4.4

Understanding the Principles 4.1 4.2 What is a random variable? How do discrete and continuous random variables differ? Applying the Concepts Basic 4.3 4.4 4.5 4.6 Type of Random Variable. Classify the following ran dom variables according to whether they are discrete or continuous: a. The number of words spelled correctly by a student on a spelling test b. The amount of water flowing through the Hoover Dam in a day 1:. The length of time an employee is late for work d. The number of bacteria in a particular cubic centimeter of drinking water e. The amount of carbon monoxide produced per gallon of unleaded gas f. Your weight Type of Random Variable. Identify the following random variables as discrete or continuous: a. The amount of flu vaccine in a syringe b. The heart rate (number of beats per minute) of an American male 1:. The time it takes examination d. The barometric pressure at a given location e. The number of registered voters who vote in a national election f. Your score on the either the SAT or ACT Type of Random Variable. Identify the following variables as discrete or continuous: a. The difference in reaction time to the same stimulus before and after training b. The number of violent crimes committed per month in your community 1:. The number of corru'nercial aircraft near-misses per month d. The number of winners each week in a state lottery e. The number of free throws made per game by a basket ball team f. The distance traveled by a school bus each day NHTSA crash tests. The National Highway Trafc Safety Administration (NHTSA) has developed a driver-side \"star\" scoring system for crashtesting new cars. Each crashtested car is given a rating ranging from one star (3) to ve stars [**\"\" )1 the more stars in the rating, the bet ter is the level of crash protection in a headon collision. Suppose that a car is selected and its driverside star rating :. .1-.___:___r 1- .. _. ___._r .L- __._r___ _r _.___: a student to complete an _ .L _ __x:__ 4.7 MW 4.8 4.11] Customers in line at a Subway shop. The number of cus tomers, x, waiting in line to order sandwiches at a Subway shop at noon is of interest to the store manager. What val ues can it assume? ls x a discrete or continuous random variable? Sound waves from a basketball. Refer to the American Journal oanyrr'cs (June 2010) experiment on sound waves produced from striking a basketball, Exercise 2.43 (p. 52). Recall that the frequencies of sound wave echoes resulting from striking a hanging basketball with a metal rod were re corded. Classify the random variable, frequency (measured in hertz) of an echo, as discrete or continuous. Mongolian desert ants. Refer to the Journal of Blogeogmpny (Dec. 2003) study of ants in Mongolia, presented in Exercise 2.68 (p. 63). Two of the several vari ables recorded at each of 11 study sites were annual rainfall (in millimeters) and number of ant species. Identify these variables as discrete or continuous. Motivation of drug dealers. Refer to the Applied Psychology in Criminal Justice (Sept. 2009} study of the personalin characteristics of drug dealers, Exercise 2.102 (p. Tl}. For each of 1C0 convicted drug dealers, the resea rchers measured several variables, including the number of prior felony arrests I. Is I a discrete or continuous random variable? Explain. Applying the Concepts lnterrnediate 4.11 4.12 4.13 4.14 4.15 4.16 Psychology. Give an example of a discrete random variable of interest to a psychologist. Sociology. Give an example of a discrete random variable of interest to a sociologist. Nursing. Give an example of a discrete random variable of interest to a hospital nurse. Art history. Give an example of a discrete random variable of interest to an art historian. Irrelevant speech effects. Refer to the Acoustical Science & Technology (Vol. 35, 2014) study of the degree to which the memorization prm 'E impaired by irrelevant backgroiurd speech (called irrelevant speech effects), Exeuc'se 2.34 (p. 49). Recall that subjects performed a memorization task under two conditions: (1) with irrelevant backgromd speech and (2) in silence. Leth represent the different: in the error rates for the two conditionscalled the relative difference in error rate (RDER). Explain why I 'E a continuous random variable. Shaft graves in ancient Greece. Refer to the American Journal of Archaeology (Jan. 2014) study of shaft graves in ancient Greece, Exercise 2.3? (p. 50). Let Jr represent the SECTION 4.6 I The Hypergeomotnc Random Variable [Optional] 223 college newspaper. If the two students are selected at ran dom, what is the probability that at least one plagiarized. on the essay?. 4.119 On-s'rte treatment of hazardous waste. The Resource Conservation and Recovery Act mandates the tracking and disposal of hazardous waste produced at US. facilities. Professional Geographer (Feb. 2000) reported the hazard- ouswaste generation and disposal characteristics of 209 facilities. Only 8 of these facilities treated hazardous waste onsite. a. In a random sample of 10 of the 209 facilities, what is the expected number in the sample that treat hazard ous waste onsite'EI Interpret this result. b. Find the probability that 4 of the 10 selected facilities treat hazardous waste onsite. 