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modern mathematical statistics with applications

Modern Mathematical Statistics With Applications 1st Edition Jay L Devore - Solutions

18. The article Determination of Most Representative Subdivision (J. Energy Engrg., 1993: 43—55) gave data on various characteristics of subdivisions that could be used in deciding whether to provide electrical power using overhead lines or underground lines.Here are the values of the variable
19. The article cited in Exercise 18 also gave the following values of the variables y number of culsde-sac and z number of intersections:y 1 0 1 0 0 2 0 1 1 1 2 1 0 0 1 1 0 1 1 z 1 8 6 1 1 5 3 0 0 4 4 0 0 1 2 1 4 0 4 y 1 1 0 0 0 1 1 2 0 1 2 2 1 1 0 2 1 1 0 z 0 3 0 1 1 0 1 3 2 4 6 6 0 1 1 8 3 3
20. How does the speed of a runner vary over the course of a marathon (a distance of 42.195 km)? Consider determining both the time to run the rst 5 kmand the time to run between the 35-km and 40-km points, and then subtracting the former time from the latter time.A positive value of this
21. In a study of warp breakage during the weaving of fabric (Technometrics, 1982: 63), 100 specimens of yarn were tested. The number of cycles of strain to breakage was determined for each yarn specimen, resulting in the following data:86 146 251 653 98 249 400 292 131 169 175 176 76 264 15 364
22. The accompanying data set consists of observations on shear strength (lb) of ultrasonic spot welds made on a certain type of alclad sheet. Construct a relative frequency histogram based on ten equal-width classes with boundaries 4000, 4200, . . . . [The histogram will agree with the one in
23. A transformation of data values by means of some mathematical function, such as or 1/x, can often yield a set of numbers that has nicer statistical properties than the original data. In particular, it may be possible to nd a function for which the histogram of transformed values is more
24. The clearness index was determined for the skies over Baghdad for each of the 365 days during a particular year ( Contribution to the Study of the Solar Radiation Climate of the Baghdad Environment, Solar Energy, 1990: 7—12). The accompanying table gives the results.a. Determine relative
25. The paper Study on the Life Distribution of Microdrills(J. Engrg. Manufacture, 2002: 301—305) reported the following observations, listed in increasing order, on drill lifetime (number of holes that a drill machines before it breaks) when holes were drilled in a certain brass alloy.11 14 20
26. Consider the following data on type of health complaint(Jjoint swelling,Ffatigue,Bback pain, Mmuscle weakness,Ccoughing,Nnose running/irritation,Oother) made by tree planters. Obtain frequencies and relative frequencies for the various categories, and draw a histogram. (The data is
29. Fire load (MJ/m2) is the heat energy that could be released per square meter of oor area by combustion of contents and the structure itself. The article Fire Loads in Of ce Buildings (J. Struct. Engrg., 1997:365—368) gave the following cumulative percentages(read from a graph) for re loads in
30. The article The Pedaling Technique of Elite Endurance Cyclists (Int. J. Sport Biomechanics, 1991: 29—53) reported the accompanying data on single-leg power at a high workload:244 191 160 187 180 176 174 205 211 183 211 180 194 200a. Calculate and interpret the sample mean and median.b.
31. In Superbowl XXXVII, Michael Pittman of Tampa Bay rushed (ran with the football) 17 times on rst down, and the results were the following gains in yards:23 1 4 1 6 5 9 6 21 3 2 0 2 24 1 1a. Determine the value of the sample mean.b. Determine the value of the sample median. Why is it so
32. The minimum injection pressure (psi) for injection molding specimens of high amylose corn was determined for eight different specimens (higher pressure corresponds to greater processing dif culty), resulting in the following observations (from Thermoplastic Starch Blends with a
33. A sample of 26 offshore oil workers took part in a simulated escape exercise, resulting in the accompanying data on time (sec) to complete the escape( Oxygen Consumption and Ventilation During Escape from an Offshore Platform, Ergonomics, 1997: 281—292):389 356 359 363 375 424 325 394 402 373
34. The article Snow Cover and Temperature Relationships in North America and Eurasia (J. Climate Appl. Meteorol., 1983: 460—469) used statistical techniques to relate the amount of snow cover on each continent to average continental temperature.Data presented there included the following ten
36. The propagation of fatigue cracks in various aircraft parts has been the subject of extensive study in recent years. The accompanying data consists of propagation lives ( ight hours/104) to reach a given crack size in fastener holes intended for use in military aircraft( Statistical Crack
37. Compute the sample median, 25% trimmed mean, 10% trimmed mean, and sample mean for the microdrill data given in Exercise 25, and compare these measures.
