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inferential statistics
Probability And Statistics For Engineers And Scientists 9th Global Edition Ronald E. Walpole, Raymond Myers, Sharon L. Myers, Keying E. Ye - Solutions
5.55 The probability that a student pilot passes the written test for a private pilot’s license is 0.7. Find the probability that a given student will pass the test(a) on the third try;(b) before the fourth try.
5.54 According to a study published by a group of University of Massachusetts sociologists, about twothirds of the 20 million persons in this country who take Valium are women. Assuming this figure to be a valid estimate, find the probability that on a given day the fifth prescription written by a
5.53 An inventory study determines that, on average, demands for a particular item at a warehouse are made 5 times per day. What is the probability that on a given day this item is requested(a) more than 5 times?(b) not at all?
5.52 A scientist inoculates mice, one at a time, with a disease germ until he finds 2 that have contracted the disease. If the probability of contracting the disease is 1/6, what is the probability that 7 mice are required?
5.51 Three people toss a fair coin and the odd one pays for coffee. If the coins all turn up the same, they are tossed again. Find the probability that fewer than 3 tosses are needed.
5.50 Find the probability that a person flipping a coin gets(a) the third head on the seventh flip;(b) the first head on the fourth flip.
5.49 The probability that a person living in a certain city owns a dog is estimated to be 0.4. Find the probability that the tenth person randomly interviewed in that city is the fifth one to own a dog.
5.48 Every hour, 10,000 cans of soda are filled by a machine, among which 200 underfilled cans are produced.Each hour, a sample of 30 cans is randomly selected and the number of ounces of soda per can is checked. Denote by X the number of cans selected that are underfilled. Find the probability
5.47 A government task force suspects that some manufacturing companies are in violation of federal pollution regulations with regard to dumping a certain type of product. Twenty firms are under suspicion but not all can be inspected. Suppose that 3 of the firms are in violation.(a) What is the
5.46 A large company has an inspection system for the batches of small compressors purchased from vendors.A batch typically contains 15 compressors. In the inspection system, a random sample of 5 is selected and all are tested. Suppose there are 2 faulty compressors in the batch of 15.(a) What is
5.45 Biologists doing studies in a particular environment often tag and release subjects in order to estimate the size of a population or the prevalence of certain features in the population. Ten animals of a certain population thought to be extinct (or near extinction)are caught, tagged, and
5.44 An urn contains 3 green balls, 2 blue balls, and 4 red balls. In a random sample of 5 balls, find the probability that both blue balls and at least 1 red ball are selected.
5.43 A foreign student club lists as its members 2 Canadians, 3 Japanese, 5 Italians, and 2 Germans. If a committee of 4 is selected at random, find the probability that(a) all nationalities are represented;(b) all nationalities except Italian are represented.
5.42 Find the probability of being dealt a bridge hand of 13 cards containing 5 spades, 2 hearts, 3 diamonds, and 3 clubs.
5.41 A nationwide survey of 17,000 college seniors by the University of Michigan revealed that almost 70%disapprove of daily pot smoking. If 18 of these seniors are selected at random and asked their opinion, what is the probability that more than 8 but fewer than 14 disapprove of smoking pot daily?
5.40 It is estimated that 4000 of the 10,000 voting residents of a town are against a new sales tax. If 15 eligible voters are selected at random and asked their opinion, what is the probability that at most 7 favor the new tax?
5.39 An annexation suit against a county subdivision of 1200 residences is being considered by a neighboring city. If the occupants of half the residences object to being annexed, what is the probability that in a random sample of 10 at least 4 favor the annexation suit?
5.38 Among 150 IRS employees in a large city, only 30 are women. If 10 of the employees are chosen at random to provide free tax assistance for the residents of this city, use the binomial approximation to the hypergeometric distribution to find the probability that at least 2 women are selected.
5.37 Suppose that the manufacturing company of Exercise 5.36 decides to change its acceptance scheme.Under the new scheme, an inspector takes 1 item at random, inspects it, and then replaces it in the box;a second inspector does likewise. Finally, a third inspector goes through the same procedure.
5.36 A manufacturing company uses an acceptance scheme on items from a production line before they are shipped. The plan is a two-stage one. Boxes of 25 items are readied for shipment, and a sample of 3 items is tested for defectives. If any defectives are found, the entire box is sent back for
5.35 A company is interested in evaluating its current inspection procedure for shipments of 50 identical items. The procedure is to take a sample of 5 and pass the shipment if no more than 2 are found to be defective. What proportion of shipments with 20% defectives will be accepted?