4.12I] Guilt in decision making. Refer to theJonrnal ofenaviorol Decision Making (Jan. 200'?) study of how guilty feelings impact decisions, Exercise 3.59 (p. 142). Recall that 5? students were assigned to a guilty state through a reading.)I writing task. Immediater after the task, the students were presented with a decision problem where the stated option has predominantly negative features (e.g., spending money on repairing a very old car). Of these 5? students, 45 chose the stated option. Suppose 10 of the 5? guiltystate stu dents are selected at random. Define I as the number in the sample of 10 who chose the stated option. a. Fll'lCl P[r = 5). b. Fll'lCl P(x = 8). 4.124 Cell phone handofl' behavior. Refer to the Journal of Engineering, Compmmg and Architecture (Vol. 3., 2009) study of cell phone handoff behavior1 Exercise 3.62 (p. 143). Recall that a \"handoft\" describes the process of a cell phone moving from one base channel (identied by a color code) to another. During a particular driving trip a cell phone changed channels (color codes) 85 times. Color code \"b\" was accessed 40 times on the trip. You randomly select T of the 85 handoffs. How likely is it that the cell phone accesses color code \"b\" only twice for these 2 handolzfs? 4.125 Establishing boundaries in academic engineering. How academic engineers establish boundaries (e.g., differen tiating between engineers and other scientists, defining the different disciplines within engineering, determining the quality of journal publications) was investigated in Engineering Studies (Aug. 2012). Participants were 10 ten ured or tenureearning engineering faculty members in a School of Engineering at a large researchoriented univer sity. The table gives a breakdown of the department affili ation of the engineers. Each participant was interviewed at length, and responses were used to establish boundaries. Suppose we randomly select 3 participants from the orig inal 10 to form a university committee charged with de veloping boundary guidelines. Let .1; represent the number of conunittee members who are from the Department of Engineering Physics. Identify the probability distribution for x, and give a formula for the distribution. c. What is the expected value (mean) of 1? Department Number of Participants 4.121 Contaminated gun cartridges. Refer to the investigation of Chemical Engineeng 1 contaminated gun cartridges at a weapons manufacturer, CiVil Engineering 2 presented in Exercise 4.33 (p. 194). [n a sample of 158 Engineering PhXSiCS _ 4 cartridges from a certain lot. 36 were found to be contam- Mecwlcal Eligmermg 2 Industrlal Engineenng 1 inated and. 122 were \"clean.\" If you randomly select 5 of these 1.58 cartridges, what is the probability that all 5 will be Applying the ConceptsAdvanced 4.126 Gender discrimination suit. The Jamal of Business dir. Economic Statistics (July 201]) presented a case in which a \"clean\"? Applying the Concepts Intermediate 4.122 Lot hispection sampling. Imagine that you are purchasing small lots of a manufactured product. If it is very costly to test a single item, it may be desirable to test a sample of items from the lot instead of testing every item in the lot. Suppose each lot contains 10 items. You decide to sample 4 items per lot and reject the lot if you observe 1 or more defectives. a. If the lot contains 1 defective item, what is the proba charge of gender discrimination was led against the U.S. Postal Service. At the time, there were 332 US. Postal Service employees (229 men and 23 women) who applied for promotion. 0f the 3'2 employees who were awarded pro motion, 5 were female. Make an inference about whether or not females at the [1.5. Postal Service were promoted fairly. 4.127 Awarding of home improvement grants. The Minneapolis Conunuru'ty Development Agency (MCDA) makes home 14 Learning the Mechanics p(x) .2 .2 1 4.18 Consider the following probability distribution: Since the values that x can assume are mutually exclusive X events, the event {x = 12 } is the union of three mutually p(x) .2 A exclusive events: a. List the values that x may assume. {x = 10} U {x = 11} U {x =12} b. What value of x is most probable? a. Find P(x = 12). c. What is the probability that x is greater than 0? b. Find P(x > 12). d. What is the probability that x = -2? c. Find P(x = 14). 4.19 A discrete random variable x can assume five possible d. Find P(x = 14). NW values: 20, 21, 22, 23, and 24. The MINITAB histogram at e. Find P(x = 11 or x > 12). the bottom of the page shows the likelihood of each value. 4.22 The random variable x has the discrete probability distribu- a. What is p(22)? tion shown here: b. What is the probability that x equals 20 or 24? C. What is P(x = 23)? 