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applied statistics and probability for engineers
Applied Statistics For Engineers And Scientists 3rd Edition Jay L. Devore, Nicholas R. Farnum, Jimmy A. Doi - Solutions
=+Surface Roughness in Ultrasonic Vibration Turning” (J. of Engr. Manuf., 2009: 641–652). The response variable is surface roughness (mm), and the independent variables are vibration amplitude(mm), depth of cut (mm), feed rate (mm/rev), and cutting speed (m/min), respectively.The regression
=+36. The accompanying Minitab regression output is based on data that appeared in the article “Application of Design of Experiments for Modeling
=+b. Residual and total sums of squares are .03836 and 5.1109, respectively. What proportion of observed variation in deposition rate can be attributed to the stated approximate relationship between deposition rate and the two predictor variables?
=+a. A least squares fit of y 5 a 1 b1x1 1 b2x2 to this data gave a 5 .0558, b1 5 .3749, and b2 5.0028. What value of deposition rate would you predict when wire feed rate 5 11.5 and welding speed 5 40? What is the value of the corresponding residual?
=+Stainless Steel Claddings Deposited by FCAW”(J. Mater. Engr. Perform., 2012: 1862–1872) investigated how y 5 deposition rate was influenced by x1 5 wire feed rate (Wf, in m/min) and x2 5 welding speed (S, in cm/min). The following 22 observations correspond to the experiment condition where
=+35. Recently there has been increased use of stainless steel claddings in industrial settings. Claddings are used to finish the exterior walls of a building and help weatherproof the structure. To ensure the quality of claddings, it is essential to know how welding parameters impact the cladding
=+b. What are the values of SSResid and SSTo?Verify that these values are consistent with the value of R-sq given on the output. Do you think the fit of the quadratic is good? Explain.
=+a. What is the equation of the best-fit quadratic?Use this quadratic to predict yield strength when temperature is 110.
=+regression function (a graph in the cited paper suggests that the authors did this):Predictor Coef SE Coef T P Constant 111.277 2.100 52.98 0.000 temp 0.32845 0.03303 9.94 0.010 tempsqd -0.0010050 0.0001213 -8.29 0.014 S=3.44398 R–Sq=98.1% R–Sq(adj)=96.3%Analysis of Variance Source DF SS MS
=+34. The accompanying data was extracted from the article “Effects of Cold and Warm Temperatures on Springback of Aluminum-Magnesium Alloy 5083-H111” (J. Engr. Manuf., 2009: 427–431). The response variable is yield strength (MPa), and the predictor is temperature ( C).x: 250 25 100 200 300
=+b. Use a statistical computer package to fit a quadratic function to this data and then predict bond strength when thickness is 500. Assess the fit of the quadratic to the data.
=+a. Is it possible to transform this data as described in this section so that there is an approximate linear relationship between the transformed variables? Why or why not?
=+Thickness: 220 220 220 220 370 Strength: 24.0 22.0 19.1 15.5 26.3 Thickness: 370 370 370 440 440 Strength: 24.6 23.1 21.2 25.2 24.0 Thickness: 440 440 680 680 680 Strength: 21.7 19.2 17.0 14.9 13.0 Thickness: 680 860 860 860 860 Strength: 11.8 12.2 11.2 6.6 2.8
=+33. The article “Residual Stresses and Adhesion of Thermal Spray Coatings” (Surface Engr., 2005:35–40) considered the relationship between the thickness (mm) of NiCrAl coatings deposited on stainless steel substrate and corresponding bond strength (MPa). The following data was read from a
=+b. Plot the residuals from your linear fit in part (a)and look for any patterns that might suggest an inappropriate choice of transformation. If necessary, return to part (a) and try a different transformation.
=+a. The authors were interested in predicting PLR based on the pile inner diameter. Transform only the independent variable x so that a scatterplot of the transformed data shows a substantial linear pattern. Then fit a straight line to this data, use the line to establish an approximate
=+diameter, d (mm), of nine test piles used in case studies. The data is given here:d: 691.0 292.0 83.7 37.2 78.9 PLR: 1.00 0.82 0.76 0.44 0.76 d: 107.9 82.5 1444.0 1444.0 PLR: 0.88 0.75 1.00 1.00
=+indicator of the degree of plugging, researchers often use the plug length ratio (PLR), which is the ratio of the plug length at the end of pile installation to the length of the pile. The article “Base Capacity of Open-Ended Steel Pipe Piles in Sand”(J. Geotech. Geoenviron. Engr., 2012:
=+32. There has been an increasing demand for openended steel pipe piles to be used as deep foundations for offshore and onshore structures. When an open-ended pile is driven into the ground, a soil plug often forms within the pile. The driving resistance and the base capacity of the pile are
=+c. Use the selected transformation to predict amplitude when cycles to failure 5 5000.
