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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
The accompanying table gives summary data on cube compressive strength (N/mm2) for concrete specimens made with a pulverized fuel-ash mix (“A Study of Twenty-Five-Year-Old Pulverized Fuel Ash Concrete Used in Foundation Structures,” Proc. Inst. Civ. Engrs., Mar. 1985: 149–165):Age Sample
a. Does the data provide compelling evidence for concluding that true average strength for the 1078 grade exceeds that uestion posed in part (a) by carrying out a formal test of hypotheses. Is the result different from what you conjectured in part (a)?c. Does it appear that ICU stay for patients
Tensile strength tests were carried out on two different grades of wire rod (“Fluidized Bed Patenting of Wire Rods,” Wire J., June 1977: 56–61), resulting in the accompanying data.Sample Sample Mean Sample Grader Size (kg/mm2) SD AISI 1064 m 129 x 107.6 s1 1.3 AISI 1078 n 129 y
Are male college students more easily bored than their female counterparts? This question was examined in the article “Boredom in Young Adults—Gender and Cultural Comparisons” (J. of Cross-Cultural Psych., 1991: 209–223).The authors administered a scale called the Boredom Proneness Scale to
Persons having Reynaud’s syndrome are apt to suffer a sudden impairment of blood circulation in fingers and toes. In an experiment to study the extent of this impairment, each subject immersed a forefinger in water and the resulting heat output(cal/cm2/min) was measured. For m 10 subjects with
a. Use the data of Exercise 2 to compute a 95% CI for 1 2. Does the resulting interval suggest that 1 2 has been precisely estimated?b. Use the data of Exercise 3 to compute a 95% upper confidence bound for 1 2.
Let 1 denote true average tread life for a premium brand of P205/65R15 radial tire and let 2 denote the true average tread life for an economy brand of the same size. Test H0:1 2 5000 versus Ha: 1 2 5000 at level .01 using the following data: m 45, x 42,500, s1 2200, n 45, y 36,800,
Let 1 and 2 denote true average tread lives for two competing brands of size P205/65R15 radial tires. Test H0: 1 2 0 versus Ha: 1 2 0 at level .05 using the following data:m 45, x 42,500, s1 2200, n 45, y 40,400, and s2 1900.
60. The relative conductivity of a semiconductor device is determined by the amount of impurity “doped” into the device during its manufacture. A silicon diode to be used for a specific purpose requires an average cut-on voltage of .60 V, and if this is not achieved, the amount of impurity must
82. When the population distribution is normal and n is large, the sample standard deviation S has approximately a normal distribution with E(S) and V(S) 2/(2n). We already know that in this case, for any n, X is normal with E(X) and V(X) 2/n.a. Assuming that the underlying distribution is
67. A new method for measuring phosphorus levels in soil is described in the article “A Rapid Method to Determine Total Phosphorus in Soils” (Soil Sci. Amer. J., 1988: 1301–1304).Suppose a sample of 11 soil specimens, each with a true phosphorus content of 548 mg/kg, is analyzed using the new
e. For a level .05 test, what conclusion would you reach?
d. What is the P-value for the value of the test statistic computed in part (c)?
c. What is the value of the test statistic for this data?
64. In Exercise 63, suppose the experimenter had believed before collecting the data that the value of was approximately .30. If the experimenter wished the probability of a type II error to be .05 when 3.00, was a sample size 50 unnecessarily large?heses in part (a)?
4. If a test is derived under specific assumptions about the distribution or population being sampled, how will the test perform when the assumptions are violated?
69. The article “Caffeine Knowledge, Attitudes, and Consumption in Adult Women” (J. of Nutrition Educ., 1992: 179–184)reports the following summary data on daily caffeine consumption for a sample of adult women: n 47, x 215 mg, s 235 mg, and range 5–1176.a. Does it appear plausible
70. The accompanying output resulted when MINITAB was used to test the appropriate hypotheses about true average activation time based on the data in Exercise 57. Use this information to reach a conclusion at significance level .05 and also at level .01.TEST OF MU 25.000 VS MU G.T. 25.000 N MEAN
81. Referring to Exercise 80, suppose an investigator wishes to test H0: 2 .04 versus Ha: 2 .04 based on a sample of 21 observations. The computed value of 20s2/.04 is 8.58.Place bounds on the P-value and then reach a conclusion at level .01.
