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applied statistics and probability for engineers

Questions and Answers of Applied Statistics And Probability For Engineers

Kyphosis refers to severe forward flexion of the spine following corrective spinal surgery. A study carried out to determine risk factors for kyphosis reported the accompanying ages (months) for 40
Suppose x and y are related according to a probabilistic exponential model Y ex !, with V(!) a constant independent of x (as was the case in the simple linear model Y 0 1x !). Is V(Y) a
In each of the following cases, decide whether the given function is intrinsically linear. If so, identify x and y , and then explain how a random error term ! can be introduced to yield an
A plot in the article “Thermal Conductivity of Polyethylene:The Effects of Crystal Size, Density, and Orientation on the Thermal Conductivity” (Polymer Eng. and Science, 1972: 204–208) suggests
Exercise 14 presented data on body weight x and metabolic clearance rate/body weight y. Consider the following intrinsically linear functions for specifying the relationship between the two
Thermal endurance tests were performed to study the relationship between temperature and lifetime of polyester enameled wire (“Thermal Endurance of Polyester Enameled Wires Using Twisted Wire
The following data on mass rate of burning x and flame length y is representative of that which appeared in the article “Some Burning Characteristics of Filter Paper” (Combustion Science and
No tortilla chip aficionado likes soggy chips, so it is important to find characteristics of the production process that produce chips with an appealing texture. The following data on x frying time
If there is at least one x value at which more than one observation has been made, there is a formal test procedure for testing H0: Yx 01x for some values 0, 1 (the true regression function is
Recall that ˆ0 ˆ1x has a normal distribution with expected value 0 1x and variance2so that Z has a standard normal distribution. If S SSE/(n2) is substituted for , the resulting variable
a. Could a linear regression result in residuals 23, 27, 5, 17, 8, 9, and 15? Why or why not?b. Could a linear regression result in residuals 23, 27, 5, 17, 8, 12, and 2 corresponding to x values 3,
a. Express the ith residual Yi Yˆi (where Yˆi ˆ0 ˆ1xi) in the form cj Yj, a linear function of the Yjs. Then use rules of variance to verify that V(Yi Yˆi) is given by Expression
a. Show that n i1 ei 0 when the ei s are the residuals from a simple linear regression.b. Are the residuals from a simple linear regression independent of one another, positively correlated, or
Consider the following four (x, y) data sets; the first three have the same x values, so these values are listed only once Frank Anscombe, “Graphs in Statistical Analysis,” Amer. Statistician,
Continuous recording of heart rate can be used to obtain information about the level of exercise intensity or physical strain during sports participation, work, or other daily activities. The article
The article “Effects of Gamma Radiation on Juvenile and Mature Cuttings of Quaking Aspen” (Forest Science, 1967:240–245) reports the following data on exposure time to radiation (x, in kr/16
The accompanying data on x true density (kg/mm3) and y moisture content (% d.b.) was read from a plot in the article “Physical Properties of Cumin Seed” (J. Agric. Engr.Res., 1996:
Wear resistance of certain nuclear reactor components made of Zircaloy-2 is partly determined by properties of the oxide layer. The following data appears in an article that proposed a new
Example 12.6 presented the residuals from a simple linear regression of moisture content y on filtration rate x.a. Plot the residuals against x. Does the resulting plot suggest that a straight-line
Suppose the variables x commuting distance and y commuting time are related according to the simple linear regression model with 10.a. If n 5 observations are made at the x values x1 5,
87. Reconsider the situation of Exercise 73, in which x retained austenite content using a garnet abrasive and y abrasive wear loss were related via the simple linear regression model Y 0 1x .
b. Use the data to develop a way of predicting an HW measurement from a BOD POD measurement, and investigate the effectiveness of such predictions.
a. Use various methods to decide whether it is plausible that the two techniques measure on average the same amount of fat.
86. The article “Evaluating the BOD POD for Assessing Body Fat in Collegiate Football Players” (Medicine and Science in Sports and Exericse, 1999: 1350–1356) reports on a new air displacement
83. Use the formula for computing SSE to verify that r 2 1 SSE/SST.
81. Let sx and sy denote the sample standard deviations of the observed x’s and y’s, respectively [so s 2 x (xi x)2/(n 1)and similarly for s 2 y ].a. Show that an alternative expression
79. Show that SSE Syy ˆ1Sxy, which gives an alternative computational formula for SSE.80. Suppose that x and y are positive variables and that a sample of n pairs results in r 1. If the
d. Predict peak photovoltage when % absorption is 19.1, and compute the value of the corresponding residual.e. The article’s authors claim that there is a useful linear relationship between %
c. What proportion of the observed variation in peak photovoltage can be explained by the model relationship?
b. Assuming that the simple linear regression model is appropriate, obtain the equation of the estimated regression line.