=+b. Which transformation from part (a) does the best job of producing an approximate linear relationship?
=+in order to predict strain amplitude from cycles to failure.Obs Cycfail Strampl Obs Cycfail Strampl 1 1326 .01495 11 7356 .00576 2 1593 .01470 12 7904 .00580 3 4414 .01100 13 79 .01212 4 5673 .01190 14 4175 .00782 5 29,516 .00873 15 34,676 .00596 6 26 .01819 16 114,789 .00600 7 843 .00810 17 2672
=+31. Failures in aircraft gas turbine engines due to high cycle fatigue is a pervasive problem. The article “Effect of Crystal Orientation on Fatigue Failure of Single Crystal Nickel Base Turbine Blade Superalloys” (J. of Engr. for Gas Turbines and Power, 2002: 161–176) gave the
=+c. Plot the residuals from your linear fit in part (b)and look for any patterns that might suggest an inappropriate choice of transformation. If necessary, return to part (b) and try a different transformation.
=+b. Transform only the dependent variable y so that a scatterplot of the transformed data shows a substantial linear pattern. Then fit a straight line to this data, use the line to establish an approximate relationship between x and y, and predict the dynamic shear modulus when the temperature
=+a. Construct a scatterplot of y 5 dynamic shear modulus versus x 5 temperature. Would it be reasonable to characterize the relationship between the two variables as approximately linear?
=+Temp: 54.4 46.1 43.3 29.4 G : 9.28 32.47 46.98 344.36 Temp: 21.1 12.7 4.4 G : 1,030.38 4,870.00 18,300.00
=+is a measure of the stiffness or resistance of the asphalt binder to deformation under load. In one experiment, the researchers measured the dynamic shear modulus of the asphalt binder samples over a range of testing temperatures (°C). The following is the corresponding data for binder type
=+30. In the article “Sensitivity of Oklahoma Binders on Dynamic Modulus of Asphalt Mixes and Distress Functions” (J. Mater. Civ. Engr., 2012:1076–1088), researchers measured various physical characteristics of performance grade asphalt binders commonly used in Oklahoma. One important
=+b. Find a transformation that produces an approximate linear relationship between the transformed values. Then fit a line to the transformed data and use it to obtain an equation that describes approximately the relationship between the untransformed variables.
=+a. Would you fit a straight line to the data and use it as a basis for predicting nondimensionalized total bed load from the unsteadiness parameter?Why or why not?
=+simulation, the article reported the computed value of the unsteadiness parameter Pgt and the nondimensionalized total bed load, Wt . One aim of the study was to investigate the behavior of y 5 Wt as a function of x 5 Pgt. Data from the experiment is given here:x: 0.0021 0.0041 0.0045 0.0046 y:
=+29. The authors of “Experimental and Numerical Investigation of Bed-Load Transport Under Unsteady Flows” (J. Hydraul. Engr., 2011: 1276–1282)simulated sediment yield of a gravel bed load under varying rates of water flow. The researchers wanted to mathematically model the behavior of
=+c. Fit a straight line to the (x, y=) data. Assess the quality of the fit. Finally, based on the linear fit, predict the value of failure time from a load of 85%.
=+. Would it be reasonable to characterize the relationship between these two variables to be linear?
=+b. Transform the response variable by computing y= 5 log(y). Construct a scatterplot of x and y=
=+Would it be reasonable to characterize the relationship between the two variables to be linear?