80. Chapter 7 presented a CI for the variance 2 of a normal population distribution. The key result there was that the rv 2 (n 1)S2/2 has a chi-squared distribution with n 1 df. Consider the null hypothesis H0: 2 2 0 (equivalently, 0). Then when H0 is true, the test statistic 2 (n 1)
79. A hot-tub manufacturer advertises that with its heating equipment, a temperature of 100°F can be achieved in at most 15 min. A random sample of 32 tubs is selected, and the time necessary to achieve a 100°F temperature is determined for each tub. The sample average time and sample standard
73. The incidence of a certain type of chromosome defect in the U.S. adult male population is believed to be 1 in 75.A random sample of 800 individuals in U.S. penal institutions reveals 16 who have such defects. Can it be concluded
72. The accompanying observations on residual flame time(sec) for strips of treated children’s nightwear were given in the article “An Introduction to Some Precision and Accuracy of Measurement Problems” (J. of Testing and Eval., 1982: 132–140). Suppose a true average flame time of at most
3. When two or more tests are appropriate in a given situation, how can the tests be compared to decide which should be used?
2. Does there exist a general principle, not dependent just on intuition, that can be used to obtain best or good test procedures?
1. What are the practical implications and consequences of choosing a particular level of significance once the other aspects of a test have been determined?
39. A university library ordinarily has a complete shelf inventory done once every year. Because of new shelving rules instituted the previous year, the head librarian believes it may be possible to save money by postponing the inventory. The librarian decides to select at random 1000 books from
48. Newly purchased tires of a certain type are supposed to be filled to a pressure of 30 lb/in2. Let denote the true average pressure. Find the P-value associated with each given z statistic value for testing H0: 30 versus Ha: 30.a. 2.10b. 1.75c. .55d. 1.41e. 5.3 49. Give as much information as
47. Let denote the mean reaction time to a certain stimulus.For a large-sample z test of H0: 5 versus Ha: 5, find the P-value associated with each of the given values of the z test statistic.a. 1.42b. .90c. 1.96d. 2.48e. .11
46. Pairs of P-values and significance levels, , are given. For each pair, state whether the observed P-value would lead to rejection of H0 at the given significance level.a. P-value .084, .05b. P-value .003, .001c. P-value .498, .05d. P-value .084, .10e. P-value .039,
45. For which of the given P-values would the null hypothesis be rejected when performing a level .05 test?a. .001b. .021c. .078d. .047e. .148
44. Scientists think that robots will play a crucial role in factories in the next several decades. Suppose that in an experiment to determine whether the use of robots to weave computer cables is feasible, a robot was used to assemble 500 cables.The cables were examined and there were 15
43. A manufacturer of plumbing fixtures has developed a new type of washerless faucet. Let p P(a randomly selected faucet of this type will develop a leak within 2 years under normal use). The manufacturer has decided to proceed with production unless it can be determined that p is too large; the
42. Each of a group of 20 intermediate tennis players is given two rackets, one having nylon strings and the other synthetic gut strings. After several weeks of playing with the two rackets, each player will be asked to state a preference for one of the two types of strings. Let p denote the
41. A plan for an executive traveler’s club has been developed by an airline on the premise that 5% of its current customers would qualify for membership. A random sample of 500 customers yielded 40 who would qualify.
51. Let denote true average serum receptor concentration for all pregnant women. The average for all women is known to be 5.63. The article “Serum Transferrin Receptor for the Detection of Iron Deficiency in Pregnancy” (Amer. J.Clinical Nutr., 1991: 1077–1081) reports that P-value .10 for a
52. The article “Analysis of Reserve and Regular Bottlings: Why Pay for a Difference Only the Critics Claim to Notice?”(Chance, Summer 2005, pp. 9–15) reported on an experiment to investigate whether wine tasters could distinguish between more expensive reserve wines and their regular
Are you particularly impressed with the ability of tasters to distinguish between the two types of wine?
59. A spectrophotometer used for measuring CO concentration[ppm (parts per million) by volume] is checked for accuracy by taking readings on a manufactured gas (called span gas)in which the CO concentration is very precisely controlled at 70 ppm. If the readings suggest that the spectrophotometer
58. A certain pen has been designed so that true average writing lifetime under controlled conditions (involving the use of a writing machine) is at least 10 hours. A random sample of 18 pens is selected, the writing lifetime of each is determined, and a normal probability plot of the resulting
57. The times of first sprinkler activation for a series of tests with fire prevention sprinkler systems using an aqueous film-forming foam were (in sec)27 41 22 27 23 35 30 33 24 27 28 22 24(see “Use of AFFF in Sprinkler Systems,” Fire Technology, 1976: 5). The system has been designed so that
56. A random sample of soil specimens was obtained, and the amount of organic matter (%) in the soil was determined for each specimen, resulting in the accompanying data (from“Engineering Properties of Soil,” Soil Science, 1998: 93–102).1.10 5.09 0.97 1.59 4.60 0.32 0.55 1.45 0.14 4.47 1.20
55. Many consumers are turning to generics as a way of reducing the cost of prescription medications. The article“Commercial Information on Drugs: Confusing to the Physician?” (J. of Drug Issues, 1988: 245–257) gives the results of a survey of 102 doctors. Only 47 of those surveyed knew the
If the true percentage of “early yields” is actually 50%(so that the theoretical point is the median of the yield distribution) and a level .01 test is used, what is the probability that the company concludes a modification of the process is necessary?
what would you advise the company to do?
