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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
8.4 The dotplot of Figure 8.8 shows the median values of single-family housing (in thousands of dollars) in 152 U.S. metropolitan areas across the United States (http://www.realtor.org/research/research/metroprice). These data have a mean of 167.19, a median of 141.80, and a
8.3 Obtain a standard six-sided die.a If the die is balanced, what will the outcomes of an ideal sample of 30 tosses look like?b Find the mean and variance of the ideal sample outcomes portrayed in part (a).c Toss your die 30 times and record the upperface outcomes. How does the distribution of
8.2 What’s the average number of hours that students in your statistics class study in a week?Estimate this average by selecting a random sample of five students and asking each sampled student how many hours he or she studies in a typical week. Use a random-number table in selecting the sample.
8.1 Concentrations of uranium 238 were measured in 12 soil samples from a certain region, with the following results in pCi/g (picoCuries per gram):0.76, 1.90, 1.84, 2.42, 2.01, 1.77, 1.89, 1.56, 0.98, 2.10, 1.41, 1.32 a Construct a boxplot.b Calculate and s 2 for these data.c If another soil
7.62 A box contains four balls, numbered 1 through 4.One ball is selected at random from this box.Let and the Xi’s are zero otherwise. Show that any two of the random variables X1, X2, and X3 are independent, but the three together are not.
7.61 The negative binomial variable X was defined as the number of the trial on which the r th success occurs in a sequence of independent trials with constant probability p of success on each trial. Let Xi denote a geometric random variable, defined as the number of the trial on which the first
7.60 Let X be a continuous random variable with distribution function F(x) and density function f (x).We can then write, for x1 x2, As a function of x2 for fixed x1, the right-hand side of this expression is called the conditional distribution function of X given that X x1. On taking the
7.59 Refer to Exercise 7.45. If the supplier stocks an amount equal to , what is the expected amount sold during the week?
7.58 Refer to Exercise 7.41 and 7.44.a Find b Use the result about expected value expectation to find E(X2).c Find E(X2) directly from the marginal density of X2.
7.57 A retail grocery merchant figures that her daily gain from sales X is a normally distributed random variable with 50 and 2 10 (measurements in dollars). X could be negative if she is forced to dispose of perishable goods. Also, she figures daily overhead costs Y to have a gamma
7.56 The lifelength X of a fuse has a probability density Three such fuses operate independently. Find the joint density of their lifelengths, X1, X2, and X3.
7.55 A quality control plan calls for randomly selecting three items from the daily production (assumed to be large) of a certain machine and observing the number of defectives. However, the proportion p of defectives produced by the machine varies from day to day and is assumed to have a uniform
7.54 Refer to Exercise 7.48.a Find E(X1 2X2).b Find V(X1 2X2).
7.53 Refer to Exercise 7.47.a Find Cov(X1, X2).b Find E(3X1 2X2).c Find V(3X1 2X2).
7.52 Refer to Exercise 7.42.a Find Cov(X1, X2).bFind by finding the probability distribution of .c Find and by using Theorem 5.2.
7.51 Refer to Exercise 7.41. Find Cov (X1, X2).
7.50 Refer to Exercise 7.40.a Find E(X1).b Find V(X1).c Find Cov(X1, X2).
7.48 Let X1 and X2 have the joint density function given by a Find the value of K that makes the function a probability density function.b Find the marginal densities of X1 and X2.c Find the conditional density of X1 given .d Find the conditional density of X2 given .e Find .f Find .
7.47 Let X1 and X2 have the joint density function given by a Find the marginal density functions of X1 and X2.b Are X1 and X2 independent?c Find the conditional density of X1 given .
7.46 Let (X1, X2) denote the coordinates of a point selected at random inside a unit circle with center at the origin. That is, X1 and X2 have the joint density function given by a Find the marginal density function of X1.b Find P(X1 X2).
7.45 Let X1 denote the amount of a certain bulk item stocked by a supplier at the beginning of a week, and suppose that X1 has a uniform distribution over the interval 0 X1 1. Let X2 denote the amount of this item sold by the supplier during the week, and suppose that X2 has a uniform
7.44 For Exercise 7.41.a Find the conditional density of X1 given .b Find the conditional density of X2 given .c Show that X1 and X2 are dependent.d Find .
