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modern mathematical statistics with applications

Modern Mathematical Statistics With Applications 3rd Edition Jay L. Devore, Kenneth N. Berk, Matthew A. Carlton - Solutions

An insurance company issues a policy covering losses up to 5 (in thousands of dollars). The loss, X, follows a distribution with density function ƒ(x) = 3/x4 for x ≥ 1 and = 0 otherwise. What is the expected value of the amount paid under the policy?
Let X have the pdf ƒ(x) = 1/[π(1 + x2)] for–∞ < x < ∞ (a central Cauchy distribution), and show that Y = 1/X has the same distribution.
Human body temperatures for healthy individuals have approximately a normal distribution with mean 98.25°F and standard deviation .75°F. (The past accepted value of 98.6 °F was obtained by converting the Celsius value of 37°, which is correct to the nearest integer.)a. Find the 90th
If adult female heights are normally distributed, what is the probability that the height of a randomly selected woman isa. Within 1.5 SDs of its mean value?b. Farther than 2.5 SDs from its mean value?c. Between 1 and 2 SDs from its mean value?
Let X be the birth weight, in grams, of a randomly selected full-term baby. The article ““Fetal Growth Parameters and Birth Weight: Their Relationship to Neonatal Body Composition” (Ultrasound Obstetrics Gynecol. 2009: 441–446) suggests that X is normally distributed with mean 3500 and
The following proof that the normal pdf integrates to 1 comes courtesy of Professor Robert Young, Oberlin College. Let f(z) denote the standard normal pdf, and consider the function of two variablesLet V denote the volume under the graph of g(x, y) above the xy-plane.a. Let A denote the area under
Based on extensive data from an urban freeway near Toronto, Canada, “it is assumed that free speeds can best be represented by a normal distribution” [“Impact of Driver Compliance on the Safety and Operational Impacts of Freeway Variable Speed Limit Systems” (J. Transp. Engr. 2011:
Suppose only 70% of all drivers in a state regularly wear a seat belt. A random sample of 500 drivers is selected. What is the probability thata. Between 320 and 370 (inclusive) of the drivers in the sample regularly wear a seat belt?b. Fewer than 325 of those in the sample regularly wear a seat
Suppose X ∼ N(μ, σ).a. Show via integration that E(X) = μ.b. Show via integration that V(X) =σ2.
The moment generating function can be used to find the mean and variance of the normal distribution.a. Use derivatives of MX(t) to verify that E(X) = μ and V(X) = σ2.b. Repeat (a) using RX(t) = ln[MX(t)], and compare with part (a) in terms of effort.
Let X have a standard gamma distribution with α = 7. Evaluate the following:a. P(X ≤ 5)b. P(X < 5)c. P(X > 8)d. P(3 ≤ X ≤ 8)e. P(3 < X < 8)f. P(X < 4 or X > 6)
Suppose the time spent by a randomly selected student at a campus computer laboratory laboratory has a gamma distribution with mean 20 min and variance 80 min2.a. What are the values of α and β?b. What is the probability that a student uses the laboratory for at most 24 min?c. What is the
The article “Numerical Prediction of Surface Wear and Roughness Parameters During Running-In for Line Contacts Under Mixed Mixed Lubrication” (J. Tribol., Nov. 2018) proposes probability models for several variables that arise in studying the wear of mechanical components like gears and piston
In studies of anticancer drugs it was found that if if mice are injected with cancer cells, the survival time can be modeled with the exponential distribution. Without treatment the expected survival time was 10 h. What is the probability thata. A randomly selected mouse will survive at least 8 h?
Determine the mean and variance of the gamma distribution by differentiating the moment generating function MX(t).
Many U.S. railroad tracks were built using A7 steel, and there is renewed interest in the properties of this metal. The article “Stress-State, Temperature, and Strain Rate Dependence of Vintage ASTM A7 Steel” (J. Engr. Mater. Tech. 2019) describes, among other things, the distribution of
In recent years the Weibull distribution has been used to model engine emissions of various pollutants. Let X denote the amount of NOx emission (g/gal) from a randomly selected four-stroke engine of a certain type, and suppose that X has a Weibull distribution with α = 2 and β = 10 (suggested by
The accompanying normal probability plot was constructed from a sample of 30 readings on tension for mesh screens behind the surface of video display tubes. Does it appear plausible that the tension distribution is normal? Explain. z percentile 0 -2 200 ***** 250 ..... 300 350 Tension
Let X denote the ultimate tensile strength (ksi) at −200° of a randomly selected steel specimen of a certain type that exhibits “cold brittleness” at low temperatures. Suppose that X has a Weibull distribution with α = 20 and β = 100.a. What is the probability that X is at most 105 ksi?b.
