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applied statistics and probability for engineers

Probability And Statistics For Engineering And The Sciences 7th Edition Dave Ellis, Jay L Devore - Solutions

77. The authors of the paper from which the data in Exercise 1.27 was extracted suggested that a reasonable probability model for drill lifetime was a lognormal distribution with 4.5 and .8.
76.a. In Exercise 72, what is the median lifetime of such tubes? [Hint: Use Expression (4.12).]b. In Exercise 74, what is the median return time?c. If X has a Weibull distribution with the cdf from Expression(4.12), obtain a general expression for the (100p)th percentile of the distribution.d. In
75. Let X have a Weibull distribution with the pdf from Expression (4.11). Verify that (1 1/). [Hint: In the integral for E(X), make the change of variable y (x/), so that x y1/.]
11. An automobile service facility specializing in engine tune-ups knows that 45% of all tune-ups are done on four-cylinder automobiles, 40% on six-cylinder automobiles, and 15% on eight-cylinder automobiles. Let X the number of cylinders on the next car to be tuned.a. What is the pmf of X?b.
74. Let X the time (in 101 weeks) from shipment of a defective product until the customer returns the product. Suppose that the minimum return time is 3.5 and that the excess X 3.5 over the minimum has a Weibull distribution with parameters 2 and 1.5 (see the Industrial Quality
c. What value is such that exactly 50% of all specimens have lifetimes exceeding that value?
b. What is the probability that a specimen’s lifetime is between 100 and 250?
a. What is the probability that a specimen’s lifetime is at most 250? Less than 250? More than 300?
73. The authors of the article “A Probabilistic Insulation Life Model for Combined Thermal-Electrical Stresses” (IEEE Trans. on Elect. Insulation, 1985: 519–522) state that “the Weibull distribution is widely used in statistical problems relating to aging of solid insulating materials
72. The lifetime X (in hundreds of hours) of a certain type of vacuum tube has a Weibull distribution with parameters a 2 and b 3. Compute the following:a. E(X) and V(X)b. P(X 6)c. P(1.5 X 6)(This Weibull distribution is suggested as a model for time in service in “On the Assessment of
71.a. The event {X2 y} is equivalent to what event involving X itself?b. If X has a standard normal distribution, use part (a) to write the integral that equals P(X2 y). Then differentiate this with respect to y to obtain the pdf of X2 [the square of a N(0, 1) variable]. Finally, show that X2
70. If X has an exponential distribution with parameter , derive a general expression for the (100p)th percentile of the distribution. Then specialize to obtain the median.
69. A system consists of five identical components connected in series as shown:As soon as one component fails, the entire system will fail.Suppose each component has a lifetime that is exponentially distributed with .01 and that components fail independently of one another. Define events Ai
68. The special case of the gamma distribution in which is a positive integer n is called an Erlang distribution. If we replace by 1/ in Expression (4.8), the Erlang pdf is f(x; , n) ((n x)n1 1e)!x x 0 0 x 0 It can be shown that if the times between successive events are
67. Suppose that when a transistor of a certain type is subjected to an accelerated life test, the lifetime X (in weeks) has a gamma distribution with mean 24 weeks and standard deviation 12 weeks.a. What is the probability that a transistor will last between 12 and 24 weeks?b. What is the
66. Suppose the time spent by a randomly selected student who uses a terminal connected to a local time-sharing computer facility has a gamma distribution with mean 20 min and variance 80 min2.a. What are the values of and ?b. What is the probability that a student uses the terminal for at most
12. Airlines sometimes overbook flights. Suppose that for a plane with 50 seats, 55 passengers have tickets. Define the random variable Y as the number of ticketed passengers who actually show up for the flight. The probability mass function of Y appears in the accompanying table.
64. Evaluate the following:a. (6)b. (5/2)c. F(4; 5) (the incomplete gamma function)d. F(5; 4)e. F(0; 4)65. Let X have a standard gamma distribution with 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)
63. A consumer is trying to decide between two long-distance calling plans. The first one charges a flat rate of 10¢ per minute, whereas the second charges a flat rate of 99¢ for calls up to 20 minutes in duration and then 10¢ for each additional minute exceeding 20 (assume that calls lasting a
b. What is the probability that the extent of daily seaice change is within 1 standard deviation of the mean value?
a. What is the value of the parameter ?
62. The paper “Microwave Obsevations of Daily Antarctic SeaIce Edge Expansion and Contribution Rates” (IEEE Geosci.and Remote Sensing Letters, 2006: 54–58) states that “The distribution of the daily sea-ice advance/retreat from each sensor is similar and is approximately double
61. Extensive experience with fans of a certain type used in diesel engines has suggested that the exponential distribution provides a good model for time until failure. Suppose the mean time until failure is 25,000 hours. What is the probability thata. A randomly selected fan will last at least
c. What is the value of the median distance?
a. What is the probability that the flight will accommodate all ticketed passengers who show up?
b. What is the probability that distance exceeds the mean distance by more than 2 standard deviations?
a. What is the probability that the distance is at most 100 m?At most 200 m? Between 100 and 200 m?