54. Because of variability in the manufacturing process, the actual yielding point of a sample of mild steel subjected to increasing stress will usually differ from the theoretical yielding point. Let p denote the true proportion of samples that yield before their theoretical yielding point. If on
40. The article “Statistical Evidence of Discrimination” (J.Amer. Stat. Assoc., 1982: 773–783) discusses the court case Swain v. Alabama (1965), in which it was alleged that there was discrimination against blacks in grand jury selection.Census data suggested that 25% of those eligible for
83. Let X1, X2, . . . , Xn be a random sample from an exponential distribution with parameter . Then it can be shown that 2Xi has a chi-squared distribution with 2n (by first showing that 2Xi has a chi-squared distribution with 2).a. Use this fact to obtain a test statistic and rejection
14. Reconsider the situation of Exercise 11 and suppose the rejection region is {x: x 10.1004 or x 9.8940} {z: z 2.51 or z 2.65}.a. What is for this procedure?b. What is when 10.1? When 9.9? Is this desirable?
10. A mixture of pulverized fuel ash and Portland cement to be used for grouting should have a compressive strength of more than 1300 KN/m2
11. The mixture will not be used unless experimental evidence indicates conclusively that the strength specification has been met. Suppose compressive strength for he scale is to be recalibrated if either x 10.1032 or x 9.8968. What is the probability that recalibration is carried out when it is
12. A new design for the braking system on a certain type of car has been proposed. For the current system, the true average braking distance at 40 mph under specified conditions is known to be 120 ft. It is proposed that the new design be implemented only if sample data strongly indicates a
13. Let X1, . . . , Xn denote a random sample from a normal population distribution with a known value of .a. For testing the hypotheses H0: 0 versus Ha:0 (where 0 is a fixed number), show that the test with test statistic X and rejection region x 0 2.33/n has significance level .01.b. Suppose
15. Let the test statistic Z have a standard normal distribution when H0 is true. Give the significance level for each of the following situations:a. Ha: 0, rejection region z 1.88b. Ha: 0, rejection region z 2.75c. Ha: 0, rejection region z 2.88 or z 2.88
16. Let the test statistic T have a t distribution when H0 is true.Give the significance level for each of the following situations:a. Ha: 0, df 15, rejection region t 3.733b. Ha: 0, n 24, rejection region t 2.500c. Ha: 0, n 31, rejection region t 1.697 or t 1.697
Using the selected region, what would you conclude if 6 of the 25 queried favored company 1?
Compute the probability of a type II error for the selected region when p .3, again when p .4, and also for both p .6 and p .7.
1. For each of the following assertions, state whether it is a legitimate statistical hypothesis and why:a. H: 100b. H: x~ 45c. H: s .20d. H: 1/2 1e. H: X Y 5f. H: .01, where is the parameter of an exponential distribution used to model component lifetime
3. To determine whether the pipe welds in a nuclear power plant meet specifications, a random sample of welds is selected, and tests are conducted on each weld in the sample. Weld strength is measured as the force required to break the weld.Suppose the specifications state that mean strength of
4. Let denote the true average radioactivity level (picocuries per liter). The value 5 pCi/L is considered the dividing line between safe and unsafe water. Would you recommend testing H0: 5 versus Ha: 5 or H0: 5 versus Ha: 5? Explain your reasoning. [Hint: Think about the consequences of a
7. Water samples are taken from water used for cooling as it is being discharged from a power plant into a river. It has been determined that as long as the mean temperature of the discharged water is at most 150°F, there will be no negative effects on the river’s ecosystem. To investigate
9. Two different companies have applied to provide cable television service in a certain region. Let p denote the proportion of all potential subscribers who favor the first company over the second. Consider testing H0: p .5 versus Ha: p .5 based on a random sample of 25 individuals. Let X denote
What is the probability distribution of the test statistic X when H0 is true? Use it to compute the probability of a type I error.