7.43 For Exercise 7.40, find the conditional density of X1 given , Are X1 and X2 independent?
7.42 From a legislative committee consisting of four Republicans, three Democrats, and two Independents, a subcommittee of three persons is to be randomly selected to discuss budget compromises. Let X1 denote the number of Republicans and X2 the number of Democrats on the subcommittee.a Find the
7.41 Let X1 and X2 have the joint density function given by a Find the marginal density functions of X1 and X2.b Find .c Find .
7.40 Let X1 and X2 have the joint probability density function given by a Find the value of K that makes this a probability density function.b Find the marginal densities of X1 and X2.c Find the joint distribution of X1 and X2.d Find .e Find .
7.39 A certain type of elevator has a maximum weight capacity X1, which is normally distributed with a mean and standard deviation of 5,000 and 300 pounds, respectively. For a certain building equipped with this type of elevator, the elevator loading, X2, is a normally distributed random variable
7.38 Resistors of a certain type have resistances that are normally distributed with a mean of 100 ohms and a standard deviation of 10 ohms. Two such resistors are connected in series, which causes the total resistance in the circuit to be the sum of the individual resistances. Find the probability
7.37 Let X1 and X2 denote independent normally distributed random variables, not necessarily having the same mean or variance. Show that, for any constants a andb, Y aX1 bX2 is normally distributed.
7.36 There are two entrances to a parking lot. Cars arrive at entrance I according to a Poisson distribution with an average of three per hour and at entrance II according to a Poisson distribution with an average of four per hour. Find the probability that exactly three cars arrive at the parking
7.35 Find the moment-generating function for the negative binomial random variable. Use it to derive the mean and variance of that distribution.
7.34 Vehicles arriving at an intersection can turn right or left or can continue straight ahead. In a study of traffic patterns at this intersection over a long period of time, engineers have noted that 40% of the vehicles turn left, 25% turn right, and the remainder continue straight ahead.a For
7.33 Refer to Exercise 7.32. If Y denotes the number of items containing at least one defect among the 10 sampled items, find the probability that a Y is exactly 2.b Y is at least 1.
7.32 In a large lot of manufactured items, 10% contain exactly one defect and 5% contain more than one defect. If 10 items are randomly selected from this lot for sale, the repair costs total (Y1 3Y2), where Y1 denotes the number among the 10 having one defect and Y2 denotes the number with two
7.31 Among a large number of applicants for a certain position, 60% have only a high school education, 30% have some college training, and 10% have completed a college degree. If five applicants are selected to be interviewed, find the probability that at least one will have completed a college
7.30 Customers leaving a subway station can exit through any one of three gates. Assuming that each customer is equally likely to select any one of the three gates, find the probability that, among four customers, a Two select gate A, one selects gate B, and one selects gate C.b All four select the
7.29 The U.S. Bureau of Labor Statistics reports that approximately 21% of the adult population under age 65 is between 18 and 24 years of age, 28% is between 25 and 34, 19% is between 35 and 44, and 32% is between 45 and 64. An automobile manufacturer wants to obtain opinions on a new design from
7.28 Given the recent emphasis on solar energy, solar radiation has been carefully monitored at various sites in Florida. For typical July days in Tampa, 30%have total radiation of at most 5 calories, 60% have total radiation of at most 6 calories, and 100% have total radiation of at most 8
7.27 Wing cracks, in the inspection of commercial aircraft, are reported as nonexistent, detectable, or critical. The history of a certain fleet shows that 70% of the planes inspected have no wing cracks, 25% have detectable wing cracks, and 5% have critical wing cracks. For the next five planes
7.26 The typical cost of damages for a fire in a family home is $20,000, the typical cost for an apartment fire is $10,000, and the typical cost for a fire in other dwellings is only $2,000. Using the information in Exercise 7.25, find the expected total damage cost for four independently reported
7.25 The National Fire Incident Reporting Service reports that, among residential fires, approximately 73% are in family homes, 20% are in apartments, and the other 7% are in other types of dwellings. If four fires are independently reported in one day, find the probability that two are in family
7.24 Refer to Exercise 7.21. Suppose a customer spends a length of time y1 at the store. Find the probability that this customer spends less than half of that time at the service window.
7.23 Refer to Exercise 7.21. The random variable Y1 Y2 represents the time spent at the service window.a Find E(Y1 Y2).b Find V(Y1 Y2).c Is it highly likely that a customer would spend more than 2 minutes at the service window?
