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Ch06 Discrete Probability Distributions.docx

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Chapter 6 Discrete Probability Distributions True/False 1. The Poisson probability distribution is always negatively skewed. Answer: False 2. A random variable is assigned numerical values based on the outcomes of an experiment. Answer: True 3. A random variable represents the outcomes of an experiment. Answer: True 4. The probability of a particular outcome, designated X, must always be between 0 and 100 inclusive. Answer: False 5. If we toss two coins and count the number of heads, there could be 0, 1 or 2 heads. Since the exact number of heads resulting from this experiment is due to chance, the number of heads appearing is a random variable. Answer: True 6. If Unique Buying Services has 100 employees, there might be 0, 1, 2, 3 up to 100 employees absent on Monday. In this case, the day of the week is the random variable. Answer: False 7. A discrete random variable can have only certain clearly separated values. Answer: True 8. A discrete variable may assume fractional or decimal values, but they must have distance between them. Answer: True 9. For a binomial distribution, each trial has a known number of successes. For example, a four question multiple-choice test can only have zero, one, two, three and four successes (number correct). Answer: True 10. The random variable for a Poisson probability distribution can assume an infinite number of values. Answer: True 11. A probability distribution is a mutually exclusive listing of experimental outcomes that can occur by chance and their corresponding probabilities. Answer: True 12. To construct a binomial probability distribution, the number of trials and the probability of success must be known. Answer: True 13. The Poisson probability distribution is a continuous probability distribution. Answer: False 14. If we measure the weight of a car, the variable is a discrete random variable. Answer: False 15. A binomial distribution has a characteristic that the trials are independent, which means that the outcome of one trial does not affect the outcome of any other trial. Answer: True 16. For a binomial distribution, outcomes of an experiment are classified into one of two mutually exclusive categories (a success or a failure). Answer: True 17. For a binomial distribution, the probability of a success stays the same for each trial, but the probability of a failure varies from trial to trial. Answer: False 18. For a binomial distribution, the data collected are the result of counts. Answer: True 19. When sampling is done without replacement and the outcomes are measured as either a success or failure, the hypergeometric distribution should be applied. Answer: True 20. A Poisson distribution is a discrete probability distribution. It has the same four characteristics as the binomial, but in addition, the probability of a success is small and the number of trials is relatively large. Answer: True 21. A random variable may be either discrete or continuous. Answer: True 22. The mean of a binomial probability distribution can be determined by multiplying the probability of a failure by the number of trials. Answer: False 23. A binomial distribution is a continuous probability distribution. Answer: False 24. The binomial and Poisson distributions are discrete probability distributions. Answer: True Goal: 4, 6 25. To construct a binomial distribution, it is necessary to know the total number of trials and the probability of success on each trial. Answer: True 26. If the probability of success () remains the same, but n increases, the shape of the binomial distribution becomes more symmetrical. Answer: True 27. The mean of a probability distribution is called its expected value. Answer: True 28. The mean of a binomial distribution is the product of n and . Answer: True 29. The variance of a binomial distribution is found by (1 - ). Answer: True 30. As a general rule of thumb, if the items selected for a sample are not replaced and the sample size is less than 5 percent of the population, the binomial distribution can be used to approximate the hypergeometric distribution. Answer: True 31. In a Poisson distribution, the probability of success may vary from trial to trial. Answer: False 32. A mail-order house, advertising "same day" service, received a large number of complaints. A change in the handling of orders was made with less than 5 unfilled orders on hand (per picker) at the end of 95 out of every 100 working days. Frequent checks of unfilled orders at the end of the day revealed that the distribution of unfilled orders approximated