5.34 What is the probability that a waitress will refuse to serve alcoholic beverages to only 2 minors if she randomly checks the IDs of 5 among 9 students, 3 of whom are minors?
5.33 If 7 cards are dealt from an ordinary deck of 52 playing cards, what is the probability that(a) exactly 2 of them will be face cards?(b) at least 1 of them will be a queen?
5.32 From a lot of 10 missiles, 3 are selected at random and fired. If the lot contains 4 defective missiles that will not fire, what is the probability that(a) all 3 will fire?(b) at most 2 will not fire?
5.31 A random committee of size 4 is selected from 4 doctors and 2 nurses. Write a formula for the probability distribution of the random variable X representing the number of doctors on the committee. Find P(2 ≤ X ≤ 3).
5.30 To avoid detection at customs, a traveler places 5 narcotic tablets in a bottle containing 10 vitamin tablets that are similar in appearance. If the customs official selects 2 of the tablets at random for analysis, what is the probability that the traveler will be arrested for illegal
5.29 A homeowner plants 5 bulbs selected at random from a box containing 5 tulip bulbs and 4 daffodil bulbs. What is the probability that he planted 2 daffodil bulbs and 3 tulip bulbs?
5.28 A manufacturer knows that on average 20% of the electric toasters produced require repairs within 1 year after they are sold. When 20 toasters are randomly selected, find appropriate numbers x and y such that(a) the probability that at least x of them will require repairs is less than 0.5;(b)
5.27 If the probability that a fluorescent light has a useful life of at least 800 hours is 0.9, find the probabilities that among 20 such lights(a) exactly 18 will have a useful life of at least 800 hours;(b) at least 15 will have a useful life of at least 800 hours;(c) at least 2 will not have a
5.26 Assuming that 6 in 10 automobile accidents are due mainly to a speed violation, find the probability that among 8 automobile accidents, 6 will be due mainly to a speed violation(a) by using the formula for the binomial distribution;(b) by using Table A.1.
5.25 Suppose that for a very large shipment of integrated-circuit chips, the probability of failure for any one chip is 0.10. Assuming that the assumptions underlying the binomial distributions are met, find the probability that at most 3 chips fail in a random sample of 20.
5.24 A safety engineer claims that only 40% of all workers wear safety helmets when they eat lunch at the workplace. Assuming that this claim is right, find the probability that 4 of 6 workers randomly chosen will be wearing their helmets while having lunch at the workplace.
5.23 The probabilities are 0.4, 0.1, 0.3, and 0.2, respectively, that a delegate to a certain convention arrived by air, bus, automobile, or train. What is the probability that among 10 delegates randomly selected at this convention, 3 arrived by air, 3 arrived by bus, 2 arrived by automobile, and
5.22 According to a genetics theory, a certain cross of guinea pigs will result in red, black, and white offspring in the ratio 8:4:4. Find the probability that among 8 offspring, 5 will be red, 2 black, and 1 white.
5.21 The surface of a circular dart board has a small center circle called the bull’s-eye and 20 pie-shaped regions numbered from 1 to 20. Each of the pie-shaped regions is further divided into three parts such that a person throwing a dart that lands in a specific region scores the value of the
5.20 According to USA Today (March 18, 1997), of 4 million workers in the general workforce, 5.8% tested positive for drugs. Of those testing positive, 22.5%were cocaine users and 54.4% marijuana users.(a) What is the probability that of 10 workers testing positive, 2 are cocaine users, 5 are
5.19 As a student drives to school, he encounters a traffic signal. This traffic signal stays green for 35 seconds, yellow for 5 seconds, and red for 60 seconds. Assume that the student goes to school each weekday between 8:00 and 8:30 a.m. Let X1 be the number of times he encounters a green light,
5.18 (a) In Exercise 5.9, how many of the 15 trucks would you expect to have blowouts?(b) What is the variance of the number of blowouts experienced by the 15 trucks? What does that mean?
5.17 If X represents the number of people in Exercise 5.13 who believe that antidepressants do not cure but only cover up the real problem, find the mean and variance of X when 7 people are selected at random.
5.16 Suppose that airplane engines operate independently and fail with probability equal to 0.4. Assuming that a plane makes a safe flight if at least one-half of its engines run, determine whether a 4-engine plane or a 2-engine plane has the higher probability for a successful flight.