7.22 Refer to Exercise 7.21. If a customer’s total waiting time plus service time is known to be more than 2 minutes, find the probability that the customer waited less than 1 minute to be served.
7.21 A particular fast-food outlet is interested in the joint behavior of the random variables Y1, the total time between a customer’s arrival at the store and his leaving the service window, and Y2, the time that the customer waits in line before reaching the service window. Since Y1 contains
7.20 For a sheet-metal stamping machine in a certain factory, the time between failures, X1, has a mean (MTBF) of 56 hours and a variance of 16.The repair time, X2, has a mean (MTTR) of 5 hours and a variance of 4.a If X1 and X2 are independent, find the expected value and variance of Y X1 X2,
7.19 The proportions X1 and X2 of two chemicals found in samples of an insecticide have the joint probability density function The random variable Y X1 X2 denotes the proportion of the insecticide due to both chemicals combined.a Find E(Y) and V(Y ).b Find an interval in which values of Y
7.18 In a study of particulate pollution in air samples over a smokestack, X1 represents the amount of pollutant per sample when a cleaning device is not operating, and X2 represents the amount when the cleaning device is operating. Assume that (X1, X2) has the joint probability density function=
7.17 Table 7.2 shows the joint distribution of fatalities and number of seat belts used by children under age 5. What is the “average” behaviour of X1 and X2, and how are they associated? Follow the steps below to answer this question.a Find E(X1), V(X1), E(X2), and V(X2).b Find Cov(X1, X2).
7.16 A bombing target is in the center of a circle with a radius of 1 mile. A bomb falls at a randomly selected point inside that circle. If the bomb destroys everything within mile of its landing point, what is the probability that it destroys the target?
7.15 Two telephone calls come into a switchboard at random times in a fixed 1-hour period. If the calls are made independently of each other, a find the probability that both calls are made in the first half hour.b find the probability that the two calls are made within 5 minutes of each other.
7.14 Each of two quality-control inspectors interrupts a production line at randomly, but independently, selected times within a given workday (of 8 hours).Find the probability that the two interruptions will be more than 4 hours apart.
7.13 Two friends are to meet at a library. Each arrives at independently and randomly selected times within a fixed 1-hour period. Each agrees to wait no more than 10 minutes for the other.Find the probability that they meet. Discuss how your solution generalizes to the study of the distance
7.12 A bus arrives at a bus stop at a randomly selected time within a 1-hour period. A passenger arrives at the bus stop at a randomly selected time within the same hour. The passenger will wait for the bus up to one-quarter of an hour. What is the probability that the passenger will catch the bus?
7.11 An electronic surveillance system has one of each of two different types of components in joint operation. Letting X1 and X2 denote the random life lengths of the components of type I and type II, respectively, we have the joint probability density function given by(Measurements are in
7.10 Refer to Exercise 7.9. Find the probability that employee I spends more than 75% of the week on her assigned task, given that employee II spends exactly 50% of the work week on his assigned task.
7.9 Let X1 and X2 denote the proportions of time, out of one workweek, that employees I and II, respectively, actually spend performing their assigned tasks. The joint relative frequency behavior of X1 and X2 is modeled by the probability density function a Find b Find c Are X1 and X2 independent?
7.8 Refer to Exercise 7.5.a Find the marginal density functions for X1 and X2.b Are X1 and X2 independent?c Find
7.7 Let X1 and X2 denote the proportion of two different chemicals found in a sample mixture of chemicals used as an insecticide. Suppose X1 and X2 have joint probability density given by
