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 random 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 In January the U.S. unemployment rate dropped to 8.3% (U.S. Department of Labor website, February 10, 2012). The Census Bureau includes nine states in the Northeast region. Assume that the random variable of interest is the number of Northeastern states with an unemployment rate in January that was less than 8.3%. What values may this random variable assume?
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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 random 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 an empirical discrete 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
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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 randomly selected senior executive.
b. Develop a probability distribution for the job satisfaction score of a randomly selected middle manager.
c. What is the probability a randomly selected senior executive will report a job satisfaction score of 4 or 5?
d. What is the probability a randomly selected 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 randomly selected 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 Time Warner Cable provides television and Internet service to over 15 million people (Time Warner Cable website, October 24, 2012). Suppose that the management of Time Warner Cable subjectively assesses a probability distribution for the number of new subscribers next year in the state of New York as follows....
a. Is this probability distribution valid? Explain.
b. What is the probability Time Warner will obtain more than 400,000 new subscribers?
c. What is the probability Time Warner will obtain fewer than 200,000 new subscribers?
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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 times that owner-occupied and renter-occupied units had a water supply stoppage lasting 6 or more hours in the past 3 months (U.S. Census Bureau website, October 2012)....
a. Define a random variable x = number of times that owner-occupied units had a water supply stoppage lasting 6 or more hours in the past 3 months and develop a probability distribution for the random variable. (Let x = 4 represent 4 or more times.)
b. Compute the expected value and variance for x.
c. Define a random variable y = number of times that renter-occupied units had a water supply stoppage lasting 6 or more hours in the past 3 months and develop a probability distribution for the random variable. (Let y = 4 represent 4 or more times.)
d. Compute the expected value and variance for y.
e. What observations can you make from a comparison of the number of water supply stoppages reported by owner-occupied units versus renter-occupied units?
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5.19 West Virginia has one of the highest divorce rates in the nation, with an annual rate of approximately 5 divorces per 1000 people (Centers for Disease Control and Prevention website, January 12, 2012). The Marital Counseling Center, Inc. (MCC) thinks that the high divorce rate in the state may require them to hire additional staf
f. Working with a consultant, the management of MCC has developed the following probability distribution for x = the number of new clients for marriage counseling for the next year....
a. Is this probability distribution valid? Explain.
b. What is the probability MCC will obtain more than 30 new clients?
c. What is the probability MCC will obtain fewer than 20 new clients?
d. Compute the expected value and variance of x.
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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 In Gallup’s Annual Consumption Habits Poll, telephone interviews were conducted for a random sample of 1014 adults aged 18 and over. One of the questions was, “How man; cups of coffee, if any, do you drink on an average day?” The following table shows the results obtained (Gallup website, August 6, 2012)....Define a random variable x = number of cups of coffee consumed on an average day. Let x = 4 represent four or more cups.
a. Develop a probability distribution for x.
b. Compute the expected value of x.
c. Compute the variance of x.
d. Suppose we are only interested in adults who drink at least one cup of coffee on an average day. For this group, let y = the number of cups of coffee consumed on an average day. Compute the expected value of y and compare it to the expected value of x.
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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
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5.26
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5.26
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5.28
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5.29
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5.30
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5.31 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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5.32 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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5.33 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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5.34 For its Music 360 survey, Nielsen Co. asked teenagers and adults how each group has listened to music in the past 12 months. Nearly two-thirds of U.S. teenagers under the age of 18 say they use Google Inc.’s video-sharing site to listen to music and 35% of the teenagers said they use Pandora Media Inc.’s custom online radio service (The Wall Street Journal, August 14, 2012). Suppose 10 teenagers are selected randomly to be interviewed about how they listen to music.
a. Is randomly selecting 10 teenagers and asking whether or not they use Pandora Media Inc.’s online service a binomial experiment?
b. What is the probability that none of the 10 teenagers use Pandora Media Inc.’s online radio service?
c. What is the probability that 4 of the 10 teenagers use Pandora Media Inc.’s online radio service?
d. What is the probability that at least 2 of the 10 teenagers use Pandora Media Inc.’s online radio service?
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5.35 The Center for Medicare and Medical Services reported that there were 295,000 appeals for hospitalization and other Part A Medicare service. For this group, 40% of first-round appeals were successful (The Wall Street Journal, October 22, 2012). Suppose 10 first-round appeals have just been received by a Medicare appeals office.
a. Compute the probability that none of the appeals will be successful.
b. Compute the probability that exactly one of the appeals will be successful.
c. What is the probability that at least two of the appeals will be successful?
d. What is the probability that more than half of the appeals will be successful?
