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introduction to probability statistics

Probability And Statistics For Engineering And The Sciences 8th Edition Jay L Devore, Roger Ellsbury - Solutions

78. A boiler has five identical relief valves. The probability that any particular valve will open on demand is .95. Assuming independent operation of the valves, calculate P(at least one valve opens) and P(at least one valve fails to open).
74. The proportions of blood phenotypes in the U.S. population are as follows:A B AB O.40 .11 .04 .45 Assuming that the phenotypes of two randomly selected individuals are independent of one another, what is the probability that both phenotypes are O? What is the probability that the phenotypes of
73. If A and B are independent events, show that A and B are also independent. [Hint: First establish a relationship between , P(B), and .]
72. In Exercise 13, is any Ai independent of any other Aj?Answer using the multiplication property for independent events.
71. An oil exploration company currently has two active projects, one in Asia and the other in Europe. Let A be the event that the Asian project is successful and B be the event that the European project is successful. Suppose that A and B are independent events with and .a. If the Asian project is
69. In Exercise 59, consider the following additional information on credit card usage:70% of all regular fill-up customers use a credit card.50% of all regular non-fill-up customers use a credit card.60% of all plus fill-up customers use a credit card.50% of all plus non-fill-up customers use a
68. A friend who lives in Los Angeles makes frequent consulting trips to Washington, D.C.; 50% of the time she travels on airline #1, 30% of the time on airline #2, and the remaining 20% of the time on airline #3. For airline #1, flights are late into D.C. 30% of the time and late into L.A. 10% of
65. A large operator of timeshare complexes requires anyone interested in making a purchase to first visit the site of interest. Historical data indicates that 20% of all potential purchasers select a day visit, 50% choose a one-night visit, and 30% opt for a two-night visit. In addition, 10%of day
63. For customers purchasing a refrigerator at a certain appliance store, let A be the event that the refrigerator was manufactured in the U.S., B be the event that the refrigerator had an icemaker, and C be the event that the customer purchased an extended warranty. Relevant probabilities area.
62. 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
61. 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(A2 ¨
58. Show that for any three events A, B, and C with ,.
57. If , show that . [Hint: Add to both sides of the given inequality and then use the result of Exercise 56.]
56. For any events A and B with , show that.
54. In Exercise 13, , for .Use the probabilities given there to compute the following probabilities, and explain in words the meaning of each one.a. b.c.d. .
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 and .What is ?
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 first box and placed in the second box. Then a ball is randomly selected from the second box and placed in the first box.a. 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
49. The accompanying table gives information on the type of coffee selected by someone purchasing a single cup at a particular airport kiosk.Consider randomly selecting such a coffee purchaser.a. What is the probability that the individual purchased a small cup? A cup of decaf coffee?b. If we learn
48. Reconsider the system defect situation described in Exercise 26 (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
47. Return to the credit card scenario of Exercise 12 (Section 2.2), where , , ,, and . 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 card, what is the probability that he or she has a Visa
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
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 O A B AB 1 .082
44. Show that . Give an interpretation involving subsets.An k B 5 A n n 2 k B
42. A starting lineup in basketball consists of two guards, two forwards, and a center.a. A certain college team has on its roster three centers, four guards, four forwards, and one individual (X) who can play either guard or forward. How many different starting lineups can be created? [Hint:
40. 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
39. Fifteen telephones have just been received at an authorized service center. Five of these telephones are cellular, five are cordless, and the other five are corded phones. Suppose that these components are randomly allocated the numbers 1, 2, . . . , 15 to establish the order in which they will
38. A box in a certain supply room contains four 40-W lightbulbs, five 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
35. A production facility employs 20 workers on the day shift, 15 workers on the swing shift, and 10 workers on the graveyard shift. A quality 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
32. A stereo store is offering a special price on a complete set of components (receiver, compact disc player, speakers, turntable). A purchaser is offered a choice of manufacturer for each component:Receiver: Kenwood, Onkyo, Pioneer, Sony, Sherwood Compact disc player: Onkyo, Pioneer, Sony,
29. As of April 2006, roughly 50 million .com web domain names were registered (e.g., yahoo.com).a. How many domain names consisting of just two letters in sequence can be formed? How many domain names of length two are there if digits as well as letters are permitted as characters? [Note: A
28. In Exercise 5, 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 are
27. An academic department with five faculty members—Anderson, Box, Cox, Cramer, and Fisher—must select two of its members to serve on a personnel review committee.Because the work will be time-consuming, no one is anxious to serve, so it is decided that the representative will be selected by
26. A certain system can experience three different types of defects. Let denote the event that the system has a defect of type i. Suppose thata. What is the probability that the system does not have a type 1 defect?b. What is the probability that the system has both type 1 and type 2 defects?c.