5.15 It is known that 60% of mice inoculated with a serum are protected from a certain disease. If 5 mice are inoculated, find the probability that(a) none contracts the disease;(b) fewer than 2 contract the disease;(c) more than 3 contract the disease.
5.14 The percentage of wins for the Chicago Bulls basketball team going into the playoffs for the 1996–97 season was 87.7. Round the 87.7 to 90 in order to use Table A.1.(a) What is the probability that the Bulls sweep (4-0)the initial best-of-7 playoff series?(b) What is the probability that the
5.13 A national study that examined attitudes about antidepressants revealed that approximately 70% of respondents believe “antidepressants do not really cure anything, they just cover up the real trouble.” According to this study, what is the probability that at least 3 of the next 5 people
5.12 In a survey of customers for a departmental store, it is reported that 75% of the customers are from a high economic group. What is the probability that fewer than 4 of the next 9 customers do not belong to this income group?
5.11 The probability that a patient recovers from a delicate heart operation is 0.9. What is the probability that exactly 4 of the next 6 patients having this operation survive?
5.10 In a certain fitness test for athletes, it is found that 10% of the athletes fail to complete the test. Of the next 15 athletes tested, find the probability that(a) from 3 to 6 fail;(b) fewer than 4 fail;(c) more than 5 fail.
5.9 In testing a certain kind of truck tire over rugged terrain, it is found that 25% of the trucks fail to complete the test run without a blowout. Of the next 15 trucks tested, find the probability that(a) from 3 to 6 have blowouts;(b) fewer than 4 have blowouts;(c) more than 5 have blowouts.
5.8 According to a study published by a group of University of Massachusetts sociologists, approximately 60% of the Valium users in the state of Massachusetts first took Valium for psychological problems. Find the probability that among the next 8 users from this state who are interviewed,(a)
5.7 One prominent physician claims that 70% of those with lung cancer are chain smokers. If his assertion is correct,(a) find the probability that of 10 such patients recently admitted to a hospital, fewer than half are chain smokers;(b) find the probability that of 20 such patients recently
5.6 According to a survey by the Administrative Management Society, one-half of U.S. companies give employees 4 weeks of vacation after they have been with the company for 15 years. Find the probability that among 8 companies surveyed at random, the number that give employees 4 weeks of vacation
5.5 According to Chemical Engineering Progress(November 1990), approximately 30% of all pipework failures in chemical plants are caused by operator error.(a) What is the probability that out of the next 20 pipework failures at least 10 are due to operator error?(b) What is the probability that no
5.4 In a certain city district, the need for money to buy drugs is stated as the reason for 75% of all thefts.Find the probability that among the next 5 theft cases reported in this district,(a) exactly 2 resulted from the need for money to buy drugs;(b) at most 3 resulted from the need for money
5.3 A student is selected from a class of 100 students to represent the class for a competitive event, by selecting a tag at random, from a box containing 100 tags numbered 1 to 100. Find the formula for the probability distribution of X representing the number on the tag drawn. What is the
5.2 Twelve people are given two identical speakers, which they are asked to listen to for differences, if any.Suppose that these people answer simply by guessing.Find the probability that three people claim to have heard a difference between the two speakers.
5.1 A random variable X that assumes the values x1, x2 , . . . , xk is called a discrete uniform random variable if its probability mass function is f(x) = 1 k for all of x1, x2 , . . . , xk and 0 otherwise. Find the mean and variance of X.