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5.36 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 5.4 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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5.37
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5.38 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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5.39
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5.40 A study conducted by the Pew Research Center showed that 75% of 18-to 34-year-olds living with their parents say they contribute to household expenses (The Wall Street Journal, October 22, 2012). Suppose that a random sample of fifteen 18-to 34-year-olds living with their parents is selected and asked if they contribute to household expenses.
a. Is the selection of the fifteen 18-to 34-year-olds living with their parents a binomial experiment? Explain.
b. If the sample shows that none of the fifteen 18-to 34-year-olds living with their parents contribute to household expenses, would you question the results of the Pew Research Study? Explain.
c. What is the probability that at least 10 of the fifteen 18-to 34-year-olds living with their parents contribute to household expenses?
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5.41 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 2 or fewer will withdraw.
b. Compute the probability that exactly 4 will withdraw.
c. Compute the probability that more than 3 will withdraw.
d. Compute the expected number of withdrawals.
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5.42 A Gallup Poll showed that 30% of Americans are satisfied with the way things are going in the United States (Gallup website, September 12, 2012). Suppose a sample of 20 Americans is selected as part of a study of the state of the nation.
a. Compute the probability that exactly 4 of the 20 Americans surveyed are satisfied with the way things are going in the United States.
b. Compute the probability that at least 2 of the Americans surveyed are satisfied with the way things are going in the United States.
c. For the sample of 20 Americans, compute the expected number of Americans who are satisfied with the way things are going in the United States.
d. For the sample of 20 Americans, compute the variance and standard deviation of the number of Americans who are satisfied with the way things are going in the United States.
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5.43
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5.44 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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5.45 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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5.46 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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5.47 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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5.48 In 2011, New York City had a total of 11,232 motor vehicle accidents that occurred on Monday through Friday between the hours of 3 P.M. and 6 P.M. (New York State Department of Motor Vehicles website, October 24, 2012). This corresponds to mean of 14.4 accidents per hour.
a. Compute the probability of no accidents in a 15-minute period.
b. Compute the probability of at least one accident in a 15-minute period.
c. Compute the probability of four or more accidents in a 15-minute period.
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5.49 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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5.50 According to the National Oceanic and Atmospheric Administration (NOAA), the state of Colorado averages 18 tornadoes every June (NOAA website, November 8, 2012). (Note: There are 30 days in June.)
a. Compute the mean number of tornadoes per day.
b. Compute the probability of no tornadoes during a day.
c. Compute the probability of exactly one tornado during a day.
d. Compute the probability of more than one tornado during a day.
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5.51
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5.52 Suppose N = 10 and r = 3. Compute the hypergeometric probabilities for the following values of n and x.
a. n = 4, x = 1.
b. n = 2, x = 2.
c. n = 2, x = 0.
d. n = 4, x = 2.
e. n = 4, x = 4.
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5.53 Suppose N = 15 and r = 4. What is the probability of x = 3 for n = 10?
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5.54 A recent survey showed that a majority of Americans plan on doing their holiday shopping online because they don’t want to spend money on gas driving from store to store (SOASTA website, October 24, 2012). Suppose we have a group of 10 shoppers; 7 prefer to do their holiday shopping online and 3 prefer to do their holiday shopping in stores. A random sample of 3 of these 10 shoppers is selected for a more in-depth study of how the economy has impacted their shopping behavior.
a. What is the probability that exactly 2 prefer shopping online?
b. What is the probability that the majority (either 2 or 3) prefer shopping online?
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5.55 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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5.56 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 1 of the employees in the sample works at the plant in Hawaii?
c. What is the probability that 2 or more of the employees in the sample work at the plant in Hawaii?
d. What is the probability that 9 of the employees in the sample work at the plant in Texas?
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5.57 The Zagat Restaurant Survey provides food, decor, and service ratings for some of the top restaurants across the United States. For 15 restaurants located in Boston, the average price of a dinner, including one drink and tip, was $48.60. You are leaving on a business trip to Boston and will eat dinner at three of these restaurants. Your company will reimburse you for a maximum of $50 per dinner. Business associates familiar with these restaurants have told you that the meal cost at one-third of these restaurants will exceed $50. Suppose that you randomly select three of these restaurants for dinner.
a. What is the probability that none of the meals will exceed the cost covered by your company?
b. What is the probability that one of the meals will exceed the cost covered by your company?
c. What is the probability that two of the meals will exceed the cost covered by your company?
d. What is the probability that all three of the meals will exceed the cost covered by your company?