77. The article “Photocharge Effects in Dye Sensitized Ag[Br,I]Emulsions at Millisecond Range Exposures” (Photographic Sci. and Engr., 1981: 138–144) gives the accompanying data on x % light
75. An investigation was carried out to study the relationship between speed (ft/sec) and stride rate (number of steps taken/sec) among female marathon runners. Resulting summary quantities included
73. The presence of hard alloy carbides in high chromium white iron alloys results in excellent abrasion resistance, making them suitable for materials handling in the mining and materials processing
e. Would you use the simple linear regression model to draw conclusions when the salinity level is 40?
d. What is the value of the sample correlation coefficient?
c. Does there appear to be a useful linear relationship between the two variables?
b. Calculate a point estimate of expected nitrate level when salinity level is 35.5.
a. What is the sample size n? [Hint: Look for degrees of freedom for SSE.]
72. The SAS output on the next page is based on data from the article “Evidence for and the Rate of Denitrification in the Arabian Sea” (Deep Sea Research, 1978: 431–435). The variables under
71. The accompanying data on x diesel oil consumption rate measured by the drain–weigh method and y rate measured by the CI-trace method, both in g/hr, was read from a graph in the article “A
70. Forensic scientists are often interested in making a measurement of some sort on a body (alive or dead) and then using that as a basis for inferring something about the age of the body. Consider
68. The appraisal of a warehouse can appear straightforward compared to other appraisal assignments. A warehouse appraisal involves comparing a building that is primarily an open shell to other such
c. Now suppose a sample of n 10,000 (x, y) pairs resulted in r .022. Test H0: 0 versus Ha: 0 at level .05.Is the result statistically significant? Comment on the practical significance of
b. Does this small P-value indicate that there is a very strong linear relationship between x and y (a value of that differs considerably from 0)? Explain.
a. What conclusion would be appropriate at level of significance .001?
66. Consider a time series—that is, a sequence of observations X1, X2, . . . obtained over time—with observed values x1, x2, . . . , xn. Suppose that the series shows no upward or downward trend
65. The article “Chronological Trend in Blood Lead Levels”(N. Engl. J. Med., 1983: 1373–1377) gives the following data on y average blood lead level of white children age 6 months to 5 years
64. The article “Increases in Steroid Binding Globulins Induced by Tamoxifen in Patients with Carcinoma of the Breast”(J. Endocrinology, 1978: 219–226) reports data on the effects of the drug
60. The article “A Dual-Buffer Titration Method for Lime Requirement of Acid Mine-soils” (J. of Environ. Qual., 1988:452–456) reports on the results of a study relating to revegetation of soil
58. The Turbine Oil Oxidation Test (TOST) and the Rotating Bomb Oxidation Test (RBOT) are two different procedures for evaluating the oxidation stability of steam turbine oils.
56. The article “Bone Density and Insertion Torque as Predictors of Anterior Cruciate Ligament Graft Fixation Strength” (The Amer. J. of Sports Med., 2004: 1421–1429)gave the accompanying data
55. Verify that V(ˆ0 ˆ1x) is indeed given by the expression in the text. [Hint: V(di Yi) d2 i V(Yi).]
54. The decline of water supplies in certain areas of the United States has created the need for increased understanding of relationships between economic factors such as crop yield and hydrologic
53. Consider the following four intervals based on the data of Example 12.4 (Section 12.2):a. A 95% CI for mean porosity when unit weight is 110b. A 95% PI for porosity when unit weight is 110c. A
52. Plasma etching is essential to the fine-line pattern transfer in current semiconductor processes. The article “Ion Beam-Assisted Etching of Aluminum with Chlorine”(J. Electrochem. Soc., 1985:
51. Refer to Example 12.12 in which x volume fraction of oxides/inclusions and y % elongation.a. MINITAB gave sˆ0ˆ1(.40) .0311 and sˆ0ˆ1(1.20) .0352.Why is the former estimated
c. Obtain an interval of plausible values for position resulting from a single observation to be made when concentration is .50, interpret your interval, and compare to the interval of (b).d. Obtain
b. Obtain an interval estimate for mean Fermi level position when concentration is .50, and interpret your estimate.
a. Obtain an interval estimate of the expected change in Fermi level position associated with an increase of .1 in Ge concentration, and interpret your estimate.
50. Silicon-germanium alloys have been used in certain types of solar cells. The paper “Silicon-Germanium Films Deposited by Low-Frequency Plasma-Enhanced Chemical Vapor Deposition” (J. of
48. The catch basin in a storm sewer system is the interface between surface runoff and the sewer. The catch basin insert is a device for retrofitting catch basins to improve pollutant removal
d. Suppose it had previously been believed that when crystallization temperature was 400°C, true average silver content would be .25. Carry out a test at significance level .05 to decide whether the
c. Calculate a 95% CI for the true average change in silver content associated with a 1°C increase in temperature.
b. How would the width of a 95% CI for true average silver content when temperature is 400°C compare to the width of the interval in part (a)? Answer without computing this new interval.
a. Estimate true average silver content when temperature is 500°C using a 95% confidence interval.
