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modern mathematical statistics with applications
Modern Mathematical Statistics With Applications 1st Edition Jay L Devore - Solutions
19. The following data is representative of that reported in the article An Experimental Correlation of Oxides of Nitrogen Emissions from Power Boilers Based on Field Data (J. Engrg. Power, July 1973: 165—170), with x burner area liberation rate (MBtu/hr-ft2) and y NOx emission rate (ppm):x
17. The article Characterization of Highway Runoff in Austin, Texas, Area (J. Environ. Engrg., 1998:131—137) gave a scatter plot, along with the least squares line, of x rainfall volume (m3) and y runoff volume (m3) for a particular location. The accompanying values were read from the plot.x 5
16. As an alternative to the use of father s height to predict son s height, Galton also used the midparent height, the average of the father s and mother s heights. Here are the heights of nine female students along with their midparent heights in inches:Midparent 66.0 65.5 71.5 68.0 70.0 65.5
15. Refer to the data given in Exercise 1 on tank temperature and ef ciency ratio.a. Determine the equation of the estimated regression line.b. Calculate a point estimate for true average ef ciency ratio when tank temperature is 182.c. Calculate the values of the residuals from the least squares
14. The accompanying data on x current density(mA/cm2) and y rate of deposition (mm/min) appeared in the article Plating of 60/40 Tin/Lead Solder for Head Termination Metallurgy (Plating and Surface Finishing, Jan. 1997: 38—40). Do you agree with the claim by the article s author that a
13. Exercise 4 gave data on x BOD mass loading and yBODmass removal.Values of relevant summary quantities are n 14 xi 517 yi 346 xiyi 25,825a. Obtain the equation of the least squares line.b. Predict the value of BOD mass removal for a single observation made when BOD mass loading is 35,
12. In Example 12.4 the probability of cancer metastasizing was p(x) e2.5x/(1 e2.5x).a. Tabulate values of x, p(x), the odds p(x)/[1p(x)], and the log odds ln{p(x)/[1 p(x)]} for x 0, 1, 2, 3, . . . , 10.b. Explain what happens to the odds when x is increased by 1. Your explanation
9. The ow rate y (m3/min) in a device used for airquality measurement depends on the pressure drop x (in. of water) across the device s lter. Suppose that for x values between 5 and 20, the two variables are related according to the simple linear regression model with true regression line y .12
8. Referring to Exercise 7, suppose that the standard deviation of the random deviation e is 350 psi.a. What is the probability that the observed value of 28-day strength will exceed 5000 psi when the value of accelerated strength is 2000?b. Repeat part (a) with 2500 in place of 2000.c. Consider
6. One factor in the development of tennis elbow, a malady that strikes fear in the hearts of all serious tennis players, is the impact-induced vibration of the racket-and-arm system at ball contact. It is well known that the likelihood of getting tennis elbow depends on various properties of the
5. The article Objective Measurement of the Stretchability of Mozzarella Cheese (J. Texture Stud., 1992: 185—194) reported on an experiment to investigate how the behavior of mozzarella cheese varied with temperature. Consider the accompanying data on xtemperature and yelongation (%) at failure
4. A study to assess the capability of subsurface ow wetland systems to remove biochemical oxygen demand(BOD) and various other chemical constituents resulted in the accompanying data on x BOD mass loading (kg/ha/d) and y BOD mass removal(kg/ha/d) ( Subsurface Flow Wetlands A Performance
3. Bivariate data often arises from the use of two different techniques to measure the same quantity. As an example, the accompanying observations on x hydrogen concentration (ppm) using a gas chromatography method and y concentration using a new sensor method were read from a graph in the
2. The article Exhaust Emissions from Four-Stroke LawnMower Engines (J. AirWater Manag. Assoc., 1997: 945—952) reported data from a study in which both a baseline gasoline mixture and a reformulated gasoline were used. Consider the following observations on age (yr) and NOx emissions
1. The ef ciency ratio for a steel specimen immersed in a phosphating tank is the weight of the phosphate coating divided by the metal loss (both in mg/ft2).The article Statistical Process Control of a Phosphate Coating Line (Wire J. Internat., May 1997:78—81) gave the accompanying data on tank
