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Statistics And Probability With Applications For Engineers And Scientists Using MINITAB R And JMP 2nd Edition Irwin Guttman, Kalanka P. Jayalath, Bhisham C. Gupta - Solutions
3. A sole of a running shoe is built by randomly selecting one layer of material I, one layer of material II, and two layers of material III. The thicknesses of individual 244 6 Distribution of Functions of Random Variables layers of material I, II, and III have distributions with means 0.20, 0.30,
2. In filling soft drinks into 12-oz cans, assume the population of net amounts of drinks generated by the automatic filling machine, adequately calibrated, has a distribution with mean of 12.15 oz and standard deviation 0.1 oz. Assume that the population of aluminum cans used for fillings have a
1. .(a) X, Y, and Z are independent, Poisson random variables with mean 2, 7, and 9, respectively. Determine the mean and variance of the random variable U =5X + 3Y + 8Z.(b) Let (X1, . . . , X5), (Y1, . . . , Y3), and (Z1, . . . , Z8) be random samples from the three Poisson populations with mean
69. The following data give time T to failure of a drug in suspension form:5 7 9 13 24 32 35 38 42 47 49 52 The lifetime of the suspension follows the lognormal distribution with parameters μand σ2. Estimate the values of μ and σ2.
68. Refer to Problem 67. Determine the probability that the time to failure of a machine is: (a) less than 200 hours, (b) between 300 and 500 hours, (c) more than 600 hours
67. Suppose that time (in hours) to failure of a machine is modeled by the lognormal with parameters μ and σ2. The failure time of 12 such machines are as follows:269 207 214 254 739 580 267 725 154 306 439 215 Estimate the values of μ and σ2.
66. Assume that the log of failure times are normally distributed, with parameters μ andσ2. A sample of 10 parts selected at random has failure times whose logs are 7.77 8.45 7.59 7.03 7.17 6.46 7.46 9.09 7.81 7.47 Use normal probability paper to determine the approximate values of μ and σ2.
65. Referring to Problem 64:(a) Find the mean and the variance of the random variable X(b) What percentage of the rods is within two standard deviations of the mean?
64. Suppose that X is the length of a rod that is uniformly distributed over the specification limits 19 and 20 cm. Find the probabilities: (a) P(19.2 < X < 19.5), (b) P(X 19.7).
63. A random variable X is distributed uniformly over the interval [0, 20]. Determine the probabilities: (a) P(X 12).
62. The fluid volume of a popular drink in a 12-oz can is normally distributed with mean 12.3 oz and standard deviation 0.2 oz.(a) What is the probability that a randomly selected can has less than 12 oz of drink?(b) What is the probability that a randomly selected can has more than 12.5 oz of
61. Suppose that a random variable X is distributed as binomial with n = 225 and θ = 0.2.Using the normal approximation, find the following probabilities: (a) P(X ≤ 60), (b)P(X ≥ 57), (c) P(80 ≤ X ≤ 100).
60. The time between arrivals of cars in a garage is exponentially distributed with a mean time between arrivals of cars of 30 minutes.(a) What is the probability that the time between arrivals of two successive cars is more than 45 minutes?(b) What is the probability that the time between arrivals
59. Referring to Problem 58:(a) What is the probability that more than two breakdowns occur in 1000 hours?(b) What is the probability that at least two breakdowns occur in 1000 hours?(c) What is the probability that less than two breakdowns occur in 1000 hours?
