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statistics for engineers and scientists
Probability And Statistics For Engineers 5th Edition Richard L Scheaffer, Madhuri Mulekar, James T McClave, Cecie Starr - Solutions
5.55 The National Maximum Speed Limit (NMSL) of 55 miles per hour has been in force in the United States since early 1974. The benefits of this law have been studied by D.B. Kamerud(Transportation Research, 17A, no. 1, 1983, pp.51–64), who reports that the fatality rate for interstate highways
5.54 The quality of computer disks is measured by sending the disks through a certifier that counts the number of missing pulses. A certain brand of computer disk has averaged 0.1 missing pulse per disk.a Find the probability that the next inspected disk will have no missing pulse.b Find the
5.53 The number of telephone calls coming into the central switchboard of an office building averages four per minute.a Find the probability that no calls will arrive in a given 1-minute period.b Find the probability that at least two calls will arrive in a given 1-minute period.c Find the
5.52 Let Y denote a random variable having a Poisson distribution with mean 2. Find the following.a P(Y 4) b P (Y 4)c P(Y 4) d P(Y 4|Y 2)
5.51 An appliance comes in two colors, white and brown, which are in equal demand. A certain dealer in these appliances has three of each color in stock, although this is not known to the customers. Customers arrive and independently order these appliances. Find the probability that a The third
5.50 The telephone lines coming into an airline reservation office are all occupied about 60% of the time.a If you are calling this office, what is the probability that you complete your call on the first try? The second try? The third try?b If you and a friend both must complete separate calls to
5.49 A large lot of tires contains 10% defectives. Four are to be chosen for placement on a car.a Find the probability that six tires must be selected from the lot to get four good ones.b Find the expected value and variance of the number of selections that must be made to get four good tires.
5.48 In the setting of Exercise 5.47, suppose a company wants to set up three producing wells. Find the expected value and variance of the number of wells that must be drilled to find three successful ones.
5.47 A geological study indicates that an exploratory oil well drilled in a certain region should strike oil with probability 0.2. Find the probability that a The first strike of oil comes on the third well drilled.b The third strike of oil comes on the fifth well drilled.What assumptions are
5.46 If one-third of the persons donating blood at a clinic have O blood, find the probability that a The first O donor is the fourth donor of the day.b The second O donor is the fourth donor of the day.
5.45 Refer to Exercise 5.44. If each test costs $20, find the expected value and variance of the total cost of conducting the tests to locate three positives.Do you think it is highly likely that the cost of completing these tests would exceed $350?
5.44 The employees of a firm that manufactures insulation are being tested for indications of asbestos in their lungs. The firm is requested to send three employees who have positive indications of asbestos to a medical center for further testing. If 40% of the employees have positive indications
5.43 Refer to Exercise 5.40. Find the mean and variance of the number of the trial on which a The first non-defective engine is found.b The third non-defective engine is found.
5.42 Refer to Exercise 5.40. Given that the first two engines are defective, find the probability that at least two more engines must be tested before the first nondefective is found.
5.41 Refer to Exercise 5.40. Find the probability that the third nondefective engine is found a On the fifth trial.b On or before the fifth trial.
5.40 Suppose 10% of the engines manufactured on a certain assembly line are defective. If engines are randomly selected one at a time and tested, find the probability that the first nondefective engine is found on the second trial.
5.39 Let Y denote a negative binomial random variable with p 0.4. Find P(Y 4) if a r 2 b r 4
5.38 Let Y denote a random variable having a geometric distribution, with probability of success on any trial denoted by p.a Find P(Y 2) if p 0.1.b Find P(Y 4 |Y 2) for general p. Compare the result with the unconditional probability P(Y 2).