25. The three most popular options on a certain type of new car are a built-in GPS (A), a sunroof (B), and an automatic transmission (C). If 40% of all purchasers request A, 55%request B, 70% request C, 63% request A or B, 77% request A or C, 80% request B or C, and 85% request A or B or C,
24. Show that if one event A is contained in another event B(i.e., A is a subset of B), then . [Hint: For such A and B, A and are disjoint and , as can be seen from a Venn diagram.] For general A and B, what does this imply about the relationship among, and ?
21. 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
20. 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
18. A box contains six 40-W bulbs, five 60-W bulbs, and four 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?
17. Let A denote the event that the next request for assistance from a statistical software consultant relates to the SPSS package, and let B be the event that the next request is for help with SAS. Suppose that and .a. Why is it not the case that ?b. Calculate .c. Calculate .d. Calculate .
15. Consider the type of clothes dryer (gas or electric) purchased by each of five 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 and P(all five
14. Suppose that 55% of all adults regularly consume coffee, 45% regularly consume carbonated soda, and 70% regularly consume at least one of these two products.a. What is the probability that a randomly selected adult regularly consumes both coffee and soda?b. What is the probability that a
13. A computer consulting firm presently has bids out on three projects. Let , for , and suppose that , , ,, , ,. Express in words each of the following events, and compute the probability of each event:a.b. [Hint: ]c. d.e. f.
12. 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 , , and P(A ¨ B) 5 .25.P(A) 5 .5 P(B) 5 .4a. Compute the probability that the selected
11. A mutual fund company offers its customers a variety of 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 in the
10.a. In Example 2.10, identify three events that are mutually exclusive.b. Suppose there is no outcome common to all three of the events A, B, and C. Are these three events necessarily mutually exclusive? If your answer is yes, explain why;if your answer is no, give a counterexample using the
9. Use Venn diagrams to verify the following two relationships for any events A and B (these are called De Morgan’s laws):a.b.[Hint: In each part, draw a diagram corresponding to the left side and another corresponding to the right side.]
6. A college library has five copies of a certain text on reserve.Two copies (1 and 2) are first printings, and the other three (3, 4, S A ´ C A ¨ C B ´ C B ¨ C and 5) are second printings. A student examines these books in random order, stopping only when a second printing has been selected.
3. 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 C ´ D C ¨ D functions. For the entire system to function,
2. Suppose that vehicles taking a particular freeway exit can turn right (R), turn left (L), or go straight (S). Consider A ´ B A ¨ B Sobserving the direction for each of three successive vehicles.a. List all outcomes in the event A that all three vehicles go in the same direction.b. List all
83. Consider numerical observations . It is frequently of interest to know whether the xi 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-and-leaf display or histogram.However, if n is
82. The sample data sometimes represents a time series, where value of a response variable x at time t. Often the observed series shows a great deal of random variation, which makes it difficult to study longerterm behavior. In such situations, it is desirable to produce a smoothed version of the
81. A study carried out to investigate the distribution of total braking time (reaction time plus accelerator-to-brake movement time, in ms) during real driving conditions at 60 km/hr gave the following summary information on the distribution of times (“A Field Study on Braking Responses During
80. 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. of the Institution of Engineers, 1995: 211–215) gives the following information on lengths (km) for one particular system:Length Frequency 6
79. Let and denote the sample mean and variance for the sample and let and denote these quantities when an additional observation is added to the sample.a. Show how can be computed from and .b. Show that so that can be computed from , , and .c. Suppose that a sample of 15 strands of drapery yarn
78. Consider a sample and suppose that the values of , s2, and s have been calculated.a. Let for . How do the values of s2 and s for the yi’s compare to the corresponding values for the xi’s? Explain.b. Let for . What are the values of the sample variance and sample standard deviation for the
77. The authors of the article “Predictive Model for Pitting Corrosion in Buried Oil and Gas Pipelines” (Corrosion, 2009: 332–342) provided the data on which their investigation was based.a. Consider the following sample of 61 observations on maximum pitting depth (mm) of pipeline specimens
75. 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. 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 73.b. For a categorical sample, how would you define the modal category?