58. The time (in units of 500 hours) between the two consecutive breakdowns of a machine is modeled by an exponential distribution with mean of 1.2.(a) What is the probability that the second breakdown does not occur for at least 600 hours after the first breakdown? [Note: 600 = (1.2) × 500.](b)
57. Suppose that a random variable X is distributed as an exponential with mean 20.Find the following probabilities:(a) P(X >25)(b) P(15 < X < 25)(c) P(X ≥ 20)
56. The time needed (in hours) for a worker to finish a job is modeled by a lognormal distribution with parameters μ = 2 and σ2 = 4. Find the following probabilities:(a) The worker needs at least 50 hours to finish the job.(b) The worker needs more than 60 hours to finish the job.(c) The worker
55. The life (in hours) of a catalytic converter of a passenger car is modeled by a Weibull distribution with parameters α = 2000 and β = 0.4. (Assume τ = 0).(a) What is the probability that the catalytic converter needs to be replaced before 14,000 hours?(b) What is the probability that the
54. Referring to Problem 53:(a) What is the probability that the dehumidifier fails before 5000 hours?(b) What is the probability that the dehumidifier lasts between 8000 to 12,000 hours?(c) What is the probability that the dehumidifier lasts at least 7000 hours?
53. The life (in hours) of a domestic dehumidifier is modeled by a Weibull distribution with parameters α = 300 and β = 0.25 hour. Assuming τ = 0,(a) What is the mean life of the dehumidifier?(b) What is the variance of the life of the dehumidifier?
52. Suppose that random variable X is distributed as normal with parameters μ = 2 andσ2 = 4. Find the value of x such that (a) P(X ≤ x) = 0.05, (b) P(X ≥ x) = 0.33.
51. Suppose that random variable X is distributed as lognormal with parameters μ = 2 and σ2 = 4 Find the mean and variance of X.226 5 Continuous Random Variables
50. Suppose that random variable X (in hundreds) is distributed as lognormal with parametersμ = 2 and σ2 = 4. Determine the following probabilities: (a) P(X ≤ 750), (b)P(X ≥ 1500).
49. Suppose that random variable X (in thousands) is distributed as lognormal with parameters μ = 5 and σ2 = 9. Determine the following probabilities: (a) P(3500 ≤X ≤ 9500), (b) P(1500 ≤ X ≤ 2500).
48. Suppose that a random variable X (in thousands) is distributed as lognormal with parameters μ = 3 and σ2 = 4. Determine the following probabilities: (a) P(X ≤ 5500),(b) P(X ≥ 2000).
47. Referring to Problem 46:(a) Determine the probability that at least two accidents take place in three months.(b) Determine the probability that less than two accidents take place in two months.
46. The time lapse between two accidents in a large manufacturing plant has an approximately exponential distribution with a mean of two months.(a) What is the probability that the time lapse between two accidents is less than three months?(b) What is the probability that the time lapse between two
45. Referring to Problem 44:(a) Use Chebyshev’s theorem to find an interval that contains at least 88.8% of the waiting times.(b) Determine the actual probability of waiting times to fall in the interval you determined in part (a).
44. The waiting time, say X hours in an emergency room, is distributed as a gamma with mean μ = 2 and variance σ2 = 3.(a) Determine the probability density function for the waiting time.(b) Determine the probability that randomly selected patient has to wait more than 2.5 hours.
43. The time between arrivals (in minutes) of customers at a teller’s window in a bank, is a gamma random variable with γ = 1, λ = 0.1. Find the following probabilities:(a) The time between arrivals of two customers is more than 10 minutes.(b) The time between arrivals of two customers is less
42. Referring to Problem 40, suppose that a customer bought a new car with tires of that brand. Find the probabilities of the following events:(a) All the tires will last at least 50,000 miles.(b) At least one tire will last 50,000 miles or more.(c) None of the tires will last more than 50,000
41. In Problem 39, find the mean and the variance of X, the length of life of the computer chip.
40. If the life (in thousands of miles) of a car tire follows a gamma distribution withγ = 6, λ = 0.1, determine the following probabilities:(a) P(35 < X < 85)(b) P(X >75)(c) P(X
39. Suppose that the length of life (in months) of a computer chip follows a gamma distribution with γ = 4, λ = 0.05. Determine the following probabilities:(a) P(40 < X < 120)(b) P(X >80)(c) P(X
37. Referring to Problem 36, in (a) and (c) find the exact probability P(|X − μ| ≤ 2σ).Then, find the lower bound of these probabilities using Chebyshev’s inequality. Compare the two results and comment.