The article “Reducing Uncertainty of Design Floods of Two-Component Mixture Distributions by Utilizing Flood Timescale to Classify Flood Types in Seasonally Snow Covered Region” (J. Hydrol. 2019: 588–608) reports the accompanying data on annual precipitation (mm/yr) at 34 watersheds in
Relative to the winning time, the time X of another runner in a pdf ƒX(x) = 2/x3, x > 1. The reciprocal Y = 1/X represents the ratio of the time for the winner divided by the time of the other runner. Find the pdf of Y. Explain why Y also represents the speed of the other runner relative to the
Let X be the fuel efficiency in miles per gallon of an extremely inefficient vehicle (a military tank, perhaps?), and suppose X has the pdf ƒX(x) = 2x, 0 < x < 1. Determine the pdf of Y = 1/X, which is fuel efficiency in gallons per mile.
If the length of a side of a square X is random with the pdf ƒX(x) = x/8, 0 < x < 4, and Y is the area of the square, find the pdf of Y.
Teresa and Peter each have a fair coin. Teresa tosses her coin repeatedly until obtaining the sequence HTT. Peter tosses his coin until the sequence HTH is obtained.a. Write a program to simulate Teresa’s coin tossing and, separately, Peter’s. Your program should keep track of the number of
A 40-question multiple-choice exam is sometimes administered in lower-level statistics courses. The exam has a peculiar feature: 10 of the questions have two options, 13 have three options, 13 have four options, and the other 4 have five options. (FYI, this is completely real!) What is the
Major League Baseball teams (usually) play a 162-game season, during which fans are often excited by long winning streaks and frustrated by long losing streaks. But how unusual are these streaks, really? How long a streak would you expect if the team’s performance were independent from game to
The famous Birthday Problem was presented in Example 2.22. Now suppose you have 500 Facebook friends. Make the same assumptions here as in the Birthday Problem.a. Write a program to estimate the probability that, on at least one day during the year, Facebook tells you three (or more) of your
Consider the following game: you begin with $20. You flip a fair coin, winning $10 if the coin lands heads and losing $10 if the coin lands tails. Play continues until you either go broke or have $100 (i.e., a net profit of $80). Write a simulation program to estimate:a. The probability you win the
Twenty faculty members in a certain department have just participated in a department chair election. Suppose that candidate A has received 12 of the votes and candidate B the other 8 votes. If the ballots are opened one by one in random order and the candidate selected on each ballot is recorded,
Simulation can be used to estimate numerical constants, such as π. Here’s one approach: consider the part of a disk of radius 1 that lies in the first quadrant (a quarter-circle). Imagine two random numbers, x and y, both between 0 and 1. The pair (x, y) lies somewhere in the first quadrant; let
Consider the quadratic equation ax2 + bx + c = 0. Suppose that a, b, and c are random numbers between 0 and 1 (like those produced by an RNG). Estimate the probability that the roots of this quadratic equation are real. This probability can be computed exactly using methods from Chapter 5, but a
The undergraduate statistics club at a certain university has 24 members.a. All 24 members of the club are eligible to attend a conference next week, but they can only afford to send 4 people. In how many possible ways could 4 attendees be selected?b. All club members are eligible for any of the
A small manufacturing company will start operating a night shift. There are 20 machinists employed by the company. a. If a night crew consists of 3 machinists, how many different crews are possible?b. If the machinists are ranked 1, 2, …, 20 in order of competence, how many of these crews
Refer to the previous exercise. If at each stage the person who currently “has” the rumor does not know who has already heard it and selects the next recipient at random from all five possible individuals, what is the probability that F has still not heard the rumor after it has been told ten
Consider the Coupon Collector’s Problem: 10 different coupons are distributed into cereal boxes, one per box, so that any randomly selected box is equally likely to have any of the 10 coupons inside. Write a program to simulate the process of buying cereal boxes until all 10 distinct coupons have
Consider the following famous puzzle from the early days of probability, investigatedby Pascal and Fermat. Which of the following events is more likely: to roll at least one 6 in four rolls of a fair die, or to roll at least one double-6 in 24 rolls of two fair dice?a. Write a program to simulate a
The Problem of the Points. Pascal and Fermat also explored a question concerning how to divide the stakes in a game that has been interrupted. Suppose two players, Blaise and Pierre, are playing a game where the winner is the first to achieve a certain number of points. The game gets interrupted at
Consider a woman whose brother is afflicted with hemophilia, which implies that the woman’s mother has the hemophilia gene on one of her two X chromosomes (almost surely not both, since that is generally fatal). Thus there is a 50–50 chance that the woman’s mother has passed on the bad gene
An event A is said to attract event B if P(B | A) > P(B) and repel B if P(B | A) < P(B). (This refines the notion of dependent events by specifying whether A makes B more likely or less likely to occur.)a. Show that if A attracts B, then A repels B′.b. Show that if A attracts B, then A′
A fair coin is tossed repeatedly until either the sequence TTH or the sequence THT is observed. Let B be the event that stopping occurs because TTH was observed (i.e., that TTH is observed before THT). Calculate P(B).