46. The article “The Incorporation of Uranium and Silver by Hydrothermally Synthesized Galena” (Econ. Geology, 1964: 1003–1024) reports on the determination of silver content of galena crystals
d. Interpret the hypotheses H0: 0 1251 80 and H0:0 1251 80, and then carry out a test at significance level .01.
c. How would the intervals of parts (a) and (b) compare to a CI and PI when filtration rate is 115? Answer without actually calculating these new intervals.
45. Reconsider the filtration rate–moisture content data introduced in Example 12.6 (see also Example 12.7).a. Compute a 90% CI for 0 1251, true average moisture content when the filtration rate
43. The probability of a type II error for the t test for H0: 1 10 can be computed in the same manner as it was computed for the t tests of Chapter 8. If the alternative value of1 is denoted by
40. Use the rules of expected value to show that ˆ0 is an unbiased estimator for 0 (assuming that ˆ1 is unbiased for 1).
d. Estimate the true average change in mist associated with a 1 cm/sec increase in velocity, and do so in a way that conveys information about precision and reliability.
c. The investigators were particularly interested in the impact on mist of increasing velocity from 100 to 1000(a factor of 10 corresponding to the difference between the smallest and largest x
b. What proportion of observed variation in mist can be attributed to the simple linear regression relationship between velocity and mist?
a. The investigators performed a simple linear regression analysis to relate the two variables. Does a scatter plot of the data support this strategy?
36. Mist (airborne droplets or aerosols) is generated when metal-removing fluids are used in machining operations to cool and lubricate the tool and workpiece. Mist generation is a concern to OSHA,
35. How does lateral acceleration—side forces experienced in turns that are largely under driver control—affect nausea as perceived by bus passengers? The article “Motion Sickness in Public
34. Refer to the MINITAB output of Exercise 20, in which x NO3 wet deposition and y lichen N (%).a. Carry out the model utility test at level .01, using the rejection region approach.b. Repeat
32. Exercise 16 of Section 12.2 gave data on x rainfall volume and y runoff volume (both in m3). Use the accompanying MINITAB output to decide whether there is a useful linear relationship
31. Reconsider the summary quantities given in Exercise 18 for the regression of y deflection factor on x temperature.a. Compute the estimated standard deviation sˆ1.b. Calculate a 95% CI for
29. Consider the following three data sets, in which the variables of interest are x commuting distance and y commuting time. Based on a scatter plot and the values of s and r 2, in which
27. Suppose an investigator has data on the amount of shelf space x devoted to display of a particular product and sales revenue y for that product. The investigator may wish to fit a model for which
26. Show that the “point of averages” (x, y) lies on the estimated regression line.
25. Show that b1 and b0 of expressions (12.2) and (12.3) satisfy the normal equations.
d. Because the largest x value in the sample greatly exceeds the others, this observation may have been very influential in determining the equation of the estimated line.Delete this observation and
c. What percentage of observed variation in steel weight loss can be attributed to the model relationship in combination with variation in deposition rate?
b. Calculate the equation of the estimated regression line.
a. Construct a scatter plot. Does the simple linear regression model appear to be reasonable in this situation?
24. The accompanying data was read from a graph that appeared in the article “Reactions on Painted Steel Under the Influence of Sodium Chloride, and Combinations Thereof” (Ind. Engr.Chem. Prod.
22.a. Use the summary quantities from Exercise 18 to estimate the standard deviation of the random deviation in the simple linear regression model.b. Based on the summary quantities from Exercise
21. Wrinkle recovery angle and tensile strength are the two most important characteristics for evaluating the performance of crosslinked cotton fabric. An increase in the degree of crosslinking, as
d. What is the value of total variation, and how much of it can be explained by the model relationship?
c. What is the estimate of ?
b. Predict lichen N for an NO3 deposition value of .5.
a. What are the least squares estimates of 0 and 1?
20. A number of studies have shown lichens (certain plants composed of an alga and a fungus) to be excellent bioindicators of air pollution. The article “The Epiphytic Lichen Hypogymnia Physodes as
18. The following summary statistics were obtained from a study that used regression analysis to investigate the relationship between pavement deflection and surface temperature of the pavement at
17. A regression of y calcium content (g/L) on x dissolved material (mg/cm2) was reported in the article “Use of Fly Ash or Silica Fume to Increase the Resistance of Concrete to Feed Acids”
15. Values of modulus of elasticity (MOE, the ratio of stress, i.e., force per unit area, to strain, i.e., deformation per unit length, in GPa) and flexural strength (a measure of the ability to
13. The accompanying data on x current density (mA/cm2)and y rate of deposition (m/min) appeared in the article“Plating of 60/40 Tin/Lead Solder for Head Termination Metallurgy” (Plating and

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