36. Find the mean and the variance for each of the following probability density functions:(a) f(x) = 1/2, 1 ≤ x ≤ 3, and zero elsewhere.(b) f(x) = θe−θx, x>0, θ > 0, and zero elsewhere.(c) f(x) = 12x2(1 − x), 0 ≤ x ≤ 1, and zero elsewhere.
35. Determine the value of c so that the following functions are probability density functions:(a) f(x) = cx(3 − x), 0 ≤ x ≤ 3(b) f(x) = cx2(3 − x), 0 ≤ x ≤ 3(c) f(x) = cx3(3 − x), 0 ≤ x ≤ 1
34. Determine which of the following functions are probability density functions:(a) f(x) = x(3 − x), 0 ≤ x ≤ 3(b) f(x) = x2(3 − x), 0 ≤ x ≤ 3(c) f(x) = x(3 − x), 0 ≤ x ≤ 2(d) f(x) =1λe−(x−2)/λ, x ≥ 2
33. A continuous random variable X has the probability density function f(x) =3x2, 0 < x < 1 0, otherwise(a) Find the c.d.f. F(x).(b) Find the numerical values of F(1/3), F(9/10), and P(1/3 < X ≤ 1/2).(c) Find the value of a which is such that P(X ≤a) = 1/4 (a is the 25th percentile of X).(d)
32. If a defective spot occurs in a glass disk R inches in radius, assume that it is equally likely to occur anywhere on the disk. Let X be a random variable indicating the distance between the point of occurrence of a defective spot and the center of the disk.(a) Find the expression for F(x) and
31. If X is a continuous random variable with p.d.f. f(x), the pth percentile of x (sometimes called thepth percentile of the population) is defined as that value xp for which P(X ≤ xp) = xp−∞f(x)dx = p/100 The 50th percentile is called the population median. Also, suppose that a continuous
29. Suppose that the length of time X (in months) taken by two different medications, say Lipitor and Zocor, to lower the bad cholesterol (LDL) level by 20 mg/dl can be modeled by two gamma distributions with parameters γ = 3, λ = 1 and γ = 6, λ = 1.5, respectively.(a) Find the mean and
28. Suppose that the lifetime of a serpentine belt of a car is distributed as an exponential random variable with λ = 0.00125.What is the probability that a serpentine belt lasts(a) 700 hours?(b) More than 850 hours?(c) Between 600 and 900 hours?(d) At least 650 hours?
27. The number of years a computer functions is exponentially distributed with λ = 0.1.David bought a five-year-old computer that is functioning well. What is the probability that David’s computer will last another nine years?
26. The time (in hours) needed to finish a paint job of a car is an exponentially distributed random variable with λ = 0.2.(a) Find the probability that a paint job exceeds seven hours.(b) Find the probability that a paint job exceeds seven hours but finishes before 10 hours.(c) Find the
25. In Problem 24, determine the probabilities of the following events:(a) The motor fails before 800 hours.(b) The motor lasts more than 1000 hours.(c) The motor lasts between 1000 and 1500 hours.Review Practice Problems 223
24. Suppose that the life of a motor (in hours) follows the Weibull distribution withα = 1000 and β = 2.0. Determine the mean and the variance of the random variable X.
22. Let X be a random variable distributed as the Weibull distribution with α = 100 andβ = 0.5. Determine the mean and the variance of the random variable X. (Assume that τ , the threshold parameter, has value τ = 0.)
21. Let X be a random variable distributed as an exponential distribution with λ = 2.Determine the probability that the random variable X assumes a value: (a) greater than 1, (b) greater than 2, (c) between 1 and 2, (d) greater than 0.