70. Four plots were available for an experiment to compare clover accumulation for four different sowing rates ( Performance of Overdrilled Red Clover with Different Sowing Rates and Initial Grazing Managements, New Zeal. J. Experiment. Agric., 1984:71—81). Since the four plots had been grazed
68. The results of a study on the effectiveness of line drying on the smoothness of fabric were summarized in the article Line-Dried vs. Machine-Dried Fabrics: Comparison of Appearance, Hand, and Consumer Acceptance (Home Econ. Res. J., 1984:27—35). Smoothness scores were given for nine different
67. In Example 11.10, subtract from each observation in the ith sample (i 1, . . . , 6) to obtain a set of 18 residuals. Then construct a normal probability plot and comment on the plausibility of the normality assumption.xi#acixi# aa c2i Ji b1/2 # 3 1I 12 # MSE # Fa,I1,nI 41/2 gcimi gcimi
65. Four types of mortars ordinary cement mortar(OCM), polymer impregnated mortar (PIM), resin mortar (RM), and polymer cement mortar (PCM)were subjected to a compression test to measure strength (MPa). Three strength observations for each mortar type are given in the article Polymer Mortar
64. Recall from Section 11.2 that if c1, c2, . . . , cI are numbers satisfying then c1m1. . . cImI is called a contrast in the mi s. Notice that with c1 1, c2 1, c3 . . . cI 0, m1m2, which implies that every pairwise difference between mi s is a contrast (so is, e.g., m1 .5m2
63. The critical ick er frequency (cff) is the highest frequency(in cycles/sec) at which a person can detect the ick er in a ick ering light source. At frequencies above the cff, the light source appears to be continuous even though it is actually ick ering. An investigation carried out to see
62. Four laboratories (1—4) are randomly selected from a large population, and each is asked to make three determinations of the percentage of methyl alcohol in specimens of a compound taken from a single batch. Based on the accompanying data, are differences among laboratories a source of
61. An article in the British scienti c journal Nature( Sucrose Induction of Hepatic Hyperplasia in the Rat, August 25, 1972: 461) reports on an experiment in which each of v e groups consisting of six rats was put on a diet with a different carbohydrate.At the conclusion of the experiment, the DNA
60. Numerous factors contribute to the smooth running of an electric motor ( Increasing Market Share Through Improved Product and Process Design:An Experimental Approach, Qual. Engrg., 1991:361—369).In particular, it is desirable to keep motor noise and vibration to a minimum. To study the effect
59. The article Computer-Assisted Instruction Augmented with Planned Teacher/Student Contacts(J. Experiment. Ed., Winter 1980—1981: 120—126)compared ve different methods for teaching descriptive statistics. The ve methods were traditional lecture and discussion (L/D), programmed textbook
57. When both factors are random in a two-way ANOVA experiment with K replications per combination of factor levels, the expected mean squares are E(MSE)s2,, and.a. What F ratio is appropriate for testing H0G:versus HaG: ?b. Answer part (a) for testing versus HaA: and H0B: versus HaB:s .
56. Show how a 100(1 a)% t CI for can be obtained. Then compute a 95% interval for a2 a3 using the data from Example 11.16. [Hint:With ua2 a3, the result of Exercise 55a indicates how to obtain . Then compute V( ) and and obtain an estimate of by using to estimate s(which identi es the
55.a. Show that , so that is an unbiased estimator for ai (in the xed effects model).b. With ,show that is an unbiased estimator for gij (in the xed effects model).gˆij gˆ ij Xij# Xi ## X#j# X###E1Xi ## X### 2 ai Xi## X###50,443,409 x2 x ij# 2 ijk 16,815,853
54. The accompanying data was obtained in an experiment to investigate whether compressive strength of concrete cylinders depends on the type of capping material used or variability in different batches ( The Effect of Type of Capping Material on the Compressive Strength of Concrete Cylinders,
53. A study was carried out to compare the writing lifetimes of four premium brands of pens. It was thought that the writing surface might affect lifetime, so three different surfaces were randomly selected. A writing machine was used to ensure that conditions were otherwise homogeneous (e.g.,
52. In an experiment to investigate the effect of cement factor (number of sacks of cement per cubic yard)on exural strength of the resulting concrete ( Studies of Flexural Strength of Concrete. Part 3: Effects of Variation in Testing Procedure, Proceedings ASTM, 1957: 1127—1139), I 3 different