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5.58
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5.59 The U.S. Coast guard (USCg) provides a wide variety of information on boating accidents including the wind condition at the time of the accident. The following table shows the results obtained for 4401 accidents (USCg website, November 8, 2012)....Let x be a random variable reflecting the known wind condition at the time of each accident. Set x = 0 for none, x = 1 for light, x = 2 for moderate, x = 3 for strong, and x = 4 for storm.
a. Develop a probability distribution for x.
b. Compute the expected value of x.
c. Compute the variance and standard deviation for x.
d. Comment on what your results imply about the wind conditions during boating accidents.
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5.60 The Car Repair Ratings website provides consumer reviews and ratings for garages in the United States and Canada. The time customers wait for service to be completed is one of the categories rated. The following table provides a summary of the wait-time ratings (1 = Slow/Delays; 10 = Quick/On Time) for 40 randomly selected garages located in the province of Ontario, Canada (Car Repair Ratings website, November 14, 2012)....
a. Develop a probability distribution for x = wait-time rating.
b. Any garage that receives a wait-time rating of at least 9 is considered to provide outstanding service. If a consumer randomly selects one of the 40 garages for their next car service, what is the probability the garage selected will provide outstanding wait-time service?
c. What is the expected value and variance for x?
d. Suppose that 7 of the 40 garages reviewed were new car dealerships. Of the 7 new car dealerships, two were rated as providing outstanding wait-time service. Compare the likelihood of a new car dealership achieving an outstanding wait-time service rating as compared to other types of service providers.
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5.61 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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5.62
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5.63
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5.64 The Pew Research Center surveyed adults who own/use the following technologies: Internet, smartphone, email, and land-line phone (USA Today, March 26, 2014) and asked which of these technologies would be “very hard” to give up. The following responses were obtained: Internet 53%, smartphone 49%, email 36%, and land-line phone 28%.
a. If 20 adult Internet users are surveyed, what is the probability that 3 users will report that it would be very hard to give it up?
b. If 20 adults who own a land-line phone are surveyed, what is the probability that 5 or fewer will report that it would be very hard to give it up?
c. If 2000 owners of smartphones were surveyed, what is the expected number that will report that it would be very hard to give it up?
d. If 2000 users of email were surveyed, what is expected number that will report that it would be very hard to give it up? What is the variance and standard deviation?
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5.65 The following table shows the percentage of individuals in each age group who use an online tax program to prepare their federal income tax return (CompleteTax website, November 9, 2012)....Suppose a follow-up study consisting of personal interviews is to be conducted to determine the most important factors in selecting a method for filing taxes.
a. How many 18–34-year-olds must be sampled to find an expected number of at least 25 who use an online tax program to prepare their federal income tax return?
b. How many 35–44-year-olds must be sampled to find an expected number of at least 25 who use an online tax program to prepare their federal income tax return?
c. How many 65+-year-olds must be sampled to find an expected number of at least 25 who use an online tax program to prepare their federal income tax return?
d. If the number of 18–34-year-olds sampled is equal to the value identified in part (a), what is the standard deviation of the percentage who use an online tax program?
e. If the number of 35–44-year-olds sampled is equal to the value identified in part (b), what is the standard deviation of the percentage who use an online tax program?
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5.66 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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5.67
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5.68 Mahoney Custom Home Builders, Inc. of Canyon Lake, Texas, asked visitors to their website what is most important when choosing a home builder. Possible responses were quality, price, customer referral, years in business, and special features. Results showed that 23.5% of the respondents chose price as the most important factor (Mahoney Custom Homes website, November 13, 2012). Suppose a sample of 200 potential home buyers in the Canyon Lake area was selected.
a. How many people would you expect to choose price as the most important factor when choosing a home builder?
b. What is the standard deviation of the number of respondents who would choose price as the most important factor in selecting a home builder?
c. What is the standard deviation of the number of respondents who do not list price as the most important factor in selecting a home builder?
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5.69 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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5.70 A new 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 3 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 3 or more breakdowns during a day?
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5.71 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 4 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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5.72 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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