60. Let X denote the distance (m) that an animal moves from its birth site to the first territorial vacancy it encounters.Suppose that for banner-tailed kangaroo rats, X has an exponential distribution with parameter .01386 (as suggested in the article “Competition and Dispersal from Multiple
59. Let X the time between two successive arrivals at the drive-up window of a local bank. If X has an exponential distribution with 1 (which is identical to a standard gamma distribution with 1), compute the following:a. The expected time between two successive arrivalsb. The standard
58. There is no nice formula for the standard normal cdf (z), but several good approximations have been published in articles. The following is from “Approximations for Hand Calculators Using Small Integer Coefficients” (Mathematics of Computation, 1977: 214–222). For 0 z 5.5,
b. If when measured in °C, temperature is normally distributed with mean 115 and standard deviation 2, what can be said about the distribution of temperature measured in °F?
57.a. Show that if X has a normal distribution with parameters and , then Y aX b (a linear function of X)also has a normal distribution. What are the parameters of the distribution of Y [i.e., E(Y) and V(Y)]? [Hint:Write the cdf of Y, P(Y y), as an integral involving the pdf of X, and then
56. Show that the relationship between a general normal percentile and the corresponding z percentile is as stated in this section
55. Suppose only 75% of all drivers in a certain state regularly wear a seat belt. A random sample of 500 drivers is selected.What is the probability thata. Between 360 and 400 (inclusive) of the drivers in the sample regularly wear a seat belt?b. Fewer than 400 of those in the sample regularly
54. Suppose that 10% of all steel shafts produced by a certain process are nonconforming but can be reworked (rather than having to be scrapped). Consider a random sample of 200 shafts, and let X denote the number among these that are nonconforming and can be reworked. What is the (approximate)
53. Let X have a binomial distribution with parameters n 25 and p. Calculate each of the following probabilities using the normal approximation (with the continuity correction)for the cases p .5, .6, and .8 and compare to the exact probabilities calculated from Appendix Table A.1.a. P(15 X
52. Let X denote the number of flaws along a 100-m reel of magnetic tape (an integer-valued variable). Suppose X has approximately a normal distribution with 25 and 5. Use the continuity correction to calculate the probability that the number of flaws isa. Between 20 and 30, inclusive.b. At
51. Chebyshev’s inequality, (see Exercise 44 Chapter 3), is valid for continuous as well as discrete distributions. It states that for any number k satisfying k 1, P(⏐X ⏐ k) 1/k2(see Exercise 44 in Chapter 3 for an interpretation). Obtain this probability in the case of a normal
23. A consumer organization that evaluates new automobiles customarily reports the number of major defects in each car examined. Let X denote the number of major defects in a randomly selected car of a certain type. The cdf of X is as follows:Ï 0 x 0Ô .06 0 x 1Ô .19 1 x 2 F(x) Ì .39
50. In response to concerns about nutritional contents of fast foods, McDonald’s has announced that it will use a new cooking oil for its french fries that will decrease substantially trans fatty acid levels and increase the amount of more beneficial polyunsaturated fat. The company claims that
e. If X is a random variable with a normal distribution and a is a numerical constant (a 0), then Y aX also has a normal distribution. Use this to determine the distribution of birth weight expressed in pounds (shape, mean, and standard deviation), and then recalculate the probability from part
d. How would you characterize the most extreme .1% of all birth weights?
c. What is the probability that the birth weight of a randomly selected baby of this type exceeds 7 lb?
b. What is the probability that the birth weight of a randomly selected baby of this type is either less than 2000 grams or greater than 5000 grams?
a. What is the probability that the birth weight of a randomly selected baby of this type exceeds 4000 grams? Is between 3000 and 4000 grams?
49. Consider babies born in the “normal” range of 37–43 weeks gestational age. Extensive data supports the assumption that for such babies born in the United States, birth weight is normally distributed with mean 3432 g and standard deviation 482 g. [The article “Are Babies Normal?” (The
48. Suppose Appendix Table A.3 contained (z) only for z 0.Explain how you could still computea. P(1.72 Z .55)b. P(1.72 Z .55)Is it necessary to table (z) for z negative? What property of the standard normal curve justifies your answer?