40. An experiment to study the lifetime (in hours) for a certain type of component involved putting ten components into operation and observing them for 100 hours. Eight of the components failed during that period, and those lifetimes were recorded. Denote the lifetimes of the two components still
41. The article Oxygen Consumption During Fire Suppression: Error of Heart Rate Estimation(Ergonomics, 1991: 1469—1474) reported the following data on oxygen consumption (mL/kg/min)for a sample of ten re ghters performing a resuppression simulation:29.5 49.3 30.6 28.2 28.0 26.3 33.9 29.4 23.5
42. The value of Young s modulus (GPa) was determined for cast plates consisting of certain intermetallic substrates, resulting in the following sample observations ( Strength and Modulus of a Molybdenum-CoatedTi-25Al-10Nb-3U-1Mo Intermetallic, J. Mater. Engrg. Perform., 1997: 46—50):116.4 115.9
43. The accompanying observations on stabilized viscosity(cP) for specimens of a certain grade of asphalt with 18% rubber added are from the article Viscosity Characteristics of Rubber-Modi edAsphalts(J. Mater. Civil Engrg., 1996: 153—156):2781 2900 3013 2856 2888a. What are the values of the
44. Calculate and interpret the values of the sample median, sample mean, and sample standard deviation for the following observations on fracture strength(MPa, read from a graph in Heat-Resistant Active Brazing of Silicon Nitride: Mechanical Evaluation of Braze Joints, Welding J., Aug. 1997):87 93
45. Exercise 33 in Section 1.3 presented a sample of 26 escape times for oilworkers in a simulated escape exercise. Calculate and interpret the sample standard deviation. (Hint: xi 9638 and x2 i 3,587,566).
46. A study of the relationship between age and various visual functions (such as acuity and depth perception)reported the following observations on area of scleral lamina (mm2) from human optic nerve heads( Morphometry of Nerve Fiber Bundle Pores in the Optic Nerve Head of the Human, Experiment.
47. In 1997 a woman sued a computer keyboard manufacturer, charging that her repetitive stress injuries were caused by the keyboard (Genessy v. Digital Equipment Corp.). The injury awarded about $3.5 million for pain and suffering, but the court then set aside that award as being unreasonable
48. The article AThin-Film Oxygen UptakeTest for the Evaluation of Automotive Crankcase Lubricants(Lubric. Engrg., 1984:75—83) reported the following data on oxidation-induction time (min) for various commercial oils:87 103 130 160 180 195 132 145 211 105 145 153 152 138 87 99 93 119 129a.
49. The rst four deviations from the mean in a sample of n5 reaction times were .3, .9, 1.0, and 1.3. What is the fth deviation from the mean? Give a sample for which these are the ve deviations from the mean.