51. The accompanying data resulted from an experiment to investigate whether yield from a certain chemical process depended either on the formulation of a particular input or on mixer speed.Speed 60 70 80 189.7 185.1 189.0 1 188.6 179.4 193.0 Formulation 190.1 177.3 191.1 165.1 161.7 163.3 2 165.9
50. The article Towards Improving the Properties of Plaster Moulds and Castings (J. Engrg. Manuf., x2## 4029.10, and x3## 3960.02 xi## x1## 4010.88, Carbon Sand Fiber Addition Addition Casting Wet-Mold(%) (%) Hardness Strength 0 0 61.0 34.0 0 0 63.0 16.0 15 0 67.0 36.0 15 0 69.0 19.0 30 0
49. In an experiment to assess the effects of curing time(factor A) and type of mix (factor B) on the compressive strength of hardened cement cubes, three different curing times were used in combination with four different mixes, with three observations obtained for each of the 12 curing
48. The power curves of Figures 11.5 and 11.6 can be used to obtain b P(type II error) for the F test in two-factor ANOVA. For xed values of a1, a2, . . . , aI, the quantity is computed.Then the gure corresponding to v1I1 is entered on the horizontal axis at the valuef, the power is read on the
47. Use the fact that E(Xij) m ai bj with ai bj 0 to show that , so that is an unbiased estimator for ai.
45. Suppose that in the experiment described in Exercise 40 the ve houses had actually been selected at random from among those of a certain age and size, so that factor B is random rather than xed. Test H0:versus Ha:0 using a level .01 test.
44. Check the normality and constant variance assumptions graphically for the data of Example 11.15.
43. The strength of concrete used in commercial construction tends to vary from one batch to another.Consequently, small test cylinders of concrete sampled from a batch are cured for periods up to about 28 days in temperature- and moisture-controlled environments before strength measurements are
42. The article The Effects of a Pneumatic Stool and a One-Legged Stool on Lower Limb Joint Load and Muscular Activity During Sitting and Rising (Ergonomics, 1993: 519—535) gives the accompanying data on the effort required of a subject to arise from four different types of stools (Borg scale).
41. The article Rate of Stuttering Adaptation Under Two Electro-Shock Conditions (Behav. Res. Therapy, 1967: 49—54) gives adaptation scores for three different treatments: (1) no shock, (2) shock following each stuttered word, and (3) shock during each moment of stuttering. These treatments were
39. In an experiment to assess the effect of the angle of pull on the force required to cause separation in electrical connectors, four different angles (factor A)were used and each of a sample of ve connectors(factor B) was pulled once at each angle ( A Mixed Model Factorial Experiment in Testing
38. In an experiment to see whether the amount of coverage of light-blue interior latex paint depends either on the brand of paint or on the brand of roller used, 1 gallon of each of four brands of paint was applied using each of three brands of roller, resulting in the following data (number of
37. The data set shown above is from the article Compounding of Discriminative Stimuli from the Same and Different Sensory Modalities (J. Experiment.Anal. Behav., 1971: 337—342). Rat response was maintained by xed interval schedules of reinforcement in the presence of a tone or two separate
36. Four different coatings are being considered for corrosion protection of metal pipe. The pipe will be buried in three different types of soil. To investigate whether the amount of corrosion depends either on the coating or on the type of soil, 12 pieces of pipe are selected. Each piece is
35. The number of miles of useful tread wear (in 1000 s)was determined for tires of each of ve different makes of subcompact car (factor A, with I 5) in combination with each of four different brands of radial tires (factor B, with J 4), resulting in IJ 20 observations. The values SSA 30.6,
34. Simplify E(MSTr) for the random effects model when J1 J2 . . . JI J.
33. Suppose that Xij is a binomial variable with parameters n and pi (so it is approximately normal when npi 10 and nqi 10). Then since mi npi,. How should the Xij s be transformed so as to stabilize the variance? [Hint: g(mi) mi(1 mi/n).]
32. In an experiment to compare the quality of four different brands of reel-to-reel recording tape, ve 2400-ft reels of each brand (A—D) were selected and the number of aws in each reel was determined.A: 10 5 12 14 8 B: 14 12 17 9 8 C: 13 18 10 15 18 D: 17 16 12 22 14 It is believed that the
31. When sample sizes are not equal, the noncentrality parameter is and . Referring to Exercise 22, what is the power of the test when m2 m3, m1 m2 s, and m4 m2 s?