47. The weight distribution of parcels sent in a certain manner is normal with mean value 12 lb and standard deviation 3.5 lb.The parcel service wishes to establish a weight value c beyond which there will be a surcharge. What value of c is such that 99% of all parcels are at least 1 lb under the
46. The Rockwell hardness of a metal is determined by impressing a hardened point into the surface of the metal and then measuring the depth of penetration of the point. Suppose the Rockwell hardness of a particular alloy is normally distributed with mean 70 and standard deviation 3. (Rockwell
45. A machine that produces ball bearings has initially been set so that the true average diameter of the bearings it produces is .500 in. A bearing is acceptable if its diameter is within .004 in. of this target value. Suppose, however, that the setting has changed during the course of production,
c. Between 1 and 2 SDs from its mean value?
b. Farther than 2.5 SDs from its mean value?
a. Within 1.5 SDs of its mean value?
44. If bolt thread length is normally distributed, what is the probability that the thread length of a randomly selected bolt is
51. Refer to the previous exercise.
43. The distribution of resistance for resistors of a certain type is known to be normal, with 10% of all resistors having a resistance exceeding 10.256 ohms and 5% having a resistance smaller than 9.671 ohms. What are the mean value and standard deviation of the resistance distribution?
42. The temperature reading from a thermocouple placed in a constant-temperature medium is normally distributed with mean , the actual temperature of the medium, and standard deviation . What would the value of have to be to ensure that 95% of all readings are within .1° of ?
41. The automatic opening device of a military cargo parachute has been designed to open when the parachute is 200 m above the ground. Suppose opening altitude actually has a normal distribution with mean value 200 m and standard deviation 30 m. Equipment damage will occur if the parachute opens at
c. The width of a line etched on an integrated circuit chip is normally distributed with mean 3.000 m and standard deviation .140. What width value separates the widest 10% of all such lines from the other 90%?40. The article “Monte Carlo Simulation—Tool for Better Understanding of LRFD” (J.
b. What is the 6th percentile of the distribution?
39.a. If a normal distribution has 30 and 5, what is the 91st percentile of the distribution?
38. There are two machines available for cutting corks intended for use in wine bottles. The first produces corks with diameters that are normally distributed with mean 3 cm and standard deviation .1 cm. The second machine produces corks with diameters that have a normal distribution with mean 3.04
37. Suppose that blood chloride concentration (mmol/L) has a normal distribution with mean 104 and standard deviation 5 (information in the article “Mathematical Model of Chloride Concentration in Human Blood,” J. of Med. Engr.and Tech., 2006: 25–30, including a normal probability plot as
37. The n candidates for a job have been ranked 1, 2, 3, . . . , n. Let X the rank of a randomly selected candidate, so that X has pmf p(x) { 1/n x 1, 2, 3, . . . , n 0 otherwise(this is called the discrete uniform distribution). Compute E(X) and V(X) using the shortcut formula. [Hint: The
36. Spray drift is a constant concern for pesticide applicators and agricultural producers. The inverse relationship between droplet size and drift potential is well known. The paper“Effects of 2,4-D Formulation and Quinclorac on Spray Droplet Size and Deposition” (Weed Technology,
e. If four trees are independently selected, what is the probability that at least one has a diameter exceeding 10 in.?
d. What value c is such that the interval (8.8 c, 8.8 c)includes 98% of all diameter values?
c. What is the probability that the diameter of a randomly selected tree will be between 5 and 10 in.?
b. What is the probability that the diameter of a randomly selected tree will exceed 20 in.?
a. What is the probability that the diameter of a randomly selected tree will be at least 10 in.? Will exceed 10 in.?
35. Suppose the diameter at breast height (in.) of trees of a certain type is normally distributed with 8.8 and 2.8, as suggested in the article “Simulating a Harvester-Forwarder Softwood Thinning” (Forest Products J., May 1997: 36–41).
c. How would you characterize the largest 5% of all concentration values?
b. What is the probability that the concentration is at most .10?
38. Let X the outcome when a fair die is rolled once. If before the die is rolled you are offered either (1/3.5) dollars or h(X) 1/X dollars, would you accept the guaranteed amount or would you gamble? [Note: It is not generally true that 1/E(X) E(1/X).]
a. What is the probability that the concentration exceeds .25?
34. The article “Reliability of Domestic-Waste Biofilm Reactors”(J. of Envir. Engr., 1995: 785–790) suggests that substrate concentration (mg/cm3) of influent to a reactor is normally distributed with .30 and .06.