5.37 Ten motors are packaged for sale in a certain warehouse. The motors sell for $100 each, but a “double-your-money-back” guarantee is in effect for any defectives the purchaser might receive. Find the expected net gain for the seller if the probability of any one motor being defective is
5.36 From a large lot of new tires, n are to be sampled by a potential buyer, and the number of defectives X is to be observed. If at least one defective is observed in the sample of n tires, the entire lot is to be rejected by the potential buyer. Find n so that the probability of detecting at
5.35 A firm sells four items randomly selected from a large lot known to contain 10% defectives. Let Y denote the number of defectives among the four sold. The purchaser of the item will return the defective for repair, and the repair cost is given by Find the expected repair cost.C = 3Y2 + Y + 2
5.34 An oil exploration firm is to drill 10 wells, with each well having probability 0.1 of successfully producing oil. It costs the firm $10,000 to drill each well.A successful well will bring in oil worth $500,000.a Find the firm’s expected gain from the 10 wells.b Find the standard deviation
5.33 A complex electronic system is built with a certain number of backup components in its subsystems.One subsystem has four identical components, each with probability 0.2 of failing in less than 1,000 hours. The subsystem will operate if any two or more of the four components are operating.
5.32 Refer to Exercise 5.31. How large must n be if it is desired to have probability 0.99 of detecting an aircraft entering the zone?
5.31 A missile protection system consists of n radar sets operating independently, each with probability 0.9 of detecting an aircraft entering a specified zone. (All radar sets cover the same zone.)If an airplane enters the zone, find the probability that it will be detected if a n 2 b n 4
5.30 A. Goranson and J. Hall (Aeronautical Journal, November 1980, pp. 279–280) explain that the probability of detecting a crack in an airplane wing is the product of p1, the probability of inspecting a plane with a wing crack; p2, the probability of inspecting the detail in which the crack is
5.29 According to an article by B.E. Sullivan in Transportation Research (18A, no. 2, 1984, p. 119), 55% of U.S. corporations say that one of the most important factors in locating a corporate headquarters is the “quality of life” for the employees. If five firms are contacted by the governor
5.28 The U.S. Statistical Abstract reports that the median family income in the United States for 1989 was $34,200. Among four randomly selected families, find the probability that a All four had incomes above $34,200 in 1989.b One of the four had an income below $34,200 in 1989.
5.27 Refer to Exercise 5.26. The clinic needs five Rhdonors on a certain day. How many people must donate blood to have the probability of at least five RH donors over 0.90?
5.26 Among persons donating blood to a clinic, 80%are Rh (that is, have the Rhesus factor present in their blood). Five people donate blood at the clinic on a particular day.a Find the probability that at least one of the five does not have the Rh factor.b Find the probability that at most four of
5.25 Refer to Exercise 5.24 a Find the number expected to survive out of 20 b Find the variance of the number of survivors out of 20.
5.24 In testing the lethal concentration of a chemical found in polluted water, it is found that a certain concentration will kill 20% of the fish that are subjected to it for 24 hours. If 20 fish are placed in a tank containing this concentration of chemical, find the probability that after 24
5.23 A machine that fills boxes of cereal underfills a certain proportion p. If 25 boxes are randomly selected from the output of this machine, find the probability that no more than two are underfilled when a p 0.1 b p 0.2
5.22 Let X denote a random variable having a binomial distribution with p 0.4 and n 20. Use Table 2 of the Appendix to evaluate the following.a P(X 6) b P(X 12)c P(X 8)
5.21 Let X denote a random variable having a binomial distribution with p 0.2 and n 4. Find the following.a P(X 2) b P(X 2)c P(X 2) d E(X )e V(X )
5.20 Costs of equipment maintenance are an important part of a firm’s budget. Each visit by a field representative to check out a malfunction in a word processing system costs $40. The word processing system is expected to malfunction approximately five times per month, and the standard deviation
5.19 An important feature of golf cart batteries is the number of minutes they will perform before needing to be recharged. A certain manufacturer advertises batteries that will run, under a 75-amp discharge test, for an average of 100 minutes, with a standard deviation of 5 minutes.a Find an
5.18 Keeping an adequate supply of spare parts on hand is an important function of the parts department of a large electronics firm. The monthly demand for microcomputer printer boards was studied for some months and found to average 28, with a standard deviation of 4. How many printer boards
5.17 The number of breakdowns for a university computer system is closely monitored by the director of the computing center, because it is critical to the efficient operation of the center. The number averages 4 per week, with a standard deviation of 0.8 per week.a Find an interval that must
5.16 Approximately 10% of the glass bottles coming off a production line have serious defects in the glass. If two bottles are randomly selected for inspection, find the expected value and variance of the number of inspected bottles with serious defects.