4.20 Explain why each of the following is or is not a valid prob- X .10 -1 SIN p(x) .15 40 .30 Nw ability distribution for a discrete random variable x: a. a. Find P(x = 0). WN p(x) b. Find P(x > -1). c. Find P(-1 = x = 1). d. Find P(x 1] fora = 5,p = .1 If I is abinon'Lial random variable,use Table I in Appendix B or technology.r to nd the following probabilities: :1. Pi): 5) = 1 - P(x = 5) = 1 - .951 = .049 Note from Figure 4.13 that this probability is the area in the interval u + 20, or 2.6 + 2(1.61) = (-.62, 5.82). Then the number of sightings should exceed 5-or, equivalently, should be more than 2 standard deviations from the mean-during only about 4.9% of all weeks. This percentage agrees remarkably well with that given by the empirical rule for mound-shaped distributions, which informs us to ex- pect about 5% of the measurements (values of the random variable x) to lie further than 2 standard deviations from the mean. Now Work Exercise 4.99 Exercises 4.85-4.106 Understanding the Principles 4.91 Assume that x is a random variable having a Poisson prob- 4.85 Give the four characteristics of a Poisson random variable. ability distribution with a mean of 1.5. Find the following probabilities: 4.86 Consider a Poisson random variable with probability a. P(x = 3) distribution b. P(x = 3) 10%e-10 c. P(x = 3) P(x ) = x! (x = 0, 1, 2, . .. ) d. P(x = 0) e. P(x > 0) What is the value of A? [. P(x > 6) 4.87 Consider the Poisson probability distribution shown here: 4.92 Suppose x is a random variable for which a Poisson probabil- ity distribution with A = 1 provides a good characterization. p(x ) = 34e-3 x! (x = 0, 1, 2, . .. ) a. Graph p(x) for x = 0, 1, 2, . .. .9. b. Find p and o for x, and locate & and the interval u + 20 What is the value of A? on the graph. c. What is the probability that x will fall within the interval Learning the Mechanics u + 20? 4.88 Refer to Exercise 4.86. 4.93 Suppose x is a random variable for which a Poisson probabil- a. Graph the probability distribution. ity distribution with A = 3 provides a good characterization. b. Find the mean and standard deviation of x. a. Graph p (x ) for x = 0, 1, 2, . .. .9 4.89 Refer to Exercise 4.87. b. Find p and o for x, and locate pe and the interval u + 20 a. Graph the probability distribution. on the graph. b. Find the mean and standard deviation of x. c. What is the probability that x will fall within the interval 4.90 Given that x is a random variable for which a Poisson u + 20? probability distribution provides a good approximation, 4.94 As mentioned in this section, when n is large, p is small, compute the following: and np = 7, the Poisson probability distribution provides a. P(x = 2) when A = 1 a good approximation to the binomial probability dis- b. P(x = 2) when A = 2 tribution. Since we provide exact binomial probabilities c. P(x = 2) when A = 3 (Table I in Appendix B) for relatively small values of n, d. What happens to the probability of the event you can investigate the adequacy of the approximation {x = 2} as A increases from 1 to 3? Is this intuitively for n = 25. Use Table I to find p(0), p(1), and p(2) for reasonable? n = 25 and p = .05. Calculate the corresponding Poisson218 CHAPTER 4 I Discrete Random Variables approximations, using it = p. = rip. [Note: These approxi mations are reasonably good font as small as 25, but to use such an approximation in a practical situation we would prefer to have n a 100.] Applying the Concepts Basic 4.95 4.96 4.9"! 4.93 Eye xation experiment. Cognitive scientists at the University of Massachusetts designed an experiment to measure 1, the number of times a reader's eye xated on a single word before moving past that word (Memory and Cognition, Sept. 1991'). For this experiment, Jr was found to have a mean of 1. Suppose one of the readers in the experi ment is randomly selected. and assume that x has a Poisson distribution. a. Find P(x = 0). In Find PEX :- l). c. Find P[x 5 2). Noise in laser imaging. Penumbra] imaging is a technique used by nuclear engineers for imaging objects (e.g., Xrays and lasers) that emit highenergy photons. In lEiCE Trmactiom on informtion J; System (Apr. 2&35), re searchers demonstrated that penumbrol images are always degraded by noise, where the ntu'nber Jr of noise events occurring in a unit of time follows a Poisson process with mean A. Suppose that A = 9 for a particular image. a. Find and interpret the mean of x. h. Find the standard deviation of Jr. c. The signaltonoise ratio (SNR) for a penumbrol image is dened as SNR = Info, where p. and 0' are the mean and standard deviation, respectively, of the noise pro cess. Find the SNR for 1. Spare tine replacement units. The US. Department of Defense Reliability Analysis Center publishes Selected Topita in Assurance Related Technologies (START) sheets to help improve the quality of manufactured compo nents and systems. One START sheet, titled \"Application of the Poisson Distribution" (Vol. 9. No. 1, 2032), focuses on a spare line replacement unit (LRU). The number of LRUs that fail in any 10,000hour period is assumed to fol low a Poisson distribution with a mean of 1.2. a. Find the probability that there are no LRU failures during the next 10,000 hours of operation. I). Find the probability that there are at least two LRU fail ures during the next 10,000 hours of operation. Rare planet transits. A \"planet transit\" is a rare celestial mm... :.