50. Reconsider the data on area of scleral lamina given in Exercise 46.a. Determine the lower and upper fourths.b. Calculate the value of the fourth spread.c. If the two largest sample values, 4.33 and 4.52, had instead been 5.33 and 5.52, how would this affect fs? Explain.d. By how much could the
51. Reconsider these values of rushing yardage from Exercise 31 of this chapter:23 1 4 1 6 5 9 6 21 3 2 0 2 24 1 1a. What are the values of the fourths, and what is the value of fs?b. Construct a boxplot based on the ve-number summary, and comment on its features.c. How large or small does an
52. Here is a stem-and-leaf display of the escape time data introduced in Exercise 33 of this chapter.32 55 33 49 34 35 6699 36 34469 37 03345 38 9 39 2347 40 23 41 42 4a. Determine the value of the fourth spread.b. Are there any outliers in the sample? Any extreme outliers?c. Construct a boxplot
53. The amount of aluminum contamination (ppm) in plastic of a certain type was determined for a sample of 26 plastic specimens, resulting in the following data ( The Lognormal Distribution for Modeling Quality Data when the Mean Is Near Zero, J. Qual.Tech., 1990: 105—110):30 30 60 63 70 79 87 90
55. A company utilizes two different machines to manufacture parts of a certain type. During a single shift, a sample of n 20 parts produced by each machine is obtained, and the value of a particular critical dimension for each part is determined. The comparative boxplot below is constructed from
56. Blood cocaine concentration (mg/L) was determined both for a sample of individuals who had died from cocaine-induced excited delirium (ED) and for a sample of those who had died from a cocaine overdose without excited delirium; survival time for people in both groups was at most 6 hours.The
57. Observations on burst strength (lb/in2) were obtained both for test nozzle closure welds and for production cannister nozzle welds ( Proper Procedures Are the Key to Welding Radioactive Waste Cannisters, Welding J., Aug. 1997: 61—67).Test 7200 6100 7300 7300 8000 7400 7300 7300 8000 6700 8300
58. The comparative boxplot below of gasoline vapor coef cients for vehicles in Detroit appeared in the article Receptor Modeling Approach to VOC Emission Inventory Validation (J. Environ.Engrg., 1995: 483—490). Discuss any interesting features.
59. Let x1, . . . , xn be a sample and let a and b be constants.If yi axi b for i 1, 2, . . . , n, how does fs (the fourth spread) for the yi s relate to fs for the xi s? Substantiate your assertion.
60. Consider the following information from a sample of four Wolferman s cranberry citrus English muf ns, which are said on the package label to weigh 116 g: 104.4 g; s 4.1497 g, smallest weighs 98.7 g, largest weighs 108.0 g. Determine the values of the two middle sample observations (and don t
61. Three different C2F6 ow rates (SCCM) were considered in an experiment to investigate the effect of ow rate on the uniformity (%) of the etch on a silicon wafer used in the manufacture of integrated circuits, resulting in the following data:Flow rate 125 2.6 2.7 3.0 3.2 3.8 4.6 160 3.6 4.2 4.2
62. The amount of radiation received at a greenhouse plays an important role in determining the rate of photosynthesis. The accompanying observations on incoming solar radiation were read from a graph in the article Radiation Components over Bare and Planted Soils in a Greenhouse (Solar Energy,
63. The following data on HC and CO emissions for one particular vehicle was given in the chapter introduction.HC (g/mile) 13.8 18.3 32.2 32.5 CO (g/mile) 118 149 232 236a. Compute the sample standard deviations for the HC and CO observations. Does the widespread belief appear to be justi ed?b. The
64. The accompanying frequency distribution of fracture strength (MPa) observations for ceramic bars red in a particular kiln appeared in the article Evaluating Tunnel Kiln Performance (Amer. Ceramic Soc.Bull., Aug. 1997: 59—63).Class 81—83 83—85 85—87 87—89 89—91 Freq. 6 7 17 30 43
65. Fifteen air samples from a certain region were obtained, and for each one the carbon monoxide concentrationwas determined.The results (in ppm) were 9.3 10.7 8.5 9.6 12.2 15.6 9.2 10.5 9.0 13.2 11.0 8.8 13.7 12.1 9.8 Using the interpolation method suggested in Section 1.3, compute the 10%
66.a. For what value of c is the quantity (xic)2 minimized?(Hint: Take the derivative with respect toc, set equal to 0, and solve.)b. Using the result of part (a), which of the two quantities(xi )2 and (xi m)2 will be smaller than the other (assuming that m)?
67.a. Let a and b be constants and let yi axi b for i 1, 2, . . . , n. What are the relationships between and and between sx 2 and s y 2?b. The Australian army studied the effect of high temperatures and humidity on human body temperature(Neural Network Training on Human Body Core Temperature
68. Elevated energy consumption during exercise continues after the workout ends. Because calories burned after exercise contribute to weight loss and have other consequences, it is important to understand this process. The paper Effect of Weight Training Exercise and Treadmill Exercise on
69. Anxiety disorders and symptoms can often be effectively treated with benzodiazepine medications.It is known that animals exposed to stress exhibit a decrease in benzodiazepine receptor binding in the frontal cortex. The paper Decreased Benzodiazepine Receptor Binding in Prefrontal Cortex in
70. The article Can We Really Walk Straight? (Amer.J. Phys. Anthropol., 1992: 19—27) reported on an experiment in which each of 20 healthy men was asked to walk as straight as possible to a target 60 m away at normal speed. Consider the following observations on cadence (number of strides per
71. The mode of a numerical data set is the value that occurs most frequently in the set.a. Determine the mode for the cadence data given in Exercise 70.b. For a categorical sample, how would you de ne the modal category?