18. Reconsider the paint-drying situation of Example 8.2, in which drying time for a test specimen is normally distributed with 9. The hypotheses H0: 75 versus Ha: 75 are to be tested using a random sample of n 25 observations.a. How many standard deviations (of X) below the null value is x
84. Suppose the population distribution is normal with known .Let be such that 0 . For testing H0: 0 versus Ha: 0, consider the test that rejects H0 if either z z or z z , where the test statistic is Z (X 0)/(/n).a. Show that P(type I error) .b. Derive an expression for ( ).
74. In an investigation of the toxin produced by a certain poisonous snake, a researcher prepared 26 different vials, each containing 1 g of the toxin, and then determined the amount of antitoxin needed to neutralize the toxin. The sample average amount of antitoxin necessary was found to be 1.89
29. The amount of shaft wear (.0001 in.) after a fixed mileage was determined for each of n 8 internal combustion engines having copper lead as a bearing material, resulting in x 3.72 and s 1.25.a. Assuming that the distribution of shaft wear is normal with mean , use the t test at level .05
30. The recommended daily dietary allowance for zinc among males older than age 50 years is 15 mg/day. The article“Nutrient Intakes and Dietary Patterns of Older Americans:A National Study” (J. Gerontology, 1992: M145–150)
31. In an experiment designed to measure the time necessary for an inspector’s eyes to become used to the reduced amount of light necessary for penetrant inspection, the sample average time for n 9 inspectors was 6.32 sec and the sample standard deviation was 1.65 sec. It has previously been
32. A sample of 12 radon detectors of a certain type was selected, and each was exposed to 100 pCi/L of radon. The resulting readings were as follows:105.6 90.9 91.2 96.9 96.5 91.3 100.1 105.0 99.6 107.7 103.3 92.4a. Does this data suggest that the population mean reading under these conditions
34. For a fixed alternative value , show that ( ) 0 0 as n 0 for either a one-tailed or a two-tailed z test in the case of a normal population distribution with known .
85. After a period of apprenticeship, an organization gives an exam that must be passed to be eligible for membership. Let p P(randomly chosen apprentice passes). The organization wishes an exam that most but not all should be able to pass, so it decides that p .90 is desirable. For a
28. Minor surgery on horses under field conditions requires a reliable short-term anesthetic producing good muscle relaxation, minimal cardiovascular and respiratory changes, and a quick, smooth recovery with minimal aftereffects so that horses can be left unattended. The article “A Field Trial
27. Automatic identification of the boundaries of significant structures within a medical image is an area of ongoing research. The paper “Automatic Segmentation of Medical Images Using Image Registration: Diagnostic and Simulation Applications” (J. of Medical Engr. and Tech., 2005: 53–63)
19. The melting point of each of 16 samples of a certain brand of hydrogenated vegetable oil was determined, resulting in x 94.32. Assume that the distribution of melting point is normal with 1.20.a. Test H0: 95 versus Ha: 95 using a two-tailed level .01 test.b. If a level .01 test is used,
What value of n is necessary to ensure that (94) .1 when .01?
20. Lightbulbs of a certain type are advertised as having an average lifetime of 750 hours. The price of these bulbs is very favorable, so a potential customer has decided to go ahead with a purchase arrangement unless it can be conclusively demonstrated that the true average lifetime is smaller
21. The true average diameter of ball bearings of a certain type is supposed to be .5 in. A one-sample t test will be carried out to see whether this is the case. What conclusion is appropriate in each of the following situations?a. n 13, t 1.6, .05b. n 13, t 1.6, .05c. n 25, t
25. The desired percentage of SiO2 in a certain type of aluminous cement is 5.5. To test whether the true average percentage is 5.5 for a particular production facility, 16 independently obtained samples are analyzed. Suppose that the percentage of SiO2 in a sample is normally distributed with .3
22. The article “The Foreman’s View of Quality Control”(Quality Engr., 1990: 257–280) described an investigation into the coating weights for large pipes resulting from a galvanized coating process. Production standards call for a true average weight of 200 lb per pipe. The accompanying
23. Exercise 36 in Chapter 1 gave n 26 observations on escape time (sec) for oil workers in a simulated exercise, from which the sample mean and sample standard deviation are 370.69 and 24.36, respectively. Suppose the investigators had believed a priori that true average escape time would be at
24. Reconsider the sample observations on stabilized viscosity of asphalt specimens introduced in Exercise 46 in Chapter 1(2781, 2900, 3013, 2856, and 2888). Suppose that for a particular application, it is required that true average viscosity be 3000. Does this requirement appear to have been