a Poisson distribution; that is, most of the days there were no unfilled orders; some of the days there was one order, and so on. Since the average number of unfilled orders per picker was 2.0, the mail-order house lived up to its goal. Answer: True 33. The binomial probability distribution is always negatively skewed. Answer: False 34. The expected value of summing two random variables is the sum of their expected values. Answer: True Goal: Covariance 35. If the covariance between two random variables is zero, the two variables are related. Answer: False Goal: Covariance 36. The variance of the sum of two random variables is the sum of their variances plus two times the covariance. Answer: True Goal: Covariance 37. If the covariance between two random variables is zero, the two variables are not related. Answer: True Goal: Covariance Multiple Choice 38. If the variance of a probability was computed to be 3.6 grams, what is the standard deviation? A) 0.6 B) 1.9 C) 6.0 D) 12.96 Answer: B 39. Sixty percent of the customers of a fast food chain order the Whopper, french fries and a drink. If a random sample of 15 cash register receipts is selected, what is the probability that 10 or more will show that the above three food items were ordered? A) 1,000 B) 0.186 C) 0.403 D) 0.000 Answer: C 40. Judging from recent experience, 5 percent of the computer keyboards produced by an automatic, high-speed machine are defective. What is the probability that out of six keyboards selected at random, exactly zero keyboards will be defective? A) 0.001 B) 0.167 C) 0.735 D) 0.500 Answer: C 41. The probabilities and the number of automobiles lined up at a Lakeside Olds at opening time (7:30 a.m.) for service are: On a typical day, how many automobiles should Lakeside Olds expect to be lined up at opening? A) 10.00 B) 1.00 C) 2.85 D) 1.96 Answer: C 42. On a very hot summer day, 5 percent of the production employees at Midland States Steel are absent from work. The production employees are randomly selected for a special in-depth study on absenteeism. What is the probability of randomly selecting 10 production employees on a hot summer day and finding that none of them are absent? A) 0.002 B) 0.344 C) 0.599 D) 0.100 Answer: C 43. The marketing department of a nationally known cereal maker plans to conduct a national survey to find out whether or not consumers of flake cereals can distinguish one of their favorite flake cereals. To test the questionnaire and procedure to be used, eight persons were asked to cooperate in an experiment. Five very small bowls of flake cereals were placed in front of a person. The bowls were labeled A, B, C, D and E. The person was informed that only one bowl contained his or her favorite flake cereal. Suppose that the eight persons in the experiment were unable to identify their favorite cereal and just guessed which bowl it was in. What is the probability that none of the eight guessed correctly? A) 0.168 B) 0.009 C) 0.788 D) 0.125 Answer: A 44. An insurance agent has appointments with four prospective clients tomorrow. From past experience the agent knows that the probability of making a sale on any appointment is 1 out of 5. Using the rules of probability, what is the likelihood that the agent will sell a policy to 3 of the 4 prospective clients? A) 0.250 B) 0.500 C) 0.410 D) 0.026 Answer: D 45. Sweetwater & Associates write weekend trip insurance at a very nominal charge. Records show that the probability that a motorist will have an accident during the weekend and file a claim is 0.0005. Suppose they wrote 400 policies for the coming weekend, what is the probability that exactly two claims will be filed? A) 0.8187 B) 0.2500 C) 0.0164 D) 0.0001 Answer: C 46. In which of the following discrete distribution does the probability of a success vary from one trial to the next? A) Binomial B) Poisson C) Hypergeometric D) All of the above Answer: C 47. Which of the following is a requirement for use of the hypergeometric distribution? A) Only 2 possible outcomes B) Trials are independent C) Probability of a success is greater than 1.0 D) All of the above Answer: A 48. What is a listing of all possible outcomes of an experiment and their corresponding probability of occurrence called? A) Random variable B) Probability distribution C) Subjective probability D) Frequency distribution Answer: B 49. Which one of the following is NOT a condition of the binomial distribution? A) Independent trials B) Only two outcomes C) Probability of success remains constant from trial to trial D) At least 10 observations Answer: D 50. Which is true for a binomial distribution? A) There are three or more possible outcomes B) Probability of success