20. Let X be a random variable distributed by the exponential distribution with λ = 1.5.Determine the probability that the random variable X assumes a value: (a) greater than 2, (b) less than 4, (c) between 2 and 4, (d) less than 0.
19. Let Z be a random variable distributed as the standard normal. Using the normal distribution table (Table A.4) determine the following probabilities: (a) P(Z ≤ 2.11),(b) P(Z ≥ −1.2), (c) P(−1.58 ≤ Z ≤ 2.40), (d) P(Z ≥ 1.96), (e) P(Z ≤ −1.96).
18. Let Z be a random variable distributed as the standard normal. Determine the probability that the random variable Z takes a value (a) within one standard deviation of the mean, (b) within two standard deviation of the mean, (c) within three standard deviation of the mean.
17. A resistor is composed of eight component parts soldered together in series, so that the total resistance of the resistor equals the sum of the resistances of the component parts. Three of the components are drawn from a production lot that has a mean of 200 Ω and a standard deviation of 2 Ω,
16. Suppose that items of a certain kind are counted by weighing a box of 100 of these items. The population of individual items has a mean weight of 1.45 oz and standard deviation of 0.02 oz. A batch of items weighing between 144.5 and 145.5 items is counted as 100 items. What is the
15. An article is made up of three independent parts A, B, and C. The weights of the A’s have an (approximately) normal distribution with mean 2.05 oz and standard deviation 0.03 oz. Those of the B’s have an (approximately) normal distribution with mean 3.10 oz and standard deviation 0.04 oz.
14. A mass-produced laminated item is made up of five layers. A study of the thickness of individual layers shows that each of the two outside layers have mean thickness of 0.062 in., and each of the three middle layers have mean thickness 0.042 in. The standard deviation of thickness of outside
13. A die is rolled 720 times. Using a normal distribution to approximate probabilities, estimate the probability that(a) More than 130 sixes turn up.(b) The number of sixes obtained lie between 100 and 140 inclusive.
12. It is known that the probability of dealing a bridge hand with at least one ace is approximately 0.7. If a person plays 100 hands of bridge, what is the approximate probability(a) That the number of hands he/she receives containing at least one ace is between 60 and 80 inclusive?(b) That he/she
11. If 10% of the articles produced by a given process are defective:(a) What is the probability (approximately) that more than 15% of a random sample of 400 items will be defective?(b) For what value of K is the probability (approximately) 0.90 that the number of defectives in a sample of 400 lies
10. If a sack of 400 nickels is emptied on a table and spread out, determine:(a) The probability (approximately) of getting between 175 and 225 heads (inclusive)?(b) The probability (approximately) that the number of heads is less than y? Express the answer in terms of the function Φ(z).
9. Suppose that 20% of the articles produced by a machine are defective, the defectives occurring at random during production. Using the normal distribution for determining approximations,(a) What is the approximate probability that if a sample of 400 items is taken from the production, more than
8. A fair coin is tossed 20 times. Find the exact and the normal approximation of the probability of obtaining 13 heads. Compare the two probabilities.Review Practice Problems 221
7. Suppose that the life spans (in months) of 15 patients after they are diagnosed with a particular kind of cancer are found to be as follows:47 50 37 44 37 44 38 35 40 38 49 42 39 38 44 Using MINITAB, R, or JMP, verify if it is appropriate to assume that these data come from a normal population.
6. In Example 5.6.1, if “acceptable fits” are those in which the difference between hole diameter and pin diameter lies within 0.0010 ± 0.0005 inch, what fraction of random matches would yield acceptable fits?
5. A process for making quarter inch ball bearings yields a population of ball bearings with diameters having mean 0.2497 in. and standard deviation of 0.0002 in. If we assume approximate normality of diameters and if specifications call for bearings with diameters to be within 0.2500 ± 0.0003
4. Show that P(a < X < a + l), where l is a positive constant and X has the distribution N(μ, σ2) is maximized if a = μ − l/2.