7.6 Refer to Exercise 7.4.a Find the marginal density function of X2.b Find .c Are X1 and X2 independent?d Find
7.5 Refer to Exercise 7.2.a Find the marginal density function for X1.b Find .c Are X1 and X2 independent?
7.4 An environmental engineer measures the amount(by weight) of particulate pollution in air samples (of a certain volume) collected over the smokestack of a coal-operated power plant. Let X1 denote the amount of pollutant per sample when a certain cleaning device on the stack is not operating, and
7.2 A radioactive particle is randomly located in a square area with sides one unit in length. Let X1 and X2 denote the coordinates of the particle.Since the particle is equally likely to fall in any subarea of a fixed size, a reasonable model for (X1, X2) is given by a Sketch the probability
7.1 Two construction contracts are to be randomly assigned to one or more of three firms (I, II, and III). A firm may receive more than one contract. Let X1 the number of contracts assigned to firm I X2 the number assigned to firm II.a Find the joint probability distribution for X1 and X2.b Find
10.88 Construct operating characteristic curves for the following sampling plans, with sampling by attributes.a n 10, a 2 b n 10, a 4 c n 20, a 2 d n 20, a 4
10.89 Suppose a lot of size 3,000 is to be sampled by attributes. An AQL of 4% is specified for acceptable lots. Find the appropriate level II sampling plan under a Normal inspection b Tightened inspection c Reduced inspection
10.90 Fuses of a certain type must not allow the current passing through them to exceed 20 amps. A lot of 1,200 of these fuses is to be checked for quality by a lot-acceptance plan with inspection by variables.The desired AQL is 2%. Find the appropriate sampling plan for level IV sampling and a
10.91 Refer to Exercise 10.90. If, under normal inspection, the sample mean turned out to be 19.2 amps and the sample standard deviation 0.3 amp, what would you conclude?
10.92 For a lot of 250 one-gallon cans filled with a certain industrial chemical, each gallon is specified to have a percentage of alcohol in excess of L. If the AQL is specified to be 10%, find an appropriate lot-acceptance sampling plan (level IV inspection) under a Normal inspection b Tightened
10.96 Prospective employees of an engineering firm are told that engineers in the firm work at least 45 hours per week, on the average.A random sample of 40 engineers in the firm showed that, for a particular week, they averaged 44 hours of work with a standard deviation of 3 hours. Are the
10.97 Company A claims that no more than 8% of the resistors it produced fail to meet the tolerance specifications. Tests on 100 randomly selected resistors yielded 12 that failed to meet the specifications. Is the company’s claim valid? Test at the 10% significance level.
10.98 An approved standard states that the average LC50 for DDT should be ten parts per million for a certain species of fish. Twelve experiments produced LC50 measurements of 16, 5, 21, 19, 10, 5, 8, 2, 7, 2, 4, 9 Do the data cast doubt on the standard at the 5% significance level?
10.99 A production process is supposed to be producing 10-ohm resistors. Fifteen randomly selected resistors showed a sample mean of 9.8 ohms and a sample standard deviation of 0.5 ohm. Are the specifications of the process being met? Should a two-tailed test be used here?
10.100 For the resistors of Exercise 10.99, the claim is made that the standard deviation of the resistors produced will not exceed 0.4 ohm.Can the claim be refuted at the 10% significance level?
10.101 The abrasive resistance of rubber is increased by adding a silica filler and a coupling agent to chemically bond the filler to the rubber polymer chains. Fifty specimens of rubber made with a type I coupling agent gave a mean resistance measurement of 92, with the variance of the
10.102 Two different types of coating for pipes are to be compared with respect to their ability to aid in resistance to corrosion. The amount of corrosion on a pipe specimen is quantified by measuring the maximum pit depth. For coating A, 35 specimens showed an average maximum pit depth of 0.18
10.103 Bacteria in water samples are sometimes difficult to count, but their presence can easily be detected by culturing. In 50 independently selected water samples from a lake, 43 contained certain harmful bacteria. After adding a chemical to the lake water, another 50 water samples showed only
10.104 A large firm made up of several companies has instituted a new quality-control inspection policy. Among 30 artisans sampled from Company A, only 5 objected to the new policy.Among 35 artisans sampled from Company B, 10 objected to the policy. Is there a significant difference, at the 5%
10.105 Research Quarterly, May 1979, reports on a study of impulses applied to the ball by tennis rackets of various construction. Three measurements on ball impulses were taken on each type of racket. For a Classic (wood)racket, the mean was 2.41 and the standard deviation was 0.02. For a Yamaha
10.106 For the impulse measurements of Exercise 10.105, is there sufficient evidence to say that the graphite racket gives more variable results that the wood racket? Use