A concrete beam may fail either by shear (S) or flexure (F). Suppose that three failed beams are randomly selected and the type of failure is determined for each one. Let X = the number of beams among the three selected that failed by shear. List each outcome in the sample space along with the
Jurors may be a priori biased for or against the prosecution in a criminal trial. Each juror is questioned by both the prosecution and the defense (the voir dire process), but this may not reveal bias. Even if bias is revealed, the judge may not excuse the juror for cause because of the narrow
Gambler’s Ruin. Allan and Beth currently have $2 and $3, respectively. A fair coin is tossed. If the result of the toss is H, Allan wins $1 from Beth, whereas if the coin toss results in T, then Beth wins $1 from Allan This process is then repeated, with a coin toss followed by the exchange of
The pmf for X = the number of major defects on a randomly selected appliance of a certain type isCompute the following:a. E(X)b. V(X) directly from the definitionc. The standard deviation of Xd. V(X) using the shortcut formula 0 1 2 3 4 p(x) .08 .15 .45 .27 .05
A supply company currently has in stock 500 lb of fertilizer, which it sells to customers in 10-pound bags. Let X equal the number of bags purchased by a randomly selected customer. Sales data shows that X has the following pmf:a. Compute the average number of bags bought per customer.b. Determine
In the popular game Plinko on The Price Is Right, contestants drop a circular disk (a “chip”) down a pegged board; the chip bounces down the board and lands in a slot corresponding to one of five dollar mounts. The random variable X = winnings from one chip dropped from the middle slot has
Let X be the number of pumps in use at a gas station, and suppose X has the distribution given by the accompanying table. Determine MX(t) and use it to find E(X) and V(X). X p(x) 0 1 2 .04 .34 .20 3 .20 4 5 .15 .04 6 .03
In flipping a fair coin let X be the number of tosses to get the first head Then p(x) = .5x for x = 1, 2, 3,…. Determine MX(t) and use it to get E(X) and V(X).
For the entry-level employees of a certain fast food chain, the pmf of X = highest grade level completed is specified by p(9) = .01, p(10) = .05, p(11) = .16, and p(12) = .78.a. Determine the moment generating function of this distribution.b. Use (a) to determine the mean and variance of this
Calculate the skewness coefficient for each of the distributions in the previous four exercises Do those agree with the “shape” of each distribution?
Given MX(t) = .2 + .3et + .5e3t, find p(x), E(X), V(X).
Three plants manufacture hard drives and ship them to a warehouse for distribution. Plant I produces 54% of the warehouse’s inventory with a 4% defect rate. Plant II produces 35% of the warehouse’s inventory with an 8% defect rate. Plant III produces the remainder of the warehouse’s inventory
A popular Dilbert cartoon strip (popular among statisticians, anyway) shows an allegedly “random” number generator producing the sequence 999999 with the accompanying comment, “That’s the problem with randomness: you can never be sure.” Most people would agree that 999999 seems less
The statistics department at the authors’ university participates in an annual volleyball tournament. Suppose that all 16 department members are willing to play.a. How many different six-person volleyball rosters could be generated? (That is, how many years could the department participate in the
A local bar stocks 12 American beers, 8 Mexican beers, and 9 German beers. You ask the bartender to pick out a five-beer “sampler” for you. Assume the bartender makes the five selections at random and without replacement.a. What is the probability you get at least four American beers?b. What is
A particular iPod playlist contains 100 songs, of which 10 are by the Beatles. Suppose the shuffle feature is used to play the songs in random order (the randomness of the shuffling process is investigated in “Does Your iPod Really Play Favorites?” (The Amer. Statistician 2009: 263–268)).
a. Beethoven wrote 9 symphonies and Mozart wrote 27 piano concertos. If a university radio station announcer wishes to play first a Beethoven symphony and then a Mozart concerto, in how many ways can this be done?b. The station manager decides that on each successive night (7 days per week), a
The College of Science Student Council has one representative from each of the five science departments (biology, chemistry, statistics, 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
Two voters, Al and Bill, are each choosing between one of three candidates — 1, 2, and 3 — who are running for city council. An experimental outcome specifies both Al’s choice and Bill’s choice, e.g., the pair (3, 2).a. List all elements of S.b. List all outcomes in the event A that Al and
Ann and Bev have each applied for several jobs at a local university. 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 following events:a. Ann is hired but not Bev.b. At least one of them is hired.c. Exactly one of them is hired.