5.15 Daily sales records for a computer manufacturing firm show that it will sell 0, 1, or 2 mainframe computer systems with probabilities as listed:Find the expected value, variance, and standard deviation for daily sales. Number of sales 0 1 2 Probability 0.7 0.2 0.1
5.14 How old are our drivers? Table below gives the age distribution of licensed drivers in the United States.a Describe this age distribution in terms of median, mean, and standard deviation.b Does the empirical rule work well for this distribution? Age Number (in millions) 19 and under 20-24 9.4
5.13 The graph in Figure 5.8 shows the age distribution for AIDS deaths in the United States through 1995. Approximate the mean and standard deviation of this age distribution. How does the mean age compare to the approximate median age?
5.12 The size distribution of U.S. families as estimated by the U.S. Bureau of Census is given in table below.a Calculate the mean and standard deviation of family size. Are these exact values or approximations?b How does the mean family size compare to the median family size?c Does the empirical
5.11 You are to pay $1 to play a game consisting of drawing one ticket at random from a box of numbered tickets. You win the amount (in dollars) of the number on the ticket you draw.Two boxes are available with numbered tickets as shown below:I 0, 1, 2 II 0, 0, 0, 1, 4
5.10 When turned on, each of the three switches in the diagram below works properly with probability 0.9.If a switch is working properly, current can flow through it when it is turned on. Find the probability distribution for Y, the number of closed paths from a tob, when all three switches are
5.9 Four microchips are to be placed in a computer.Two of the four chips are randomly selected for inspection before assembly of the computer. Let X denote the number of defective chips found among the two chips inspected. Find the probability distribution for X if a Two of the microchips were
5.8 Daily sales records for a computer manufacturing firm show that it will sell 0, 1, or 2 mainframe computer systems with probabilities as listed:a Find the probability distribution for X, the number of sales in a two-day period, assuming that sales are independent from day to day.b Find the
5.7 Of the people entering a blood bank to donate blood, 1 in 3 have type O blood, and 1 in 15 have type O blood. For the next three people entering the blood bank, let X denote the number with Oblood and Y the number with O blood. Assuming independence among the people with respect to blood
5.6 It was observed that 40% of the vehicles crossing a certain toll bridge are commercial trucks. Four vehicles will cross the bridge in the next minute.Find the probability distribution for X, the number of commercial trucks among the four, if the vehicle types are independent of one another.
5.5 Table 4.22 gives information on the type of injury sustained in accidents involving different types of flights. Suppose four independent flight accidents were reported in one day. Let X denote the number of accidents out of 4 that involved fatal injuries.a Find the probability distribution for
5.4 A commercial building has two entrances, numbered I and II. Three people enter the building at 9:00 A.M. Let X denote the number that select entrance I. Assuming the people choose entrances independently, find the probability distribution for X.Were any additional assumptions necessary in
5.3 Wade Boggs of the Boston Red Sox hit 0.363 in 1987. (He got a hit on 36.3% of his official times at bat.) In a typical games, he was up to bat three official times. Find the probability distribution for X, the number of hits in a typical game. What assumptions are involved in the answer? Are
5.2 The median annual income for household heads in a certain city is $28,000. Four such household heads are randomly selected for an opinion poll.a Find the probability distribution of X, the number(out of the four) that have annual incomes below $28,000.b Would you say that it is unusual to see
5.1 Among 10 applicants for an open position, 6 are females and 4 are males. Suppose three applicants are randomly selected from the applicant pool for final interviews. Find the probability distribution for X, the number of female applicants among the final three.
4.87 Does knowledge lead to the decision to be an engineer? To investigate this question, Hamill and Hodgkinson (Proceedings of ICE, 2003) surveyed 566, 11–16-year-olds, and classified them by their level of knowledge about engineering and the likelihood of becoming an engineer. The results are
4.86 There are 23 students in a classroom. What is the probability that at least two of them have the same birthday (day and month)? Assume that the year has 365 days. State your assumptions.
4.85 Using the definition of conditional probability, show that
4.84 Using Venn diagrams or similar arguments, show that for events A, B, and C
4.83 Show that the following are true for any events A and B.a b The probability that exactly one of the events occurs is .