30. Reconsider Example 11.7 involving an investigation of the effects of different heat treatments on the yield point of steel ingots.a. If J8 and s1, what is b for a level .05 F test when m1 m2, m3 m1 1, and m4 m1 1?b. For the alternative of part (a), what value of J is necessary to
29. When sample sizes are equal (Ji J), the parameters a1, a2, . . . , aI of the alternative parameterization are restricted by For unequal sample sizes, the most natural restriction is Jiai 0. Use this to show that What is E(MSTr) when H0 is true? [This expectation is correct if Jiai 0 is
28. In single-factor ANOVA with sample sizes Ji (i 1,. . . , I), show that, where n .
27. Although tea is the world s most widely consumed beverage after water, little is known about its nutritional value. Folacin is the only B vitamin present in any signi cant amount in tea, and recent advances in assay methods have made accurate determination of folacin content feasible. Consider
26. Samples of six different brands of diet/imitation margarine were analyzed to determine the level of physiologically active polyunsaturated fatty acids(PAPFUA, in percentages), resulting in the following data:Imperial 14.1 13.6 14.4 14.3 Parkay 12.8 12.5 13.4 13.0 12.3 Blue Bonnet 13.5 13.4 14.1
25. Lipids provide much of the dietary energy in the bodies of infants and young children. There is a growing interest in the quality of the dietary lipid supply during infancy as a major determinant of growth, visual and neural development, and longterm health. The article Essential Fat
24. The following partial ANOVA table is taken from the article Perception of Spatial Incongruity(J. Nervous Mental Disease, 1961: 222) in which the abilities of three different groups to identify a perceptual incongruity were assessed and compared.All individuals in the experiment had been
23. Apply the modi ed Tukey s method to the data in Exercise 22 to identify signi cant differences among the mi s.
22. The following data refers to yield of tomatoes(kg/plot) for four different levels of salinity; salinity level here refers to electrical conductivity (EC), where the chosen levels were EC 1.6, 3.8, 6.0, and 10.2 nmhos/cm:1.6 59.5 53.3 56.8 63.1 58.7 3.8 55.2 59.1 52.8 54.5 6.0 51.7 48.8 53.9
21. The article The Effect of Enzyme Inducing Agents on the Survival Times of Rats Exposed to Lethal Levels of Nitrogen Dioxide (Toxicol. Appl. Pharmacol., 1978: 169—174) reports the following data on survival times for rats exposed to nitrogen dioxide(70 ppm) via different injection regimens.
20. Refer to Exercise 19 and suppose 10, 15, and 20. Can you now nd a value of SSE that produces such a contradiction between the F test and Tukey s procedure?
19. Consider a single-factor ANOVA experiment in which I3, J5, 10, 12, and 20.Find a value of SSE for which f F.05,2,12, so that H0: m1 m2 m3 is rejected, yet when Tukey s procedure is applied none of the mi s can be said to differ signi cantly from one another.
18. Consider the accompanying data on plant growth after the application of different types of growth hormone.12 x3#x2# 1 13 17 7 14 2 21 13 20 17 Hormone 3 18 15 20 17 4 7 11 18 10 5 6 11 15 8a. Perform an F test at level a .05.b. What happens when Tukey s procedure is applied?
17. Refer to Exercise 4. Compute a 95% t CI for the contrast u (m1 m2) m3.
16. Reconsider the axial stiffness data given in Exercise 7. ANOVA output from MINITAB follows:Analysis of Variance for stiffness Source DF SS MS F P length 4 43993 10998 10.48 0.000 Error 30 31475 1049 Total 34 75468 Level N Mean StDev 4 7 333.21 36.59 6 7 368.06 28.57 8 7 375.13 20.83 10 7 407.36
15. Use Tukey s procedure on the data of Exercise 5 to identify differences in true average total Fe among the four types of formations (use MSE 15.64).
14. Use Tukey s procedure on the data in Exercise 3 to identify differences in true average ight times among the four types of mosquitos.