72. Specimens of three different types of rope wire were selected, and the fatigue limit (MPa)was determined for each specimen, resulting in the accompanying data.Type 1 350 350 350 358 370 370 370 371 371 372 372 384 391 391 392 Type 2 350 354 359 363 365 368 369 371 373 374 376 380 383 388 392
74. Consider the following data on active repair time(hours) for a sample of n 46 airborne communications receivers:.2 .3 .5 .5 .5 .6 .6 .7 .7 .7 .8 .8.8 1.0 1.0 1.0 1.0 1.1 1.3 1.5 1.5 1.5 1.5 2.0 2.0 2.2 2.5 2.7 3.0 3.0 3.3 3.3 4.0 4.0 4.5 4.7 5.0 5.4 5.4 7.0 7.5 8.8 9.0 10.3 22.0 24.5
76. Let and sn 2 denote the sample mean and variance for the sample x1, . . . , xn and let and s2 n1 denote these quantities when an additional observation xn1 is added to the sample.a. Show how can be computed from and xn1.b. Show that so that s2 n1 can be computed from xn1, , and sn 2.c.
77. Lengths of bus routes for any particular transit system will typically vary from one route to another. The article Planning of City Bus Routes (J. Institut. Engrs., 1995:211—215) gives the following information on lengths (km) for one particular system:Length 6—8 8—10 10—12 12—14
79. The sample data x1, x2, . . . , xn sometimes represents a time series, where xt the observed value of a response variable x at time t. Often the observed series shows a great deal of random variation, which makes it dif cult to study longer-term behavior. In such situations, it is desirable
80. Consider numerical observations x1, . . . , xn. It is frequently of interest to know whether the xt s are(at least approximately) symmetrically distributed about some value. If n is at least moderately large, the extent of symmetry can be assessed from a stem-andleaf display or histogram.
1. Several jobs are advertised and both Ann and Bev apply. Let A be the event that Ann is hired and let B be the event that Bev is hired. Express in terms of A and B the eventsa. Ann is hired but not Bev.b. At least one of them is hired.c. Exactly one of them is hired.
2. Each voter makes a choice from three candidates, 1, 2, and 3. We consider the votes of just two voters, Al and Bill, so the sample space S consists of all pairs, where the members of the pair are chosen from 1, 2, and 3.a. List all elements of S.b. List all outcomes in the event A that Al and
5. Three components are connected to form a system as shown in the accompanying diagram. Because the components in the 2—3 subsystem are connected in parallel, that subsystem will function if at least one of the two individual components functions. For the entire system to function, component 1
10. A construction rm is currently working on three different buildings. Let Ai denote the event that the ith building is completed by the contract date. Use the operations of union, intersection, and complementation to describe each of the following events in terms of A1, A2, and A3, draw a Venn
13. A mutual fund company offers its customers several different funds: a money-market fund, three different bond funds (short, intermediate, and long-term), two stock funds (moderate and high-risk), and a balanced fund. Among customers who own shares in just one fund, the percentages of customers
14. Consider randomly selecting a student at a certain university, and let A denote the event that the selected individual has a Visa credit card and B be the analogous event for a MasterCard. Suppose that P(A) .5, P(B) .4, and P(A B) .25.a. Compute the probability that the selected
15. A computer consulting rm presently has bids out on three projects. Let Ai {awarded project i}, for i 1, 2, 3, and suppose that P(A1) .22, P(A2) .25, P(A3) .28, P(A1 A2) .11, P(A1 A3) .05, P(A2 A3) .07, P(A1 A2 A3) .01.Express in words each of the following events, and
16. A utility company offers a lifeline rate to any household whose electricity usage falls below 240 kWh during a particular month. Let A denote the event that a randomly selected household in a certain community does not exceed the lifeline usage during January, and let B be the analogous event
17. Consider the type of clothes dryer (gas or electric)purchased by each of ve different customers at a certain store.a. If the probability that at most one of these purchases an electric dryer is .428, what is the probability that at least two purchase an electric dryer?b. If P(all ve purchase
20. A box contains four 40-W bulbs, ve 60-W bulbs, and six 75-W bulbs. If bulbs are selected one by one in random order, what is the probability that at least two bulbs must be selected to obtain one that is rated 75 W?