73. The article “Can We Really Walk Straight?” (Amer. J. of Physical Anthropology, 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
72. 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
71. Here is a description from Minitab of the strength data given in Exercise 13.d1 5 x1 2 y1,c, dn 5 xn 2 yn F 5 95C 1 32 sy s 2 x y 2 i 5 1, 2, xc, n yi 5 axi 1 b x 2 m g(xg(x i 2 m)2 i 2 x)2 g(xi 2 c)2 Variable N Mean Median TrMean StDev SE Mean strength 153 135.39 135.40 135.41 4.59 0.37
70. 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.a. Let a and b be constants and let for. What are the relationships between and and between and ?b. A sample of temperatures for initiating a certain chemical reaction yielded a sample average (°C) of 87.3 and a sample standard deviation of 1.04. What are the sample average and standard
68.a. For what value of c is the quantity 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 and will be smaller than the other (assuming that )?
67. Aortic stenosis refers to a narrowing of the aortic valve in the heart. The paper “Correlation Analysis of Stenotic Aortic Valve Flow Patterns Using Phase Contrast MRI”912,93 932,95 952,97 972,99 812,83 832,85 852,87 872,89 892,91(Annals of Biomed. Engr., 2005: 878–887) gave the following
66. A deficiency of the trace element selenium in the diet can negatively impact growth, immunity, muscle and neuromuscular function, and fertility. The introduction of selenium supplements to dairy cows is justified when pastures have low selenium levels. Authors of the paper “Effects of
65. The accompanying frequency distribution of fracture strength(MPa) observations for ceramic bars fired in a particular kiln appeared in the article “Evaluating Tunnel Kiln Performance”(Amer. Ceramic Soc. Bull., Aug. 1997: 59–63).100s/x(1024mm3)largest xi 5 77,048 x 5 76,831 s 5 180
64. Fretting is a wear process that results from tangential oscillatory movements of small amplitude in machine parts. The article “Grease Effect on Fretting Wear of Mild Steel”(Industrial Lubrication and Tribology, 2008: 67–78)included the following data on volume wear for base oils having
63. A sample of 77 individuals working at a particular office was selected and the noise level (dBA) experienced by each individual was determined, yielding the following data(“Acceptable Noise Levels for Construction Site Offices,”Building Serv. Engr. Research and Technology,
62. Consider the following information on ultimate tensile strength (lb/in) for a sample of hard zirconium copper wire specimens (from “Characterization Methods for Fine Copper Wire,” Wire J. Intl., Aug., 1997: 74–80):Determine the values of the two middle sample observations(and don’t do
60. 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
59. Blood cocaine concentration (mg/L) was determined both for a sample of individuals who had died from cocaineinduced 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. A sample of 20 glass bottles of a particular type was selected, and the internal pressure strength of each bottle was determined.Consider the following partial sample information:Three smallest observations 125.8 188.1 193.7 Three largest observations 221.3 230.5 250.2a. Are there any outliers
56. The following data on distilled alcohol content (%) for a sample of 35 port wines was extracted from the article “A Method for the Estimation of Alcohol in Fortified Wines Using Hydrometer Baumé and Refractometer Brix” (Amer.J. Enol. Vitic., 2006: 486–490). Each value is an average of
55. Here is a stem-and-leaf display of the escape time data introduced in Exercise 36 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
54. Grip is applied to produce normal surface forces that compress the object being gripped. Examples include two x|x n 5 5 gxi gx 2 5 24,657,511 i 5 20,179 people shaking hands, or a nurse squeezing a patient’s forearm to stop bleeding. The article “Investigation of Grip Force, Normal Force,
53. A mutual fund is a professionally managed investment scheme that pools money from many investors and invests in a variety of securities. Growth funds focus primarily on increasing the value of investments, whereas blended funds seek a balance between current income and growth. Here is data on
52. The first four deviations from the mean in a sample of reaction times were .3, .9, 1.0, and 1.3. What is the fifth deviation from the mean? Give a sample for which these are the five deviations from the mean.