Construct a stem-and-leaf display for the given batch of exam scores, repeating each stem twice (so, 6L through 9H). What feature of the data is highlighted by this display? 93 64 67 72 70 66 74 82 85 72 80 70 68 90 82 69 74 89 80 89 81 81 71 81 95 84 81 85 98 84 63 69 66 60 72 87 85 81 83 88
The article “Is There a Market for Functional Wines? Consumer Preferences and Willingness to Pay for Resveratrol-Enriched Red Wine” (Food Quality and Preference 2008: 360–371) included the following information for a variety of Spanish wines:a. Region of originb. Price of the wine, in eurosc.
The following data on crack depth (μm) was read from a graph in the article “Effects of Electropolishing on Corrosion and Stress Corrosion Cracking of Alloy 182 in High Temperature Water” (Corrosion Sci. 2017: 1– 10)a. Construct a stem-and-leaf display of the data.b. What is a typical, or
The authors of the article cited in the previous exercise surveyed 300 wine consumers, each of whom tasted two different wines. For each individual in the study, the following information was recorded:a. Genderb. Age, in yearsc. Monthly income, in eurosd. Educational level (primary, secondary, or
The 15th Annual Demographia International Housing Affordability Survey: 2019 reports the “median multiple,” the ratio of median home price to median household income, for 188 metropolitan areas in the United States. (A higher median multiple means that it’s harder for residents of that area
The report “Congestion Reduction Strategies” (Texas Transportation Institute, 2005) investigated how much additional time (in hours, per year per traveler) drivers spend in traffic during peak hours for a sample of urban areas. Data on “large” (e.g., Denver) and “very large” (e.g.,
Lotka’s law is used in library science to describe the productivity of authors in a given field. The article “Lotka’s Law and Productivity Patterns of Authors in Biomedical Science in Nigeria onHIV/AIDS: a Bibliometric Approach” (The Electronic Library 2016: 789–807) provides the
The article “Ecological Determinants of Herd Size in the Thorncraft’s Giraffe of Zambia” (Afric. J. Ecol. 2010: 962–971) gave the following data (read from a graph) on herd size for a sample of 1570 herds over a 34-year period.a. What proportion of the sampled herds had just one giraffe?b.
America used to be number 1 in the world for percentage of adults with four-year degrees, but it has recently dropped to 19th. Here is data on the percentage of adults age 25 or older in each state who had a four-year degree as of 2015 (listed in alphabetical order, with the District of Columbia
Tire pressure monitoring systems are increasingly common for vehicles, but such systems rarely include checking the spare tire. The article “A Statistical Study of Tire Pressures on Road-Going Vehicles” (SAE Intl. J. Passeng. Cars—Mech. Systems 2016) provided the following data on the amount
The report “Majoring in Money 2019” (Sallie Mae) provides the following relative frequency distribution for the credit card balance of a nationally representative sample of n = 464 college students:a. Approximately how many students in the survey reported a $0 credit card balance?b. What
The College Board reports the following Total SAT score distribution for 2018, the first year of the “new” SAT format:a. Create a histogram of this data. Comment on its features.b. What is a typical, or representative, Total SAT score?c. What proportion of students in 2018 scored between 800
Consider the following data on type 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 various categories, and construct a bar chart.
The report “Motorcycle Helmet Use in 2005—Overall Results” (NHTSA August 2005) included observations made on the helmet use of 1700 motorcyclists across the country. The data is summarized in the accompanying table. (A “noncompliant helmet” failed to meet U.S. Department of Transportation
The author of the article “Food and Eating on Television: Impacts and Influences” (Nutr. and Food Sci. 2000: 24–29) examined hundreds of hours of BBC television footage and categorized food images for both TV programs and commercials. The data presented here is consistent with information in
The website realtor.com listed the following sale prices (in $1000s) for a sample of 10 homes sold in 2019 in Los Osos, CA (home town of two of the authors):a. Calculate and interpret the sample mean and median.b. Suppose the second observation was 930 instead of 830. How would that affect the mean
Super Bowl LIII was the lowest scoring (and, to many, the least exciting) Super Bowl of all time. During the game, Los Angeles Rams running back Todd Gurley had just 10 rushing plays, resulting in the following gains in yards:a. Determine the value of the mean.b. Determine the value of the median.