remains the same from trial to trial C) Value of p is equal to 1.50 D) All of the above are correct Answer: B 51. Which shape describes a Poisson distribution? A) Positively skewed B) Negatively skewed C) Symmetrical D) All of the above Answer: A 52. Sponsors of a local charity decided to attract wealthy patrons to its $500-a-plate dinner by allowing each patron to buy a set of 20 tickets for the gaming tables. The chance of winning a prize for each of the 20 plays is 50-50. If you bought 20 tickets, what is the chance of winning 15 or more prizes? A) 0.250 B) 0.021 C) 0.006 D) 0.750 Answer: B 53. What kind of distribution are the binomial and Poisson distributions? A) Discrete B) Continuous C) Both discrete and continuous D) Neither discrete or continuous Answer: A AACSB: CA 54. Which of the following is correct about a probability distribution? A) Sum of all possible outcomes must equal 1 B) Outcomes must be mutually exclusive C) Probability of each outcome must be between 0 and 1 inclusive D) All of the above Answer: D 55. The weight of an offensive linesman may be 205.15 pounds, 210.23 pounds, 225.05 pounds or 219.14 pounds. What is this an illustration of? A) Continuous random variable B) Discrete random variable C) Complement rule D) All of the above Answer: A 56. Carlson Jewelers permits the return of their diamond wedding rings, provided the return occurs within two weeks of the purchase date. Their records reveal that 10 percent of the diamond wedding rings are returned. Five different customers buy five rings. What is the probability that none will be returned? A) 0.250 B) 0.073 C) 0.590 D) 0.500 E) 0.372 Answer: C 57. In a large metropolitan area, past records revealed that 30 percent of all the high school graduates go to college. From 20 graduates selected at random, what is the probability that exactly 8 will go to college? A) 0.114 B) 0.887 C) 0.400 D) 0.231 Answer: A 58. Chances are 50-50 that a newborn baby will be a girl. For families with five children, what is the probability that all the children are girls? A) 0.900 B) 0.031 C) 0.001 D) 0.250 Answer: B 59. A new car was put into production. It involved many assembly tasks. Each car was inspected at the end of the assembly line and the number of defects per unit was recorded. For the first 100 cars produced, there were 40 defective cars. Some of the cars had no defects; a few had one defect, and so on. The distribution of defects followed a Poisson distribution. Based on the first 100 produced, about how many out of every 1,000 cars assembled should have one or more defects? A) About 660 B) About 165 C) About 630 D) About 330 Answer: D 60. The production department has installed a new spray machine to paint automobile doors. As is common with most spray guns, unsightly blemishes often appear because of improper mixture or other problems. A worker counted the number of blemishes on each door. Most doors had no blemishes; a few had one; a very few had two, and so on. The average number was 0.5 per door. The distribution of blemishes followed the Poisson distribution. Out of 10,000 doors painted, about how many would have no blemishes? A) About 6,065 B) About 3,935 C) About 5,000 D) About 500 Answer: A 61. A manufacturer of headache medicine claims it is 70 percent effective within a few minutes. That is, out of every 100 users 70 get relief within a few minutes. A group of 12 patients are given the medicine. If the claim is true, what is the probability that 8 have relief within a few minutes? A) 0.001 B) 0.168 C) 0.667 D) 0.231 Answer: D 62. A true-false test consists of six questions. If you guess the answer to each question, what is the probability of getting all six questions correct? A) 0 B) 0.016 C) 0.062 D) 0.250 Answer: B 63. A farmer who grows genetically engineered corn is experiencing trouble with corn borers. A random check of 5,000 ears revealed the following: many of the ears contained no borers. Some ears had one borer; a few had two borers; and so on. The distribution of the number of borers per ear approximated the Poisson distribution. The farmer counted 3,500 borers in the 5,000 ears. What is the probability that an ear of corn selected at random will contain no borers? A) 0.3476 B) 0.4966 C) 1.000 D) 0.0631 Answer: B 64. A tennis match requires that a player win three of five sets to win the match. If a player wins the first two sets, what is the probability that the player wins the match, assuming that each player is equally likely to win each match? A) 0.5 B) 1/8 or 0.125 C) 7/8 or 0.875 D) Cannot be computed. Answer: C 65. A machine shop has 100 drill presses and other machines in constant use. The probability that a machine will