13. Repeat Exercise 12 supposing that 502.8 in addition to 427.5.
12. In Exercise 11, suppose 427.5. Now which true average spreading rates differ signi cantly from one another? Be sure to use the method of underscoring to illustrate your conclusions, and write a paragraph summarizing your results
11. An experiment to compare the spreading rates of five different brands of yellow interior latex paint available in a particular area used 4 gallons (J 4)of each paint. The sample average spreading rates(ft2/gal) for the five brands were 462.0, 512.8, 437.5, 469.3, and 532.1.The
10. In single-factor ANOVA with I treatments and J observations per treatment, let m (1/I)a. Express in terms of m. [Hint: (1/I) ]b. Compute . (Hint: For any rv Y, E(Y2) V(Y) [E(Y)]2.)c. Compute E1X2 . ## 2 E1X2i# 2 gXi#E1X## 2 X##gmi#d. Compute E(SSTr) and then show thate. Using the result
9. Derive the fundamental identity SST SSTr SSE by squaring both sides of Equation (11.2)and summing over all i and j. Hint: For any particular i, g .
8. Six samples of each of four types of cereal grain grown in a certain region were analyzed to determine thiamin content, resulting in the following data (mg/g):Wheat 5.2 4.5 6.0 6.1 6.7 5.8 Barley 6.5 8.0 6.1 7.5 5.9 5.6 Maize 5.8 4.7 6.4 4.9 6.0 5.2 Oats 8.3 6.1 7.8 7.0 5.5 7.2a. Check the ANOVA
7. A study of the properties of metal plate—connected trusses used for roof support ( Modeling Joints Made with Light-Gauge Metal Connector Plates, Forest Products J., 1979: 39—44) yielded the following observations on axial stiffness index (kips/in.)for plate lengths 4, 6, 8, 10, and 12 in.:4:
6. In an experiment to investigate the performance of four different brands of spark plugs intended for use on a 125-cc two-stroke motorcycle, ve plugs of each brand were tested for the number of miles (at a constant speed) until failure. The partial ANOVA table for the data is given here. Fill in
5. The article Origin of Precambrian Iron Formations(Econ. Geol., 1964: 1025—1057) reports the following data on total Fe for four types of iron formation(1 carbonate, 2 silicate, 3 magnetite, 4 hematite).1: 20.5 28.1 27.8 27.0 28.0 25.2 25.3 27.1 20.5 31.3 2: 26.3 24.0 26.2 20.2 23.7
4. Consider the following summary data on the modulus of elasticity ( 106 psi) for lumber of three different grades (in close agreement with values in the article Bending Strength and Stiffness of Second-Growth Douglas-Fir Dimension Lumber (Forest Products J., 1991: 35—43), except that the
3. In a study to assess the effects of malaria infection on mosquito hosts ( Plasmodium cynomolgi: Effects of Malaria Infection on Laboratory Flight Performance of Anopheles stephensi Mosquitos, Experiment.Parasitol., 1977: 397—404), mosquitos were fed on either infective or noninfective rhesus
2. The lumen output was determined for each of I 3 different brands of 60-watt soft-white lightbulbs, with J 8 bulbs of each brand tested. The sums of squares were computed as SSE 4773.3 and SSTr 591.2. State the hypotheses of interest (including word de nitions of parameters), and use the
113. Let R1 be a rejection region with signi cance level a for testingH01: u 1 versusHa1: u 1, and let R2 be a level a rejection region for testing H02: u 2 versus Ha2: u 2, where 1 and 2 are two disjoint sets of possible values of u. Now consider testing H0: u 1 2 versus the
112. Referring to Exercise 111, develop a large-sample con dence interval formula for l1 l2. Calculate the interval for the data given there using a con dence level of 95%.