4.82 A blood test for hepatitis has the following accuracy:1 2a 34 bA 12 a3 B4 bTest Result Patient with Hepatitis 0.90 0.10 Patient without Hepatitis 0.01 0.99 The disease rate in the general population is 1 in 10,000.a What is the probability that a person actually has hepatitis if he or she
4.81 Relays in a section of an electrical circuit operate independently, and each one closes properly with probability 0.9 when a switch is thrown. The following two designs, each involving four relays, are presented for a section of a new circuit. Which design has the higher probability of current
4.80 Suppose that n indistinguishable balls are to be arranged in N distinguishable boxes so that each distinguishable arrangement is equally likely. If, show that the probability that no box will be empty is given by
4.79 Eight tires of different brands are ranked from 1 to 8 (best to worst) according to mileage performance.If four of these tires are chosen at random by a customer, find the probabililty that the best tire among those selected by the customer is actually ranked third among the original eight.
4.78 A line from a to b has midpointc. A point is chosen at random on the line and marked x (the point x being chosen at random implies that x is equally likely to fall in any subinterval of fixed length l ). Find the probability that the line segments ax, bx, and ac can be joined to form a
4.77 Three events A, B, and C are said to be independent if and Suppose a balanced coin is independently tossed two times. Define the following events:A: Heads appears on the first toss.B: Heads appears on the second toss.C: Both tosses yield the same outcome.Are A, B, and C independent?
4.76 If A and B are independent events, show that A and B are also independent.
4.75 Show that for three events A, B, and C
4.74 An assembler of electric fans uses motors from two sources. Company A supplies 90% of the motors, and company B supplies the other 10%.Suppose it is known that 5% of the motors supplied by company A are defective and 3% of the motors supplied by company B are defective. An assembled fan is
4.73 An accident victim will die unless she receives in the next 10 minutes an amount of type A Rhblood, which can be supplied by a single donor. It requires 2 minutes to “type” a prospective donor’s blood and 2 minutes to complete the transfer of blood. A large number of untyped donors are
4.72 Consider two mutually exclusive events, A and B, such that P(A) 0 and P(B) 0. Are A and B independent? Give a proof with your answer.
4.71 Suppose the streets of a city are laid out in a grid, with streets running north–south and eastwest.Consider the following scheme for patrolling an area of 16 blocks by 16 blocks: A patrolman commences walking at the intersection in the center of the area. At the corner of each block he
4.69 Suppose the probability of exposure to the flu during an epidemic is 0.6. Experience has shown that a serum is 80% successful in preventing an inoculated person from acquiring the flu if exposed. A person not inoculated faces a probability of 0.90 of acquiring the flu exposed. Two persons, one
4.68 Refer to Exercise 4.67. What is the probability that the pair of coins is tossed four times before a match occurs (that is, they match for the first time on the fourth toss)?
4.67 Two men each tossing a balanced coin obtain a“match” if both coins are heads or if both are tails. The process is repeated three times.a What is the probability of three matches?b What is the probability that all six tosses(three for each man) result in “tails”?c Coin tossing provides
4.66 The quarterback on a certain football team completes 60% of his passes. If he tries three passes, assumed to be independent, in a given quarter, what is the probability that he will complete a All three?b At least one?c At least two?
4.65 a Two cards are drawn from a 52-card deck.What is the probability that the draw will yield an ace and a face card in either order?b Five cards are drawn from a 52-card deck.What is the probability that yield will be spades? that the draw will yield be of the same suit?
4.64 An inspector must perform eight tests on a randomly selected keyboard coming off an assembly line. The sequence in which the tests are conducted is important because the time lost between tests will vary. If an efficiency expert were to study all possible sequences to find the one that
4.63 An experimenter wants to investigate the effect of three variables—pressure, temperature, and type of catalyst—on the yield in a refining process. If the experimenter intends to use three settings each for temperature and pressure and two types of catalysts, how many experimental runs will
4.62 A personnel director for a corporation has hired ten new engineers. If three (distinctly different)positions are open at a particular plant, in how many ways can he fill the positions?