-. mhsnl. .. \"In...\" "mm... On "m". :.. o1"...- nrhn n|nv n. a. What is the probability that no fatalities will occur during any given month? b. What is the probability that one fatality will occur during any given month? c. Find EU) and the standard deviation ofx. Applying the ConceptsIntermediate 4.100 4.101 4.102 Trafc fatalities and sporting events. The relationship be tween close sporting events and gameday traffic fatalities was investigated in the Journal of Consumer Research (Dec. 2011). The researchers found that closer football and basketball games are associated with more trafc fatalities. The methodology used by the researchers involved mod eling the trafc fatality count for a particular game as a Poisson random variable. For games played at the winner's location (home court or home field), the mean number of traffic fatalities was .5. Use this information to nd the probability that at least 3 gameday traffic fatalities will occur at the winning team's location. LAN videoconferencing. A network administrator is in stalling a videoconferencing module in a local area network (LAN) computer system. Of interest is the capacity of the LAN to handle users who attempt to call in for videoconfer encing during peak hours. Calls are blocked if the user nds that all LAN lines are \"busy.\" The capacity is directly related to the rate at which calls are blocked ("Trafc Engineering Model for LAN Video Conferencing\" InteL 2005). Let Jr equal the number of calls blocked during the peak hour (busy) vid eoconferencing call time. The network administrator believes that J: has a Poisson dktribution with mean A = 5. a. Find the probability that fewer than 3 calls are blocked during the peak hour. b. Find E(x) and interpret its value. c. Is it likely that no calls will be blocked during the peak hour? 11. If, in fact. Jr = 0 during a randomly selected peak hour, what would you infer about the value of :1? Explain. Making high-stakes insurance decisions. The Journal of Economic Psychology (Sept. 21138) published the results of a high-stakes experiment where subjects were asked how much they would pay for insuring a valuable painting The painting was threatened by fire and theft. hence the need for insurance. To make the risk realistic. the subjects were informed that if it rained on exactly 24 days in July, the painting was considered to be stolen; and, if it rained on \"mm-I" '1': Am... :1. .il "a...\" 3|... nah-uh". ......-. mmntanmul I-n and sampling without replacement. answers. 4.108 Give the characteristics of a hypergeometric distribution. a. The sample is drawn without replacement. b. The sample is drawn with replacement. 4.109 How do binomial and hypergeometric random variables differ? In what respects are they similar? Applying the Concepts- Basic 4.116 Do social robots walk or roll? Refer to the International Learning the Mechanics Conference on Social Robotics (Vol. 6414, 2010) study 4.110 Given that x is a hypergeometric random variable with of the trend in the design of social robots, Exercise 4.25 N = 8, n = 3, and r = 5, compute the following: (p. 193). The study found that of 106 social robots, 63 were a. P(x = 1) built with legs only, 20 with wheels only, 8 with both legs b. P(x = 0) and wheels, and 15 with neither legs nor wheels. Suppose c. P(x = 3) you randomly select 10 of the 106 social robots and count d. P(x = 4) the number, x, with neither legs nor wheels. 4.111 Given that x is a hypergeometric random variable, com- a. Demonstrate why the probability distribution for x should pute p(x) for each of the following cases: not be approximated by the binomial distribution. a. N = 5, n = 3, r = 3,x = 1 b. Show that the properties of the hypergeometric proba- b. N = 9, n = 5, 1 = 3,x = 3 bility distribution are satisfied for this experiment. c. N = 4, n = 2, r = 2, x = 2 c. Find a and of for the probability distribution for x. d. N = 4, n = 2, r = 2, x = 0 NW d. Calculate the probability that x = 2. 4.112 Given that x is a hypergeometric random variable with 4.117 Mail rooms contaminated with anthrax. In Chance (Spring N = 12, n = 8, and r = 6: 2002), research statisticians discussed the problem of a. Display the probability distribution for x in tabular form. sampling mail rooms for the presence of anthrax spores. b. Compute p and o for x. Let x equal the number of mail rooms contaminated with C. Graph p(x), and locate p and the interval u + 20 on anthrax spores in a random sample of 3 mail rooms se- the graph. lected from a population of 100 mail rooms. If 20 of the d. What is the probability that x will fall within the interval 100 mail rooms are contaminated with anthrax, research- u + 20? ers showed that the probability distribution for x is given by the formula 4.113 Use the results of Exercise 4.112 to find the following probabilities: a. P(x = 1) b. P(x = 4) P(x) = c. P(x = 4) 100 d. P(x = 5) e. P(x