111. Let X1, . . . , Xm be a random sample from a Poisson distribution with parameter l1, and let Y1, . . . , Yn be a random sample from another Poisson distribution with parameter l2. We wish to test H0:l1 l2 0 against one of the three standard alternatives.Since m l for a Poisson
110. McNemar s test, developed in Exercise 55, can also be used when individuals are paired (matched) to yield n pairs and then one member of each pair is given treatment 1 and the other is given treatment 2. Then X1 is the number of pairs in which both treatments were successful, and similarly for
108. The following data refers to airborne bacteria count (number of colonies/ft3) both for m 8 carpeted hospital rooms and for n 8 uncarpeted rooms ( Microbial Air Sampling in a Carpeted Hospital, J. Environ. Health, 1968: 405). Does there appear to be a difference in true average bacteria
107. Suppose a level .05 test of H0: m1 m2 0 versus Ha: m1 m2 0 is to be performed, assuming s1 s2 10 and normality of both distributions, using equal sample sizes (m n). Evaluate the probability of a type II error when m1m21 and n 25, 100, 2500, and 10,000. Can you think of real
106. The insulin-binding capacity (pmol/mg protein)was measured for four different groups of rats:(1) nondiabetic, (2) untreated diabetic, (3) diabetic treated with a low dose of insulin, (4) diabetic treated with a high dose of insulin. The accompanying table gives sample sizes and sample standard
105. An experiment to determine the effects of temperature on the survival of insect eggswas described in the article Development Rates and a Temperature-Dependent Model of Pales Weevil (Environ. Entomology, 1987: 956—962). At 11C, 73 of 91 eggs survived to the next stage of development. At
103. How does energy intake compare to energy expenditure?One aspect of this issue was considered in the article Measurement of Total Energy Expenditure by the Doubly Labelled Water Method in Professional Soccer Players (J. Sports Sci.,2002: 391—397), which contained the accompanying data
102. In an experiment to compare bearing strengths of pegs inserted in two different types of mounts, a sample of 14 observations on stress limit for red oak mounts resulted in a sample mean and sample standard deviation of 8.48 MPa and .79 MPa, respectively, whereas a sample of 12 observations
101. The accompanying data on the alcohol content of wine is representative of that reported in a study in which wines from the years 1999 and 2000 were randomly selected and the actual content was determined by laboratory analysis (London Times, Aug. 5, 2001).Wine 1 2 3 4 5 6 Actual 14.2 14.5 14.0
100. The accompanying data on response time appeared in the article The Extinguishment of Fires Using Low-Flow Water Hose Streams Part II (Fire Techn., 1991: 291—320). The samples are independent, not paired.Good visibility .43 1.17 .37 .47 .68 .58 .50 2.75 Poor visibility 1.47 .80 1.58 1.53 4.33
99. The accompanying summary data on the ratio of strength to cross-sectional area for knee extensors is taken from the article Knee Extensor and Knee Flexor Strength: Cross-Sectional Area Ratios in Young and Elderly Men (J. Gerontol., 1992:M204—M210).Sample Sample Standard Group Size Mean Error
98. Long-term exposure of textile workers to cotton dust released during processing can result in substantial health problems, so textile researchers have been investigating methods that will result in reduced risks while preserving important fabric properties.The accompanying data on roving
97. The article Pine Needles as Sensors ofAtmospheric Pollution (Environ. Monitoring, 1982: 273—286)reported on the use of neutron-activity analysis to determine pollutant concentration in pine needles.According to the article s authors, These observations strongly indicated that for those
94. The derailment of a freight train due to the catastrophic failure of a traction motor armature bearing provided the impetus for a study reported in the article LocomotiveTraction MotorArmature Bearing Life Study (Lubricat. Engrg., Aug. 1997: 12—19).A sample of 17 high-mileage traction motors
92. The following summary data on bending strength(lb-in/in) of joints is taken from the article Bending Strength of Corner Joints Constructed with Injection Molded Splines (Forest Products J., April 1997: 89—92). Assume normal distributions.Sample Sample Sample Type Size Mean SD Without side
91. The article Quantitative MRI and Electrophysiology of Preoperative Carpal Tunnel Syndrome in a Female Population (Ergonomics, 1997: 642—649)reported that (473.3, 1691.9) was a large-sample 95% con dence interval for the difference between true average thenar muscle volume (mm3) for sufferers
90. Is the response rate for questionnaires affected by including some sort of incentive to respond along with the questionnaire? In one experiment, 110 questionnaires with no incentive resulted in 75 being returned, whereas 98 questionnaires that included a chance to win a lottery yielded 66
89. The authors of the article Dynamics of Canopy Structure and Light Interception in Pinus elliotti, North Florida (Ecol. Monogr., 1991: 33—51)planned an experiment to determine the effect of fertilizer on a measure of leaf area. A number of plots were available for the study, and half were
88. The accompanying summary data on compression strength (lb) for 12 10 8 in. boxes appeared in the article Compression of Single-Wall Corrugated Shipping Containers Using Fixed and Floating Test Platens (J. Testing Eval., 1992: 318—320). The authors stated that the difference between the
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