22. A certain factory operates three different shifts.Over the last year, 200 accidents have occurred at the factory. Some of these can be attributed at least in part to unsafe working conditions, whereas the others are unrelated to working conditions. The accompanying table gives the percentage of
23. An insurance company offers four different deductible levels none, low, medium, and high for its homeowner s policyholders and three different levels low, medium, and high for its automobile policyholders. The accompanying table gives proportions for the various categories of policyholders who
24. The route used by a certain motorist in commuting to work contains two intersections with traf c signals.The probability that he must stop at the rst signal is .4, the analogous probability for the second signal is .5, and the probability that he must stop at at least one of the two signals is
26. Use the axioms to show that if one event A is contained in another event B (i.e., A is a subset of B), then P(A)P(B). [Hint: For such A and B, A and B A are disjoint and B A (B A), as can be seen from a Venn diagram.] For general A and B, what does this imply about the relationship
27. The three major options on a certain type of new car are an automatic transmission (A), a sunroof (B), and a stereo with compact disc player (C). If 70% of all purchasers request A, 80% request B, 75% request C, 85% request A or B, 90% request A or C, 95% request B or C, and 98% request A or B
29. In Exercise 7, suppose that any incoming individual is equally likely to be assigned to any of the three stations irrespective of where other individuals have been assigned. What is the probability thata. All three family members are assigned to the same station?b. At most two family members
30. Apply the proposition involving the probability of A B to the union of the two events (A B) and C in order to verify the result for P(A B C).
31. The College of Science Council has one student representative from each of the ve science departments(biology, chemistry, geology, mathematics, physics). In how many ways cana. Both a council president and a vice president be selected?b. A president, a vice president, and a secretary be
32. A friend is giving a dinner party. Her current wine supply includes 8 bottles of zinfandel, 10 of merlot, and 12 of cabernet (she drinks only red wine), all from different wineries.a. If she wants to serve 3 bottles of zinfandel and serving order is important, how many ways are there to do
34. A chain of stereo stores is offering a special price on a complete set of components (receiver, compact disc player, speakers). A purchaser is offered a choice of manufacturer for each component:Receiver: Kenwood, Onkyo, Pioneer, Sony, Yamaha Compact Onkyo, Pioneer, Sony, disc player: Panasonic
35. Shortly after being put into service, some buses manufactured by a certain company have developed cracks on the underside of the main frame. Suppose a particular city has 25 of these buses, and cracks have actually appeared in 8 of them.a. How many ways are there to select a sample of 5 buses
36. A production facility employs 20 workers on the day shift, 15 workers on the swing shift, and 10 workers on the graveyard shift.Aquality control consultant is to select 6 of these workers for in-depth interviews.Suppose the selection is made in such a way that any particular group of 6 workers
38. An experimenter is studying the effects of temperature, pressure, and type of catalyst on yield from a certain chemical reaction. Three different temperatures, four different pressures, and ve different catalysts are under consideration.a. If any particular experimental run involves the use of
39. Refer to Exercise 38 and suppose that ve different experimental runs are to be made on the rst day of experimentation. If the ve are randomly selected from among all the possibilities, so that any group of ve has the same probability of selection, what is the probability that a different
40. A box in a certain supply room contains four 40-W lightbulbs, ve 60-W bulbs, and six 75-W bulbs.Suppose that three bulbs are randomly selected.a. What is the probability that exactly two of the selected bulbs are rated 75 W?b. What is the probability that all three of the selected bulbs have
41. Fifteen telephones have just been received at an authorized service center. Five of these telephones are cellular, ve are cordless, and the other ve are corded phones. Suppose that these components are randomly allocated the numbers 1, 2, . . . , 15 to establish the order in which they will be
42. Three molecules of type A, three of type B, three of type C, and three of type D are to be linked together to form a chain molecule. One such chain molecule is ABCDABCDABCD, and another is BCDDAAABDBCC.a. How many such chain molecules are there?(Hint: If the three A s were distinguishable from
43. Three married couples have purchased theater tickets and are seated in a row consisting of just six seats. If they take their seats in a completely random fashion (random order), what is the probability that Jim and Paula (husband and wife) sit in the two seats on the far left? What is the
