill out a benefits questionnaire.
a. What is the probability that none of the employees in the sample work at the plant in Hawaii?
b. What is the probability that one of the employees in the sample works at the plant in Hawaii?
c. What is the probability that two or more of the employees in the sample work at the plant in Hawaii?
d. What is the probability that nine of the employees in the sample work at the plant in Texas?
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5.51
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5.52
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5.53 The Barron’s Big Money Poll asked 131 investment managers across the United States about their short-term investment outlook (Barron’s, October 28, 2002). Their responses showed 4% were very bullish, 39% were bullish, 29% were neutral, 21% were bearish, and 7% were very bearish. Let x be the random variable reflecting the level of optimism about the market. Set x = 5 for very bullish down through x = 1 for very bearish.
a. Develop a probability distribution for the level of optimism of investment managers.
b. Compute the expected value for the level of optimism.
c. Compute the variance and standard deviation for the level of optimism.
d. Comment on what your results imply about the level of optimism and its variability.
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5.54 The American Association of Individual Investors publishes an annual guide to the top mutual funds (The Individual Investor’s Guide to the Top Mutual Funds, 22e, American Association of Individual Investors, 2003). Table 5.8 contains their ratings of the total risk for 29 categories of mutual funds.
a. Let x = 1 for low risk up through x = 5 for high risk, and develop a probability distribution for level of risk.
b. What are the expected value and variance for total risk?
c. It turns out that 11 of the fund categories were bond funds. For the bond funds, 7 categories were rated low and 4 were rated below average. Compare the total risk of the bond funds with the 18 categories of stock funds.
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5.55 The budgeting process for a midwestern college resulted in expense forecasts for the coming year (in $ millions) of $9, $10, $11, $12, and $13. Because the actual expenses are unknown, the following respective probabilities are assigned: .3, .2, .25, .05, and .2.
a. Show the probability distribution for the expense forecast.
b. What is the expected value of the expense forecast for the coming year?
c. What is the variance of the expense forecast for the coming year?
d. If income projections for the year are estimated at $12 million, comment on the financial position of the college.
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5.56 A survey conducted by the Bureau of Transportation Statistics (BTS) showed that the average commuter spends about 26 minutes on a one-way door-to-door trip from home to work. In addition, 5% of commuters reported a one-way commute of more than 1 hour (http://www.bts.gov, January 12, 2004).
a. If 20 commuters are surveyed on a particular day, what is the probability that 3 will report a one-way commute of more than 1 hour?
b. If 20 commuters are surveyed on a particular day, what is the probability that none will report a one-way commute of more than 1 hour?
c. If a company has 2000 employees, what is the expected number of employees that have a one-way commute of more than 1 hour?
d. If a company has 2000 employees, what is the variance and standard deviation of the number of employees that have a one-way commute of more than 1 hour?
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5.57
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5.58 Many companies use a quality control technique called acceptance sampling to monitor incoming shipments of parts, raw materials, and so on. In the electronics industry, component parts are commonly shipped from suppliers in large lots. Inspection of a sample of n components can be viewed as the n trials of a binomial experiment. The outcome for each component tested (trial) will be that the component is classified as good or defective. Reynolds Electronics accepts a lot from a particular supplier if the defective components in the lot do not exceed 1%. Suppose a random sample of five items from a recent shipment is tested.
a. Assume that 1% of the shipment is defective. Compute the probability that no items in the sample are defective.
b. Assume that 1% of the shipment is defective. Compute the probability that exactly one item in the sample is defective.
c. What is the probability of observing one or more defective items in the sample if 1% of the shipment is defective?
d. Would you feel comfortable accepting the shipment if one item was found to be defective? Why or why not?
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5.59
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5.60 A poll conducted by Zogby International showed that of those Americans who said music plays a “very important” role in their lives, 30% said their local radio stations “always” play the kind of music they like (http://www.zogby.com, January 12, 2004). Suppose a sample of 800 people who say music plays an important role in their lives is taken.
a. How many would you expect to say that their local radio stations always play the kind of music they like?
b. What is the standard deviation of the number of respondents who think their local radio stations always play the kind of music they like?
c. What is the standard deviation of the number of respondents who do not think their local radio stations always play the kind of music they like?
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5.61 Cars arrive at a car wash randomly and independently; the probability of an arrival is the same for any two time intervals of equal length. The mean arrival rate is 15 cars per hour. What is the probability that 20 or more cars will arrive during any given hour of operation?
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5.62 Anew automated production process averages 1.5 breakdowns per day. Because of the cost associated with a breakdown, management is concerned about the possibility of having three or more breakdowns during a day. Assume that breakdowns occur randomly, that the probability of a breakdown is the same for any two time intervals of equal length, and that breakdowns in one period are independent of breakdowns in other periods. What is the probability of having three or more breakdowns during a day?
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5.63 A regional director responsible for business development in the state of Pennsylvania is concerned about the number of small business failures. If the mean number of small business failures per month is 10, what is the probability that exactly four small businesses will fail during a given month? Assume that the probability of a failure is the same for any two months and that the occurrence or nonoccurrence of a failure in any month is independent of failures in any other month.
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5.64 Customer arrivals at a bank are random and independent; the probability of an arrival in any one-minute period is the same as the probability of an arrival in any other one-minute period. Answer the following questions, assuming a mean arrival rate of three customers per minute.
a. What is the probability of exactly three arrivals in a one-minute period?
b. What is the probability of at least three arrivals in a one-minute period?
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5.65 A deck of playing cards contains 52 cards, four of which are aces. What is the probability that the deal of a five-card hand provides:
a. A pair of aces?
b. Exactly one ace?
c. No aces?
d. At least one ace?
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5.66 Through the week ending September 16, 2001, Tiger Woods was the leading money winner on the PGA Tour, with total earnings of $5,517,777. Of the top 10 money winners, 7 players used a Titleist brand golf ball (http://www.pgatour.com). Suppose that we randomly select 2 of the top 10 money winners.
a. What is the probability that exactly 1 uses a Titleist golf ball?
b. What is the probability that both use Titleist golf balls?
c. What is the probability that neither uses a Titleist golf ball?
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