10.107 An interesting and practical use of the test comes about in the testing for segregation of species of plants or animals. Suppose that two species of plants, A and B, are growing on a test plot.To assess whether or not the species tend to segregate, n plants are randomly sampled from the
10.108 “Love is not blind.” An article on this topic in the Gainesville Sun, June 22, 1992, states that 72 blindfolded people tried to distinguish their partner from two decoys of similar age by feeling their foreheads. And 58% of them were correct! Is this as amazing a result as the article
10.109 Is knowledge of right-of-way laws at four-legged intersections associated with similar knowledge for T intersections? Montgomery and Carstens(Journal of Transportation Engineering, May 1987) show the results of a questionnaire designed to investigate this question (among others). How would
10.110 Chinowsky (Journal of Professional Issues in Engineering Education and Practice, 2002)researched the education background of design and construction executives. Of 111 construction executives 70 had a Bachelor of Science degree in Engineering. Of 139 design executives, 102 had a Bachelor of
10.111 Scruggs and Iwan (Journal of Structural Engineering, 2003) studied Implementation of a Brushless DC Machine (BDC) as a force actuator, for use in suppressing vibrations in civil structures.The data for semiactive BDC and magnetorheological(MR) fluid dampers are given in table below. The
10.112 Seim et al. (Journal of Structural Engineering, 2003) Studied the post-strengthening of concrete slabs with externally bonded carbon fiber reinforced polymers. The moment capacity of the slab specimens calculated according to the design guidelines at a reduced 65% level and that reached in
10.113 It is a common practice to use composite beams with formed steel deck as the floor system in steel frame structures. Park, Kim, and Yang (Journal of Structural Engineering, 2003) performed a series of tests on composite beams with web opening. The applied loads at the first yield and at the
10.114 A lack of knowledge of a job or profession often results in it being dismissed as a career option. Does knowledge lead to the decision to be an engineer? To investigate this question Hamill and Hodgkinson (Proceedings of ICE, 2003) surveyed 566 11- to 16-year-olds, and classified them by
10.115 A horizontal directional drilling training was conducted for the California Department of Transportation engineers and safety inspectors.A pretest was administered prior to the training and a final exam was given at the end of the training. Ariaratnam, Najafi, and Morones (Journal of
11.1 Use the method of least squares to fit a straight line to the following six data points:a What are the least-squares estimates of and ?b Plot the data points and graph the leastsquares line. Does the line pass through the data points? X > > y 44 1 1 32 22 2 43 3 55 6 COLO 5
11.2 Use the method of least squares to fit a straight line to the following data points:a What are the least-squares estimates of and ?b Plot the data points and graph the leastsquares line. Does the line pass through the data points? xx -2 -1 y 4 3 03 -- 1 2 1 -1
11.3 The elongation of a steel cable is assumed to be linearly related to the amount of force applied.Five identical specimens of cable gave the following results when varying forces were applied:Use the method of least squares to fit the line Y = b0 + b1x + e Force (x) 1.0 1.5 2.0 2.5 3.0
11.4 A company wants to model the relationship between its sales and the sales for the industry as a whole. For the following data, fit a straight line by the method of least squares. Company Sales Year y ($ million) Industry Sales x ($ million) 2002 0.5 10 2003 1.0 12 2004 1.0 13 2005 1.4 15 2006
11.5 It is thought that abrasion loss in certain steel specimens should be a linear function of the Rockwell hardness measure. A sample of eight specimens gave the following results:Fit a straight line to these measurements. Rochwell Hardness (x) 60 62 63 67 70 74 79 81 251 245 246 233 221 202 188
11.6 The Organization of Petroleum Exporting Countries (OPEC), a cartel of crude-oil suppliers, controls the crude oil prices and production. Do they affect the gasoline prices paid by motorists?Table on the next page gives the average annual prices of crude oil and unleaded gasoline in the United
11.7 A study is made of the number of parts assembled as a function of the time spend on the job.Twelve employees are divided into three groups, assigned to three time intervals, with the following results:Fit the model by the method of least squares. Time x 10 minutes Number of Parts Assembled y
11.8 Laboratory experiments designed to measure LC50 values for the effect of certain toxicants on fish are basically run by two different methods.One method has water continuously flowing through laboratory tanks, and the other has static water conditions. For purposes of establishing criteria for
11.9 Calculate SSE and s 2 for the data of Exercise 11.1.
11.10 Calculate SSE and s 2 for the data of Exercise 11.5.Interpret s 2 value in context of problem.
11.11 Calculate SSE and s 2 for the data of Exercise 11.6.Interpret s 2 value in context of problem.
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