=+28. Polyester fiber ropes are increasingly being used as components of mooring lines for offshore structures in deep water. The authors of the paper “Quantifying the Residual Creep Life of Polyester Mooring Ropes” (Intl. J. of Offshore and Polar Explor., 2005:223–228) used the accompanying
=+virtually the same for all four. Based on a scatterplot and a residual plot for each data set, comment on the appropriateness of fitting a straight line; include any specific suggestions for how a “straight-line analysis” might be modified or qualified.Data set: 1–3 1 2 3 4 4 Variable: x y
=+27. Consider the following four (x, y) data sets; the first three have the same x values, so these values are listed only once (from “Graphs in Statistical Analysis,” Amer. Statistician, 1973: 17–21).For each of these four data sets, the values of the summary quantities, ^xi, ^yi, and so
=+e. Personal communication with the authors of the article revealed that there was one additional observation that was not included in their scatterplot: (6.53, 96.55). What impact does this additional observation have on the equation of the least squares line and the values of se and r 2?
=+d. What proportion of observed variation in removal efficiency can be attributed to the approximate linear relationship?
=+c. Roughly what is the size of a typical deviation of points in the scatterplot from the least squares line?
=+b. Determine the equation of the least square line, obtain a point prediction of removal efficiency when temperature 5 10.50, and calculate the value of the corresponding residual.
=+a. Does a scatterplot of the data suggest appropriateness of the simple linear regression model?
=+ Calculated summary quantities are ^ xi 5 384.26,^ yi 5 3149.04, ^ x 2i 5 5099.2412, ^ xi yi 5 37,850.7762, and ^ y 2i 5 309,892.6548.
=+and Organic Vapors in a Rock Medium Biofilter”(Water Environment Research, 2001: 426–435):Removal Removal Obs Temp % Obs Temp %1 7.68 98.09 17 8.55 98.27 2 6.51 98.25 18 7.57 98.00 3 6.43 97.82 19 6.94 98.09 4 5.48 97.82 20 8.32 98.25 5 6.57 97.82 21 10.50 98.41 6 10.22 97.93 22 17.83 98.51 7
=+26. In biofiltration of wastewater, air discharged from a treatment facility is passed through a damp porous membrane that causes contaminants to dissolve in water and be transformed into harmless products. The accompanying data on x 5 inlet temperature (°C) and y 5 removal efficiency (%)was
=+d. Predict the value of the compression index when the initial void ratio is 1.10. Would you feel comfortable using the least squares line to predict the compression index when the initial void ratio is .80? Explain.
=+c. What proportion of the observed variation in the compression index can be attributed to the approximate linear relationship between the two variables?
=+b. Determine the equation of the least squares line.
=+a. Using Cc as the response and e0 as the explanatory variable, create the corresponding scatterplot. Do the values of Cc appear to be perfectly linearly related to the e0 values?Explain.
=+Soil-Bentonite Backfills” (J. Geotech. Geoenviron. Engr., 2012: 15–25) reported the following data (read from a graph) for the Cc and e0 variables for sand–bentonite backfills with varying amounts and types of zeolites.e0: 0.988 1.018 1.058 1.070 1.085 1.145 Cc: 0.19 0.20 0.20 0.22 0.23
=+25. Two important properties of a soil are its initial void ratio (e0, a measure of soil porosity) and its compression index (Cc, an indicator of soil compressibility). The article “Consolidation and Hydraulic Conductivity of Zeolite-Amended
=+c. For the PMMA lens, predict the reduction rate of transmittance when sandblast momentum is at 50 g.m/s. Do the same for the glass lens type.d. Based on your results, which lens type performed better in this experiment?
=+b. Determine the equations for the least squares line for the PMMA and glass data sets. Interpret the slope for each equation.
=+a. In one graph, overlay the scatterplots for the PMMA and the glass data sets and comment on any interesting features. Be sure to use different symbols for each data set.
=+PMMA: 10.56 20.80 15.84 31.20 48.00 PMMA: 8.56 18.93 19.35 23.65 33.05 PMMA: 21.12 41.60 64.00 16.80 33.20 PMMA: 18.53 29.21 40.39 17.21 27.21 PMMA: 51.20 13.92 27.84 42.72 PMMA: 34.74 17.40 25.89 32.82 Glass: 35.20 52.80 105.60 52.80 70.40 Glass: 5.62 8.10 31.21 13.76 15.37 Glass: 56.00 48.00
=+by the authors, compares y 5 reduction rate of transmittance (%) and x 5 sandblast momentum (g .m/s) for 14 PMMA and 8 glass substrate samples:
=+and glass Fresnel lenses used in concentrator photovoltaic modules. In the experiment, the transmittance after sandblasting of acrylic polymethylmethacrylate (PMMA) and glass Fresnel lenses were measured. The experimental data, kindly provided
=+24. By their nature, deserts are typically exposed to large amounts of solar radiation. Thus, such regions seem to be prime locations for harvesting solar energy through the installation of photovoltaic modules. These modules rely on an optical system to collect sunlight, often through some
=+d. Would you feel comfortable using the least squares line to predict the compressive strength when the fiber weight percentage is 25? Explain.Now predict the value of y when x 5 25 and interpret the result.