38. Each of n specimens is to be weighed twice on the same scale. Let Xi and Yi denote the two observed weights for the ith specimen. Suppose Xi and Yi are independent of one another, each normally distributed with mean value i (the true weight of specimen i) and variance 2.a. Show that the
35. Let X1, . . . , Xn be a random sample from a pdf that is symmetric about . An estimator for that has been found to perform well for a variety of underlying distributions is the Hodges–Lehmann estimator. To define it, first compute for each i j and each j 1, 2, . . . , n the pairwise
34. The mean squared error of an estimator ˆ is MSE(ˆ) E(ˆ )2. If ˆ is unbiased, then MSE(ˆ) V(ˆ), but in general MSE(ˆ) V(ˆ) (bias)2. Consider the estimatorˆ 2 KS2, where S2 sample variance. What value of K minimizes the mean squared error of this estimator when the
33. At time t 0, there is one individual alive in a certain population. A pure birth process then unfolds as follows. The time until the first birth is exponentially distributed with parameter . After the first birth, there are two individuals alive. The time until the first gives birth again is
32.a. Let X1, . . . , Xn be a random sample from a uniform distribution on [0, ]. Then the mle of is ˆ Y max(Xi).Use the fact that Y y iff each Xi y to derive the cdf of Y. Then show that the pdf of Y max(Xi) is fY(y) {nyn n1 0 y 0 otherwiseb. Use the result of part (a) to
31. An estimator ˆ is said to be consistent if for any ! 0, P(⏐ˆ ⏐ !) 0 0 as n 0 . That is, ˆ is consistent if, as the sample size gets larger, it is less and less likely that ˆ will be further than ! from the true value of . Show that X is a consistent estimator of when 2
30. At time t 0, 20 identical components are put on test. The lifetime distribution of each is exponential with parameter .The experimenter then leaves the test facility unmonitored.On his return 24 hours later, the experimenter immediately terminates the test after noticing that y 15 of the
29. Consider a random sample X1, X2, . . . , Xn from the shifted exponential pdf f(x; , ) {e(x) x 0 otherwise Taking 0 gives the pdf of the exponential distribution considered previously (with positive density to the right of zero). An example of the shifted exponential distribution
28. Let X1, X2, . . . , Xn represent a random sample from the Rayleigh distribution with density function given in Exercise 15. Determinea. The maximum likelihood estimator of and then calculate the estimate for the vibratory stress data given in that exercise. Is this estimator the same as the
27. Let X1, . . . , Xn be a random sample from a gamma distribution with parameters and .a. Derive the equations whose solution yields the maximum likelihood estimators of and . Do you think they can be solved explicitly?b. Show that the mle of is ˆ X.
26. Refer to Exercise 25. Suppose we decide to examine another test spot weld. Let X shear strength of the weld.Use the given data to obtain the mle of P(X 400).[Hint: P(X 400) ((400 )/).]
25. The shear strength of each of ten test spot welds is determined, yielding the following data (psi):392 376 401 367 389 362 409 415 358 375a. Assuming that shear strength is normally distributed, estimate the true average shear strength and standard deviation of shear strength using the method
24. Refer to Exercise 20. Instead of selecting n 20 helmets to examine, suppose I examine helmets in succession until I have found r 3 flawed ones. If the 20th helmet is the third flawed one (so that the number of helmets examined that were not flawed is x 17), what is the mle of p? Is this
23. Two different computer systems are monitored for a total of n weeks. Let Xi denote the number of breakdowns of the first system during the ith week, and suppose the Xis are independent and drawn from a Poisson distribution with parameter 1. Similarly, let Yi denote the number of breakdowns of
22. Let X denote the proportion of allotted time that a randomly selected student spends working on a certain aptitude test.Suppose the pdf of X is f(x; ) ( 1)x 0 x 1 0 otherwise where 1 . A random sample of ten students yields data x1 .92, x2 .79, x3 .90, x4 .65, x5 .86, x6
21. Let X have a Weibull distribution with parameters and, so E(X) (1 1/)V(X) 2{(1 2/) [(1 1/)]2}a. Based on a random sample X1, . . . , Xn, write equations for the method of moments estimators of and . Show that, once the estimate of has been obtained, the estimate of
20. A random sample of n bike helmets manufactured by a certain company is selected. Let X the number among the n that are flawed, and let p P(flawed). Assume that only X is observed, rather than the sequence of S’s and F’s.a. Derive the maximum likelihood estimator of p. If n 20 and x
19. An investigator wishes to estimate the proportion of students at a certain university who have violated the honor code.Having obtained a random sample of n students, she realizes that asking each, “Have you violated the honor code?” will probably result in some untruthful responses.
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