3. Suppose that a machine set for filling 1-lb boxes of sugar yields a population of fillings, whose weights are (approximately) normally distributed with a mean of 16.30 oz and a standard deviation of 0.15 oz. Estimate:(a) The percentage of fillings that will be underweight (i.e., less than 1
2. If X has the distribution N(15.0, 6.25), find(a) P(X 16.5)(c) C, where P(X D) = 0.025(e) E, where P(|X − 15.0| < E) = 0.99
1. A random variable X has the distribution N(1500, (200)2). Find:(a) P(X 1700)(c) A, where P(X >A) = 0.05(d) B, where P(1500 −B
10. Suppose that the lifetime T (yr) of a shock absorber in a car has the Weibull distribution with α = 1, β = 0.4.(a) What is the expected life of such a shock absorber?(b) What is the probability that such a shock absorber will be working after 10 years?
9. Suppose that in Problem 8 the lifetime of the battery has the Weibull distribution with α = 2, β = 0.5 instead of the gamma distribution of Problem 8.(a) What is the expected life of such a battery?(b) What is the probability that such a battery will be working after five years?
8. Suppose that the lifetime (in years) of a battery for a Movado watch is a random variable having a gamma distribution with γ = 2, λ = 0.5.(a) What is the expected life of such a battery?(b) What is the probability that such a battery will be working after five years?
7. Suppose that the time T (in hours) needed to repair a water pump can be modeled as gamma with γ = 2, λ = 2. What is the probability that the next repair of water pump will need:(a) at least 1 hour, (b) at most 2 hours, and (c) between 1 and 2 hours?
6. The lifetime T of certain pieces of medical equipment is distributed as Weibull with parameters α = 6, β = 0.25, and threshold parameter τ = 0.(a) Find the mean and the variance of the lifetime of the equipment.(b) Find the probability P(130 < T < 160).
5. The lifetime, in years, of a sport utility vehicle (SUV) is distributed as Weibull with parameters α = 6, β = 0.5, and threshold parameter τ = 0.(a) Find the mean and the variance of the lifetime of the SUV.(b) Find the probability that the SUV will last more than 10 years.(c) Find the
4. Suppose that a random variable T is distributed as Weibull with parameters α =250, β = 0.25, and threshold parameter τ = 0.(a) Find the mean and the variance of the random variable T.(b) Find the probability P(5000 < T < 7000).
3. The time, in hours, required to perform a quadruple bypass surgery follows the gamma distribution with mean eight hours and standard deviation two hours. Find the probability that a quadruple bypass surgery will last:(a) no more than 12 hours, (b) at least six hours, (c) between six and 10 hours.
2. The lifetime, in years, of a computer hard drive follows a gamma distribution with mean of five years and standard deviation three years. Find the probability that the hard drive will last:(a) no more than six years, (b) at least four years, (c) between five and seven years.
1. Suppose that a random variable T denoting the time, in years, to failure of a type of computer is distributed by a gamma probability distribution with γ = 6, λ = 2.Find the probability that 12 of 15 such computers are still functioning after five years.218 5 Continuous Random Variables
8. Suppose that the time between arrivals of two buses at a city bus station is exponentially distributed with a mean of five minutes. Determine the following probabilities:(a) More than two buses arrive during an interval of 5 minutes.(b) No bus arrives during an interval of 10 minutes.(c) No more
7. The time T, in minutes, between the arrival of two successive patients in an emergency room can be modeled as an exponential distribution with mean 20 minutes.Determine the following probabilities:(a) P(T > 30), (b) P(12 < T < 18), (c) P(T < 25).
6. The lifetime X, in years, of car batteries has an exponential distribution with a mean life of five years. If you buy a new car and plan to keep it for six years, then find the probability that you will change the battery during your ownership.