4.102 Project: Let X = number of hours each student in the class slept the night before. Create a discrete variable by using the following arbitrary intervals:X < 3, 3 ≤X < 6, 6 ≤X < 9, and X ≥ 9.(a) Estimate the probability distribution for X.(b) Calculate the estimated mean and variance for
4.101 Consider Review Exercise 3.73 on page 128. It involved Y , the proportion of impurities in a batch, and the density function is given by f(y) =10(1 − y)9 , 0 ≤ y ≤ 1, 0, elsewhere.(a) Find the expected percentage of impurities.(b) Find the expected value of the proportion of quality
4.100 As we shall illustrate in Chapter 12, statistical methods associated with linear and nonlinear models are very important. In fact, exponential functions are often used in a wide variety of scientific and engineering problems. Consider a model that is fit to a set of data involving measured
4.99 Consider a ferry that can carry both buses and cars across a waterway. Each trip costs the owner approximately$10. The fee for cars is $3 and the fee for buses is $8. Let X and Y denote the number of buses and cars, respectively, carried on a given trip. The joint distribution of X and Y is
4.98 A convenience store has two separate locations where customers can be checked out as they leave.These locations each have two cash registers and two employees who check out customers. Let X be the number of cash registers being used at a particular time for location 1 and Y the number being
4.97 A delivery truck travels from point A to point B and back using the same route each day. There are four traffic lights on the route. Let X1 denote the number of red lights the truck encounters going from A to B and X2 denote the number encountered on the return trip. Data collected over a long
4.96 It is known through data collection and considerable research that the amount of time in seconds that a certain employee of a company is late for work is a random variable X with density function f(x) =3(4)(503 ) (502 − x2 ), −50 ≤ x ≤ 50, 0, elsewhere.In other words, he not only is
4.95 In business, it is important to plan and carry out research in order to anticipate what will occur at the end of the year. Research suggests that the profit (loss)spectrum for a certain company, with corresponding probabilities, is as follows:Profit Probability −$15, 000 0.05 $0 0.15 $15,000
4.94 In a support system in the U.S. space program, a single crucial component works only 85% of the time.In order to enhance the reliability of the system, it is decided that 3 components will be installed in parallel such that the system fails only if they all fail. Assume the components act
4.93 A company’s marketing and accounting departments have determined that if the company markets its newly developed product, the contribution of the product to the firm’s profit during the next 6 months will be described by the following:Profit Contribution Probability−$5, 000$10, 000$30,
4.92 Consider Exercise 4.10 on page 137. Can it be said that the ratings given by the two experts are independent?Explain why or why not.
4.91 A dealer’s profit, in units of $5000, on a new automobile is a random variable X having density function f(x) =6x(1 − x), 0 ≤ x ≤ 1, 0, elsewhere.(a) Find the variance of the dealer’s profit.(b) Demonstrate that Chebyshev’s theorem holds for k = 2 with the density function
4.90 Consider random variables X and Y of Exercise 4.63 on page 158. Compute ρX Y .
4.89 Consider the joint density function f(x, y) =16y x3 , x>2, 0 < y < 1, 0, elsewhere.Compute the correlation coefficient ρX Y .
4.88 Consider the density function of Review Exercise 4.85. Demonstrate that Chebyshev’s theorem holds for k = 2 and k = 3.
4.87 Show that Cov(aX, bY ) = ab Cov(X, Y ).
4.86 Referring to the random variables whose joint density function is given in Exercise 3.40 on page 125,(a) find μX and μY ;(b) find E[(X + Y )/2].
4.85 Suppose it is known that the life X of a particular compressor, in hours, has the density function f(x) = 1 900 e−x/900, x>0, 0, elsewhere.(a) Find the mean life of the compressor.(b) Find E(X2 ).(c) Find the variance and standard deviation of the random variable X.
4.84 Referring to the random variables whose joint probability density function is given in Exercise 3.41 on page 125, find the expected weight for the sum of the creams and toffees if one purchased a box of these chocolates.
4.83 Referring to the random variables whose joint density function is given in Exercise 3.41 on page 125, find the covariance between the weight of the creams and the weight of the toffees in these boxes of chocolates.
4.82 Assume the length X, in minutes, of a particular type of telephone conversation is a random variable with probability density function f(x) = 13 e −x/3, x>0, 0, elsewhere.(a) Determine the mean length E(X) of this type of telephone conversation.(b) Find the variance and standard deviation
4.81 Referring to the random variables whose joint probability density function is given in Exercise 3.47 on page 125, find the average amount of kerosene left in the tank at the end of the day.
4.79 Prove Chebyshev’s theorem.4.80 Find the covariance of random variables X and Y having the joint probability density function f(x, y) =x + y, 0 < x < 1, 0 < y < 1, 0, elsewhere
4.78 Compute P(μ − 2σ < X < μ + 2σ), where X has the density function f(x) =30x2 (1 − x)2 , 0 < x < 1, 0, elsewhere and compare with the result given in Chebyshev’s theorem.
4.77 A random variable X has a mean μ = 10 and a variance σ2 = 4. Using Chebyshev’s theorem, find(a) P(|X − 10| ≥ 4);(b) P(|X − 10| < 4);(c) P(4 < X < 16);(d) the value of the constant c such that P(|X −10| ≥c) ≤ 0.05.