become inoperative during a given day is 0.002. During some days no machines are inoperative, but during some days, one, two, three or more are broken down. What is the probability that fewer than two machines will be inoperative during a particular day? A) 0.0200 B) 0.1637 C) 0.8187 D) 0.9824 Answer: D 66. What is the following table called? A) Probability distribution B) Cumulative frequency distribution C) Standard deviation D) Frequency table Answer: A 67. What is the only variable in the Poisson probability formula? A) B) x C) e D) P Answer: B 68. Which of the following is NOT a characteristic of a binomial probability distribution? A) Each outcome is mutually exclusive B) Each trial is independent C) Probability of success remains constant from trial to trial D) Each outcome results from two trials E) All of the above are characteristics of the binomial distribution Answer: D 69. What must you know to develop a binomial probability distribution? A) Probability of success B) Number of trials C) Number of successes D) "a" and "b" only E) "a" and "c" only Answer: D 70. In a Poisson distribution the mean is equal to A) n B) C) e - x. D) E) zero. Answer: A 71. In a Poisson distribution the variance is equal to A) n. B) C) e - x. D) E) zero. Answer: A Fill-in-the-Blank 72. How many possible experimental outcomes do the binomial distribution and the Poisson distribution have? _______ Answer: two 73. What type of population consists of a fixed number of individuals, objects, or measurements? ___________ Answer: finite 74. A probability distribution is a listing of the expected outcomes of an experiment and the probability of each outcome occurring. What is the sum of the probabilities? _____ Answer: one 75. To construct a binomial distribution we need to know the total number of ___________ and the probability of a success. Answer: trials 76. A probability distribution shows the outcomes of an experiment and the ___________________ of each one occurring. Answer: probability 77. In a binomial experiment, the probability of a _________ remains constant. Answer: success or failure 78. In a binomial experiment, what does the probability of a failure equal? _____________________________ Answer: 1 - probability of success 79. A binomial probability distribution approaches a greater degree of symmetry as the probability of success remains constant and the number of trials becomes ____________ Answer: larger or greater 80. The Poisson distribution or, the law of improbable events, is _______________ skewed. Answer: positively 81. If = 1/3 and n = 900, what is the mean of this binomial distribution? ______________ Answer: 300 82. If = 1/5 and n = 100, what is the standard deviation of this binomial distribution? _______ Answer: four 83. If n = 900 and = 1/3, what is the variance of this binomial distribution? ________ Answer: 200 84. A _______________ random variable can assume only a certain number of separated values. Answer: discrete 85. A continuous random variable can assume one of an ____________ number of values within a specific range. Answer: infinite 86. In the _______________________ distribution the probability of a success is not the same on each trail. Answer: hypergeometric 87. For the hypergeometric distribution there are ________________ possible outcomes. Answer: only 2 88. A random variable with a Poisson distribution has one of _______ mutually exclusive values. Answer: two 89. In a Poisson distribution each trial is ____________________. Answer: independent 90. The arrival of customers at a service desk follows a Poisson distribution. If they arrive at a rate of two every five minutes, what is the probability that no customers arrive in a five-minute period? ______ Answer: 0.1353 91. The arrival of customers at a service desk follows a Poisson distribution. If they arrive at a rate of four every five minutes, what is the probability that more than four customers arrive in a five minute period?_________ Answer: 0.3711 Use the following to answer questions 92-95: Elly's hot dog emporium is famous for its chilidogs. Elly's latest sales indicate that 30% of the customers ordering her chilidogs order it with hot peppers. Suppose 18 customers are selected at random. 92. What is the probability that exactly ten customers will ask for hot peppers? _______ Answer: 0.015 93. What is the probability that between two and six people inclusive want hot peppers? ________ Answer: 0.708 94. What is the probability that fifteen or more customers will want hot peppers? ____________ Answer: 0.0000 95. This situation is an example of what type of discrete probability distribution? _______________________ Answer: binomial probability distribution Multiple Choice Questions Use the following to answer questions 96-99: David's gasoline station offers 4 cents off per gallon if the customer pays in cash and does not use a credit card. Past evidence indicates that 40% of all customers pay in cash. During a one-hour period twenty-five customers buy gasoline at this station. 