4.53 A proficiency examination for a certain skill was given to 100 employees of a firm. Forty of the employees were male. Sixty of the employees passed the examination, in that they scored above a preset level for satisfactory performance.The breakdown among males and females was as
4.52 Is the race of a defendant associated with the defendant’s chance of receiving the death penalty? This is a controversial issue that has been studied by many. One important data set was collected on 326 cases in which the defendant was convicted of homicide. The death penalty was given in 36
4.51 Cholesterol level seems to be an important factor in myocardial infarctions (M.I.). In Table 4.30 the data below give the number of M.I.’s over the number in the cholesterol group for each arm of the study. In this study, patients were randomly assigned to the aspirin or placebo group.a Did
4.50 The segmented bar chart, as shown in Figure 4.19, is another way to display relative frequency data.a Explain the percentages in the bar labeled Los Angeles.b It appears that most whites live in Philadelphia. Is that a correct interpretation of these data?c Describe in words the pattern of
4.49 Study the data on employed scientists and engineers provided in Table 4.26. Think of the percentages as probabilities to aid in describing the anticipated employment pattern for scientists and engineers for this year.a Explain the conditional nature of the columns of percentages.b Suppose 3
4.48 Table 4.25 gives estimated percentages of sports footwear purchased by those in various age groups across the United States.a Explain the meaning of the 4.7 under aerobic shoes. Explain the meaning of the 67.0 under walking shoes.b Suppose the percentages provided in the table are to be used
4.47 Two methods, A and B, are available for teaching a certain industrial skill. The failure rate is 20%for A and 10% for B. However, B is more expensive and hence is used only 30% of the time. (A is used the other 70%.) A worker is taught the skill by one of the methods but fails to learn it
4.39 The data in Table 4.24 show the distribution of arrival times at work by mode of travel for workers in the central business district of a large city. The figures are percentages. (The columns should add to 100, but some do not because of rounding.) A randomly selected worker is asked about his
4.38 The data in Table 4.23 give the number of accidental deaths overall and for three specific causes for the United States in 1984. You are told that a certain person recently died in an accident. Approximate the probability that a It was a motor vehicle accident.b It was a motor vehicle accident
4.37 A purchasing office is to assign a contract for computer paper and a contract for microcomputer disks to any one of three firms bidding for these contracts. (Any one firm could receive both contracts.) Find the probability that a Firm I receives a contract given that both contracts do not go
4.36 Vehicles coming into an intersection can turn left or right or go straight ahead. Two vehicles enter an intersection in succession. Find the probability that at least one of the two vehicles turns left given that at least one of the two vehicles turns.What assumptions have you made?
4.35 A manufacturer of computer keyboards has assembly plants at two different locations, one in Iowa and the other in South Carolina. The keyboard faults in general are classified into three different categories depending on the location: fault related to a letter key, a number key, or other
4.34 A specific kind of small airplane used for dusting crops has two identical engines functioning inde-pendently of each other, but can be flown with only one engine. The past records of this kind of engine indicate a 3% mid-air failure rate for each engine. One such plane crashed while dusting
4.33 Suppose that P(A) 0.6, P(B) 0.3, and 0.15.a Determine .b Are events A and B independent?
4.32 A small store installed a diesel-powered generator for emergency power outage because they function independently of electric power.The past history of the town indicates that on about 5% of days during summer the town loses electricity due to high winds. However, high winds do not affect
4.31 The world is changing due to globalization, rapid changes, and technology integration. As a result, the university programs in civil engineering are facing challenges of preparing curriculums to meet the changing needs of industries. Once students obtain an understanding of civil engineering
4.30 A company produces two types of CDs, CD-R and CD-RW. The product CD-R constitutes 35%of total production. About 5% of the company’s production is defective, of which 40% is CD-R.Suppose a CD is selected at random from today’s production.a Display these facts using a Venn diagram.b What is
4.29 A diagnostic test for a certain disease is said to be 90% accurate in that, if a person has the disease, the test will detect it with probability 0.9. Also, if a person does not have the disease, the test will report that he or she doesn’t have it with probability 0.9. Only 1% of the
4.28 Electric motors coming off two assembly lines are pooled for storage in a common stockroom, and the room contains an equal number of motors from each line. Motors are periodically sampled from that room and tested. It is known that 10% of the motors from line I are defective and 15% of the
4.27 Nine impact wrenches are to be divided evenly among three assembly lines.a In how many ways can this be done?b Two of the wrenches are used, and seven are new. What is the probability that a particular line (line A) gets both used wrenches?
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