4.76 Suppose 120 new jobs are opening up at an automobile manufacturing plant, and 1000 applicants show up for the 70 positions. To select the best 120 from among the applicants, the company gives a test that covers mechanical skill, manual dexterity, and mathematical ability. The mean grade on
4.75 An electrical firm manufactures a 100-watt light bulb, which, according to specifications written on the package, has a mean life of 800 hours with a standard deviation of 50 hours. At most, what percentage of the bulbs fail to last even 700 hours? Assume that the distribution is symmetric
4.74 Consider again the situation of Exercise 4.72. It is required to find Var(eY ). Use Theorems 4.2 and 4.3 and define Z = eY . Thus, use the conditions of Exercise 4.73 to find Var(Z) = E(Z2 ) − [E(Z)]2 .Then do it not by using f(y), but rather by using the first-order Taylor series
4.73 For the situation in Exercise 4.72, compute E(eY ) using Theorem 4.1, that is, by using E(eY ) = 8 7ey f(y) dy.Then compute E(eY ) not by using f(y), but rather by using the second-order adjustment to the first-order approximation of E(eY ). Comment.
4.72 A manufacturing company has developed a machine for cleaning carpet that is fuel-efficient because it delivers carpet cleaner so rapidly. Of interest is a random variable Y , the amount in gallons per minute delivered. It is known that the density function is given by f(y) =1, 7 ≤ y ≤ 8,
4.71 The length of time Y , in minutes, required to generate a human reflex to tear gas has the density function f(y) =12 e−y /2, y>0, 0, elsewhere.(a) What is the mean time to reflex?(b) Find E(Y 2) and V ar(Y ).
4.70 Consider Review Exercise 3.64 on page 127.There are two service lines. The random variables X and Y are the proportions of time that line 1 and line 2 are in use, respectively. The joint probability density function for (X, Y ) is given by f(x, y) =32(x2 + y2 ), 0 ≤ x, y ≤ 1, 0,
4.69 Consider Review Exercise 3.77 on page 128. The random variables X and Y represent the number of vehicles that arrive at two separate street corners during a certain 2-minute period in the day. The joint distribution is f(x, y) =1 4(x+y )9 16, for x = 0, 1, 2, . . . and y = 0, 1, 2, . . .
4.68 The power P, in watts, which is dissipated in an electric circuit with a resistance of 50 ohms is known to be given by P = I2R, where I is current in amperes and R is the resistance. However, I is a random variable with μI = 10 amperes and σ2 I = 0.02 amperes2 .Give numerical approximations
4.67 If the joint density function of X and Y is given by f(x, y) =27(x + 2y), 0 < x < 1, 1 < y < 2, 0, elsewhere.Find the expected value of g(X, Y ) = X Y 4 + X2Y .
4.66 Let X represent the number that occurs when a green die is tossed and Y the number that occurs when a red die is tossed. Find the variance of the random variable(a) 3X − Y ;(b) X + 5Y − 5.
4.65 Let X represent the number that occurs when a red die is tossed and Y the number that occurs when a green die is tossed. Find(a) E(2X + Y );(b) E(X − 2Y );(c) E(2XY ).
4.64 Suppose that X and Y are independent random variables with probability densities and g(x) = 24 x4, x>2, 0, elsewhere, and h(y) =2y, 0 < y < 1, 0, elsewhere.Find the expected value of Z = XY .
4.63 Repeat Exercise 4.62 if X and Y are not independent and σX Y = 1.
4.62 If X and Y are independent random variables with variances σ2 X = 5 and σ2 Y = 3, find the variance of the random variable Z = −2X + 4Y − 3.
4.61 Use Theorem 4.7 to evaluate E(2XY 2 − X2Y )for the joint probability distribution shown in Table 3.1 on page 116.
4.60 Suppose that X and Y are independent random variables having the joint probability distribution xy 2 4 1 0.15 0.10 3 0.25 0.25 5 0.15 0.10 Find(a) E(2X − 3Y );(b) E(XY ).
4.59 If a random variable X is defined such that E[(X − 1)2 ] = 10 and E[(X − 2)2 ] = 5, find μand σ2 .
4.58 The total time, measured in units of 100 hours, that a teenager runs their hair dryer over a period of one year is a continuous random variable X that has the density function f(x) =⎧⎨⎩x, 0 < x < 1, 2 − x, 1 ≤ x < 2, 0, elsewhere.Use Theorem 4.6 to evaluate the mean of the random
4.57 Let X be a random variable with the following probability distribution:x −3 6 9 f(x) 16 13 12 Find E(X) and E(X2 ) and then, using these values, evaluate E[(2X + 1)2 ].
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