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
Here is the average weekday circulation (paper plus digital subscriptions) for the top 20 newspapers in the country (247wallst.- com, January 24, 2017):a. Which value, the mean or the median, do you anticipate will be higher? Why?b. Calculate the mean and median for this data. 2,237,601 512,118
An article in the Amer. J. Enol. and Viti. (2006: 486–490) includes the following alcohol content measurements (%) for a sample of n = 35 port wines.a. Graph the data. Based on the graph, what is a representative value for the alcohol content in port wines?b. Calculate the mean and the median.
Here is the data on fibrinogen levels (mg/dl) for 10 healthy dogs and 11 dogs with chronic kidney disease discussed in Example 1.18:a. For the data on the 10 healthy dogs, calculate the range, variance, standard deviation, quartiles, and interquartile range.b. Repeat part (a) for the 11 dogs with
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:Compute the following:a. The sample
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. Mater. Civil Engr. 1996: 153–156):a. What are the values of the sample mean and sample
Exercise 46 presented a sample of 26 escape times for oil workers in a simulated exercise. Calculate and interpret the sample standard deviation.Data From Exercises 46 A sample of 26 offshore oil workers took part in a simulated escape exercise, resulting in the accompanying data on time (sec)
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):
Acrylamide is a potential carcinogen that forms in certain foods, such as potato chips and French fries. The FDA analyzed McDonald’s French fries purchased at seven different locations; the following are the resulting acrylamide levels (in micrograms per kg of food):Calculate ∑xi and ∑
The US Women’s Swimming Team won the 1500 m relay at the 2016 Olympic Games. Here are the completion times, in seconds, for all eight teams that competed in the finals:a. Calculate the sample variance and standard deviation.b. If the observations were re-expressed in minutes, what would be the
Reconsider the court awards data presented in Exercise 59.a. What are the values of the quartiles, and what is the value of the iqr?b. How large or small does an observation have to be to qualify as an outlier? As an extreme outlier? c. Construct a boxplot, and comment on its features.Data
Many people who believe they may be suffering from the flu visit emergency rooms, where they are subjected to long waits and may expose others or themselves be exposed to various diseases. The article “Drive-Through Medicine: A Novel Proposal for the Rapid Evaluation of Patients During an
Here is summary information on the alcohol percentage for a sample of 25 beers:The bottom three are 3.20 (Heineken Premium Light), 3.50 (Amstel light), 4.03 (Shiner Light) and the top three are 7.50 (Terrapin All-American Imperial Pilsner), 9.10 (Great Divide Hercules Double IPA), 11.60 (Rogue
At the beginning of the 2007 baseball season each American League team had nine starting position players (this includes the designated hitter but not the pitcher). Here are the salaries for the New York Yankees and the Cleveland Indians in thousands of dollars: Construct a comparative boxplot
The article “E-cigarettes as a Source of Toxic and Potentially Carcinogenic Metals” (Environ. Res. 2017: 221–225) reports the concentration (μg/L) of cadmium, chromium, lead, manganese, and nickel in 10 cartidges for each of five e-cigarette brands. Here are the lead levels in the 50
Three different C2F6 flow rates (SCCM) were considered in an experiment to investigate the effect of flow rate on the uniformity (%) of the etch on a silicon wafer used in the manufacture of integrated circuits, resulting in the following data:Compare and contrast the uniformity observations
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
The article “Motor Vehicle Emissions Variability” (J. Air Waste Manag. Assoc. 1996: 667–675) reported the following hydrocarbon and carbon dioxide measurements using the Federal Testing Procedure for emissions-testing, applied four times each to the same car:a. Compute the sample standard
The cost-to-charge ratio for a hospital is the ratio of the actual cost of care to what the hospital charges for that care. In 2008, the Kentucky Department of Health and Family Services reported the following cost-to-charge ratios, expressed as percents, for 116 Kentucky hospitals:(For example, a
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). It was thought that private speech would be related to IQ, because IQ is supposed to
In recent years, some evidence suggests that high indoor radon concentration may be linked to the development of childhood cancers, but many health professionals remain unconvinced. The article “Indoor Radon and Childhood Cancer” (Lancet 1991: 1537–1538) presented the accompanying data on

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