5. Suppose that a computer lab in a small liberal arts college has 20 similar computers.Let the random variable T denote the time, in years, to failure of this type of computer. If the random variable T follows the exponential distribution with mean equal to three years, then find the probability
4. Let a random variable X be distributed exponentially with mean equal to 10. Find the m.g.f. of the random variable Y = 4+3X. Then, use this m.g.f. to find the mean and variance of the random variable Y.
3. Suppose that the lapse of time between two successive accidents in a paper mill is exponentially distributed with a mean of 15 days. Find the probability that the time between two successive accidents at that mill is more than 20days.
2. Suppose that a random variable X is distributed as exponential distribution with parameter λ. Show that P(X ≤ 10) = P(X ≤ 17|X ≥ 7). This property of exponential distribution is known as the “memoryless” property.
1. The waiting time T at a bank teller’s window between two successive customers is distributed as exponential with a mean of four minutes. Find the following probabilities:(a) P(T ≥ 5, (b) P(3 ≤ T ≤ 6), (c) P(T ≤ 4), and (d) P(T < 5).
6. Suppose that the lifetime X, in hours, of a surgical instrument can be modeled with lognormal distribution having parameters μ = 8 and σ2 = 2. Determine the following: (a) P(X ≥ 25,000), (b) P(X >24,000), (c) P(22,000 < X < 28,000).
5. Suppose that a random variable Y has a lognormal distribution with parametersμ = 4 and σ2 = 4. Determine the following: (a) the mean and standard deviation of Y, (b) P(Y >250), (c) P(100 < Y < 200).
4. Suppose that a random variable X is normally distributed with μ = 4 and σ2 =4. If the lifetime Y (hours), of an electronic component can be modeled with the 206 5 Continuous Random Variables distribution of the random variable Y = eX, then find the following: (a) the mean and the standard
3. Suppose that a random variable X is distributed as lognormal with parameters μ = 2 and σ = 0.5. Find the mean and the variance of the random variable X.
2. The size of a computer chip is distributed as lognormal with parameters μ and σThe following data give the sizes of eight randomly selected chips:4.18 2.83 3.76 4.79 3.59 2.98 4.16 2.12 Estimate μ and σ.
1. The lifetime, in hours, of a head gasket of a cylinder in a car engine is distributed as lognormal with mean 6000 and standard deviation 5000 hours. Find the probability that the lifetime of the head gasket is more than 8000 hours.
10. A random variable X is distributed as binomial with parameters n = 20, p = 0.4.Compute the exact and the normal approximation to P(10 ≤ X ≤ 16) and compare the two probabilities
9. An European airline company flies jets that can hold 200 passengers. The company knows from past experience that on the average, 8% of the booked passengers do not show up on time to take their flight. If the company booked 212 passengers for a particular flight, then what is the probability
8. A pathology lab does not deliver 10% of the test results in a timely manner. Suppose that in a given week, it delivered 400 test results to a certain hospital. Use the normal approximation to the binomial to find the following probabilities, assuming that test results delivered are independent
7. A semiconductor manufacturer recently found that 3% of the chips produced at its new plant are defective. Assume that the chips are independently defective or nondefective. Use a normal approximation to determine the probability that a box of 500 chips contains: (a) at least 10 defective chips
6. Find the following probabilities using the normal approximation to the binomial distribution with parameters n = 100, p = 0.8: (a) P(73 ≤ X ≤ 87), (b) P(73 < X
5. Let X be a binomial random variable with parameters n = 20 and p = 0.4.(a) Use the binomial tables (Table A.2) to determine P(4 ≤ X ≤ 12).(b) Find the mean and the standard deviation of the binomial distribution, and then use the appropriate normal approximation to find P(4 ≤ X ≤ 12).
6. Suppose in Problem 5 the total scores of a student are denoted by a random variable U. Then, find the following probabilities: (a) P(U ≥ 33), (b) P(30 ≤ U ≤ 38), (c)P(U ≥ 38).
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