96. What is the probability that at least ten pay in cash? A) 0.416 B) 0.575 C) 0.586 D) 0.425 Answer: B 97. What is the probability that no more than twenty pay in cash? A) 0.0 B) 0.1 C) 0.9 D) 1.0 Answer: D 98. What is the probability that more than ten and less than fifteen customers pay in cash? A) 0.541 B) 0.401 C) 0.380 D) 0.562 Answer: C 99. This situation is an example of what type of discrete probability distribution? A) Continuous probability distribution B) Poisson probability distribution C) Binomial probability distribution D) Hypergeometric probability distribution Answer: C Use the following to answer questions 100-103: Affirmative action commitments by industrial organizations have led to an increase in the number of women in executive positions. Satellite Office Systems has vacancies for two executives that it will fill from among four women and six men. 100. What is the probability that no woman is selected? A) 1/5 B) 1/3 C) 2/15 D) 8/15 Answer: B 101. What is the probability that at least one woman is selected? A) 8/15 B) 3/5 C) 2/3 D) 3/4 Answer: C Fill-in-the-Blank 102. What is the probability that exactly one woman is selected? ________ Answer: 8/15 or 0.5333 103. This is an example of what type of probability distribution? ___________ Answer: hypergeometric Multiple Choice Use the following to answer questions 104-106: A statistics professor receives an average of five e-mail messages per day from students. Assume the number of messages approximates a Poisson distribution. 104. What is the probability that on a randomly selected day she will have no messages? A) 0.0067 B) zero C) 0.0335 D) Impossible to have no messages Answer: A 105. What is the probability that on a randomly selected day she will have five messages? A) 0.0067 B) 0.875 C) 0.175 D) 1.0 Answer: C 106. What is the probability that on a randomly selected day she will have two messages? A) 0.0067 B) 0.0014 C) 0.420 D) 0.084 Answer: D Use the following to answer questions 107-109: A company is studying the number of monthly absences among its 125 employees. The following probability distribution shows the likelihood that people were absent 0, 1, 2, 3, 4, or 5 days last month. 107. What is the mean number of days absent? A) 1.00 B) 0.40 C) 0.72 D) 2.5 Answer: C 108. What is the variance of the number of days absent? A) 1.99 B) 1.41 C) 5.00 D) 55.52 Answer: A 109. Given the probability distribution, which of the following predictions is correct? A) 60% of the employees will have more than one day absent for a month B) There is a 0.04 probability that an employee will be absent 1 day during a month C) There is a 0.12 probability that an employee will be absent 2 days during a month D) There is a 0.50 probability that an employee will be absent 0.72 days during a month. Answer: C Fill-in-the-Blank Reference: 06_05 Use the following to answer questions 110-113: A company is studying the number of daily debit card purchases. If there were 20 purchases and the probability of a debit card purchase is 0.5, answer the following questions. 110. Of the 20 purchases, what is the expected value of the number of debit card purchases? _________________ Answer: 10 purchases 111. What is the standard deviation of the number of debit card purchases? ____ Answer: 2.24 112. What is the shape of this distribution? Explain your answer. Answer: The binomial distribution is perfectly symmetric because the probability of success is 50%. ___________ 113. Based on the shape of the distribution, approximately 99.7 % of the purchases should be between _______ and _________. Answer: 10 Plus or minus 3(2.24) or 3 and 17. Essay 114. What is the difference between a binomial and a hypergeometric distribution? Answer: For a binomial distribution, the trials are independent and the probability of a success is the same from trial to trial. For a hypergeometric distribution, the trials are not independent and the probability of a success is not the same from trial to trial. Goal: 4, 5 115. For a binomial distribution, explain why . Answer: is the mean or expected value for any discrete probability distribution. If we know that a discrete probability distribution is a binomial distribution with n independent trials and a success probability of ?, then the expected value can be computed as . Goal:2, 4

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