44. Show that ( ) ( ). Give an interpretation involving subsets.
45. The population of a particular country consists of three ethnic groups. Each individual belongs to one of the four major blood groups. The accompanying joint probability table gives the proportions of individuals in the various ethnic group—blood group combinations.Blood Group Ethnic Group O
46. Suppose an individual is randomly selected from the population of all adult males living in the United States. Let A be the event that the selected individual is over 6 ft in height, and let B be the event that the selected individual is a professional basketball player. Which do you think is
47. Return to the credit card scenario of Exercise 14(Section 2.2), where A {Visa}, B {Master-Card}, P(A) .5, P(B) .4, and P(A B) .25.Calculate and interpret each of the following probabilities(a Venn diagram might help).a. b.c. d.e. Given that the selected individual has at least one
48. Reconsider the system defect situation described in Exercise 28 (Section 2.2).a. Given that the system has a type 1 defect, what is the probability that it has a type 2 defect?b. Given that the system has a type 1 defect, what is the probability that it has all three types of defects?c. Given
49. If two bulbs are randomly selected from the box of lightbulbs described in Exercise 40 (Section 2.3)and at least one of them is found to be rated 75 W, what is the probability that both of them are 75-W bulbs? Given that at least one of the two selected is not rated 75 W, what is the
50. A department store sells sport shirts in three sizes(small, medium, and large), three patterns (plaid, print, and stripe), and two sleeve lengths (long and short).The accompanying tables give the proportions of shirts sold in the various category combinations.Short-sleeved Pattern Size Pl Pr St
51. One box contains six red balls and four green balls, and a second box contains seven red balls and three green balls. A ball is randomly chosen from the rst box and placed in the second box. Then a ball is randomly selected from the second box and placed in the rst box.a. What is the
53. A certain shop repairs both audio and video components.Let A denote the event that the next component brought in for repair is an audio component, and let B be the event that the next component is a compact disc player (so the event B is contained in A). Suppose that P(A).6 and P(B).05.What is
54. In Exercise 15, Ai {awarded project i}, for i 1, 2, 3. Use the probabilities given there to compute the following probabilities:a.b.c.d.Express in words the probability you have calculated.
55. For any events A and B with P(B)0, show that
56. If show that(Hint: Add to both sides of the given inequality and then use the result of Exercise 55.)
57. Show that for any three events A, B, and C with P(C)0,
58. At a certain gas station, 40% of the customers use regular unleaded gas (A1), 35% use extra unleaded gas (A2), and 25% use premium unleaded gas (A3).Of those customers using regular gas, only 30% ll their tanks (event B). Of those customers using extra gas, 60% ll their tanks, whereas of those
60. Components of a certain type are shipped to a supplier in batches of ten. Suppose that 50% of all such batches contain no defective components, 30% contain one defective component, and 20% contain two defective components. Two components from a batch are randomly selected and tested. What are
61. A company that manufactures video cameras produces a basic model and a deluxe model. Over the past year, 40% of the cameras sold have been of the basic model. Of those buying the basic model, 30%purchase an extended warranty, whereas 50% of all deluxe purchasers do so. If you learn that a
62. For customers purchasing a full set of tires at a particular tire store, consider the events along with A, B, and C. Assume the following unconditional and conditional probabilities:P(A) .75a. Construct a tree diagram consisting of rst-, second-, and third-generation branches and place P1C
63. In Example 2.30, suppose that the incidence rate for the disease is 1 in 25 rather than 1 in 1000. What then is the probability of a positive test result? Given that the test result is positive, what is the probability that the individual has the disease? Given a negative test result, what is

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