33. Suppose the force acting on a column that helps to support a building is normally distributed with mean 15.0 kips and standard deviation 1.25 kips. What is the probability that the forcea. Is at most 18 kips?b. Is between 10 and 12 kips?c. Differs from 15.0 kips by at most 1.5 standard
32. If X is a normal rv with mean 80 and standard deviation 10, compute the following probabilities by standardizing:a. P(X 100)b. P(X 80)c. P(65 X 100)d. P(70 X)e. P(85 X 95)f. P(⏐X 80⏐ 10)
31. Determine z for the following:a. .0055b. .09c. .663
39. A chemical supply company currently has in stock 100 lb of a certain chemical, which it sells to customers in 5-lb lots.Let X the number of lots ordered by a randomly chosen customer, and suppose that X has pmf Compute E(X) and V(X). Then compute the expected number of pounds left after the
30. Find the following percentiles for the standard normal distribution. Interpolate where appropriate.a. 91stb. 9thc. 75thd. 25the. 6th
29. In each case, determine the value of the constant c that makes the probability statement correct.a. (c) .9838b. P(0 Z c) .291c. P(c Z) .121d. P(c Z c) .668e. P(c ⏐Z⏐) .016
28. Let Z be a standard normal random variable and calculate the following probabilities, drawing pictures wherever appropriate.a. P(0 Z 2.17)b. P(0 Z 1)c. P(2.50 Z 0)d. P(2.50 Z 2.50)e. P(Z 1.37)f. P(1.75 Z)g. P(1.50 Z 2.00)h. P(1.37 Z 2.50)i. P(1.50 Z)j.
27. When a dart is thrown at a circular target, consider the location of the landing point relative to the bull’s eye. Let X be the angle in degrees measured from the horizontal, and assume that X is uniformly distributed on [0, 360].Define Y to be the transformed variable Y h(X) (2/360)X ,
26. Let X be the total medical expenses (in 1000s of dollars)incurred by a particular individual during a given year.Although X is a discrete random variable, suppose its distribution is quite well approximated by a continuous distribution with pdf f(x) k(1 x/2.5)7 for x 0.a. What is the value
25. Let X be the temperature in °C at which a certain chemical reaction takes place, and let Y be the temperature in °F(so Y 1.8X 32).a. If the median of the X distribution is , show that 1.8 32 is the median of the Y distribution.b. How is the 90th percentile of the Y distribution related to
24. Let X have the Pareto pdf f(x; k, ) k xk1 k x 0 x introduced in Exercise 10.a. If k 1, compute E(X).b. What can you say about E(X) if k 1?c. If k 2, show that V(X) k 2(k 1)2(k 2)1.d. If k 2, what can you say about V(X)?e. What conditions on k are necessary to ensure that
23. If the temperature at which a certain compound melts is a random variable with mean value 120°C and standard deviation 2°C, what are the mean temperature and standard deviation measured in °F? [Hint: °F 1.8°C 32.]
22. The weekly demand for propane gas (in 1000s of gallons)from a particular facility is an rv X with pdf f(x) 21 x 12 1 x 2 0 otherwisea. Compute the cdf of X.b. Obtain an expression for the (100p)th percentile. What is the value of ?c. Compute E(X) and V(X).d. If 1.5 thousand gallons
21. An ecologist wishes to mark off a circular sampling region having radius 10 m. However, the radius of the resulting region is actually a random variable R with pdf
20. Consider the pdf for total waiting time Y for two buses2 15 y 0 y 5 f(y) 2 5 2 15 y 5 y 10 0 otherwise introduced in Exercise 8.a. Compute and sketch the cdf of Y. [Hint: Consider separately 0 y 5 and 5 y 10 in computing F(y). A graph of the pdf should be helpful.]b.
19. Let X be a continuous rv with cdf 0 x 0 F(x) 4 x 1 ln4 x 0 x 4 1 x 4[This type of cdf is suggested in the article “Variability in Measured Bedload-Transport Rates” (Water Resources Bull., 1985: 39–48) as a model for a certain hydrologic variable.] What isa. P(X 1)?b. P(1
18. Let X denote the voltage at the output of a microphone, and suppose that X has a uniform distribution on the interval from 1 to 1. The voltage is processed by a “hard limiter”with cutoff values .5 and .5, so the limiter output is a random variable Y related to X by Y X if |X| .5, Y
17. Let X have a uniform distribution on the interval[A, B].a. Obtain an expression for the (100p)th percentile.b. Compute E(X), V(X), and X.c. For n a positive integer, compute E(Xn).
16. Answer parts (a)–(f) of Exercise 15 with X lecture time past the hour given in Exercise 5.
15. Let X denote the amount of space occupied by an article placed in a 1-ft3 packing container. The pdf of X is f(x) 90x8(1 x) 0 x 1 0 otherwisea. Graph the pdf. Then obtain the cdf of X and graph it.b. What is P(X .5) [i.e., F(.5)]?c. Using the cdf from (a), what is P(.25 X .5)?
d. What is the probability that the observed depth is within 1 standard deviation of the mean value? Within 2 standard deviations?
c. What is the probability that observed depth is at most 10?Between 10 and 15?
b. What is the cdf of depth?
a. What are the mean and variance of depth?

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