51. The article “A Thin-Film Oxygen Uptake Test for the Evaluation of Automotive Crankcase Lubricants”(Lubric. Engr., 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
50. 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 gxi
49. A study of the relationship between age and various visual functions (such as acuity and depth perception) reported the following observations on the area of scleral lamina (mm2)from human optic nerve heads (“Morphometry of Nerve Fiber Bundle Pores in the Optic Nerve Head of the
48. Exercise 34 presented the following data on endotoxin concentration in settled dust both for a sample of urban homes and for a sample of farm homes:U: 6.0 5.0 11.0 33.0 4.0 5.0 80.0 18.0 35.0 17.0 23.0 F: 4.0 14.0 11.0 9.0 9.0 8.0 4.0 20.0 5.0 8.9 21.0 9.2 3.0 2.0 0.3a. Determine the value of
47. 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 x“Heat-Resistant Active Brazing of Silicon Nitride: Mechanical Evaluation of Braze Joints,” Welding J., August,
46. 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-Modified Asphalts” (J. of Materials in Civil Engr., 1996: 153–156):2781 2900 3013 2856 2888a. What are the
44. 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 firefighters performing a fire-suppression simulation:29.5 49.3 30.6 28.2 28.0 26.3 33.9
43. 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
42.a. If a constant c is added to each xi in a sample, yielding, how do the sample mean and median of the yis relate to the mean and median of the xis? Verify your conjectures.b. If each xi is multiplied by a constantc, yielding , answer the question of part (a). Again, verify your conjectures.
41. A sample of automobiles was selected, and each was subjected to a 5-mph crash test. Denoting a car with no visible damage by S (for success) and a car with such damage by F, results were as follows:S S F S S S F F S Sa. What is the value of the sample proportion of successes x/n?b. Replace each
40. Compute the sample median, 25% trimmed mean, 10%trimmed mean, and sample mean for the lifetime data given in Exercise 27, and compare these measures.
39. 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 (flight hours/104) to reach a given crack size in fastener holes intended for use in military aircraft (“Statistical Crack
36. 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
35. 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 difficulty), resulting in the following observations(from “Thermoplastic Starch Blends with a
34. Exposure to microbial products, especially endotoxin, may have an impact on vulnerability to allergic diseases. The article “Dust Sampling Methods for Endotoxin—An Essential, But Underestimated Issue” (Indoor Air, 2006:20–27) considered various issues associated with determining
32. Fire load is the heat energy that could be released per square meter of floor area by combustion of contents and the structure itself. The article “Fire Loads in Office Buildings” (J. of Structural Engr., 1997: 365–368) gave the following cumulative percentages (read from a graph)for fire
29. Consider the following data on types of health complaint(J joint swelling, F fatigue, B back pain, M muscle weakness, C coughing, N nose running/irritation, O other) made by tree planters. Obtain frequencies and relative frequencies for the various categories, and draw a histogram. (The data is
28. Human measurements provide a rich area of application for statistical methods. The article “A Longitudinal Study of the Development of Elementary School Children’s Private Speech” (Merrill-Palmer Q., 1990: 443–463)reported on a study of children talking to themselves (private speech).
27. The paper “Study on the Life Distribution of Microdrills”(J. of Engr. 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
26. Automated electron backscattered diffraction is now being used in the study of fracture phenomena. The following information on misorientation angle (degrees) was extracted from the article “Observations on the Faceted Initiation Site in the Dwell-Fatigue Tested Ti-6242 Alloy:
25. 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 find a function for which the histogram of transformed values is more

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