=+c. Predict the value of the compressive strength when the fiber weight percentage is 6.5.
=+b. Determine the proportion of observed variation in strength that can be attributed to the approximate linear relationship between strength and fiber weight.
=+a. Determine the equation of the least squares line and interpret its slope.
=+23. Recall the data from Exercise 6 involving x 5 fiber weight (%) and y 5 compressive strength (MPa).
=+c. Construct a plot of the residuals. What does it suggest?
=+? Do these values suggest that the least squares line provides an effective summary of the relationship between the two variables?
=+b. What are the values of SSResid, SSTo, r 2, and se
=+a. Write the equation of the least squares line and use it to predict the value of recovered oil when added oil is 10 g.
=+Obs Dep Var Value Residual 1 0.6100 0.3548 0.2552 2 0.8400 0.7939 0.0461 3 1.5120 1.3209 0.1911 4 1.7920 1.9357 -0.1437 5 2.9520 2.6383 0.3137 6 2.8800 3.4287 -0.5487 7 4.4000 4.3069 0.0931 8 5.3460 5.2730 0.0730 9 6.3960 6.3269 0.0691 10 7.1890 7.4686 -0.2796 11 8.0850 8.6982 -0.6132 12 9.8400
=+Dep Mean 6.07513 Adj R-Sq 0.9953 C.V. 5.13266 Parameter Estimates Parameter Standard t Variable DF Estimate Error Value Pr > |t|Intercept 1 -0.52343 0.14528 -3.60 0.0032 oil_added 1 0.87825 0.01610 54.56
=+y: 4.33 4.44 4.40 4.26 4.32 4.34 x: 6.49 6.37 6.51 7.88 6.74 7.08 Dependent Variable: oil_recov Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 1 289.45805 289.45805 2977.07
=+22. Recall the data from Exercise 4 based on amount of oil added (in g) and the corresponding amount of oil recovered (in g) from wheat straw. Suppose that we want to use the least squares line to predict the amount of oil recovered from the wheat straw based on the initial amount of oil added.
=+d. Roughly what is the size of a typical deviation of points in the scatterplot from the least squares line?
=+c. Calculate and interpret the coefficient of determination.
=+b. Obtain the equation of the least squares line and interpret its slope.
=+a. Does a scatterplot of the data suggest an appropriate linear relationship between x and y?
=+x: 112.3 97.0 92.7 86.0 102.0 y: 75.0 71.0 57.7 48.7 74.3 x: 99.2 95.8 103.5 89.0 86.7 y: 73.3 68.0 59.3 57.8 48.5
=+21. For the past decade rubber powder has been used in asphalt cement to improve performance. The article “Experimental Study of Recycled RubberFilled High-Strength Concrete” (Magazine of Concrete Res., 2009: 549–556) included on a regression of y 5 axial strength (MPa) on x 5 cube
=+d. Does a residual plot indicate any deficiency in a straight line fit? Explain your reasoning.
=+c. Determine the proportion of observed variation in the response variable that can be attributed to the approximate linear relationship between strength and fiber weight.
=+b. Find the equation of the least squares line for this data and interpret its slope.
=+a. Does a scatterplot of the data suggest it is reasonable to assume an approximate linear relationship between x and y?
=+little research has been done on the characterization of dielectric properties of asphalt mixtures. The article “Dielectric Modeling of Asphalt Mixtures and Relationship with Density” (J. Transp. Engr., 2011: 104–111) reported on the dielectric response with percent air voids for various
=+20. Electromagnetic technologies such as ground penetrating radar offer effective nondestructive sensing techniques to determine a continuous profile of a pavement structure. The propagation of electromagnetic waves through the structure depends critically on the dielectric properties of the
=+b. The second observation has a very extreme y value (in the full data set consisting of 72 observations, there were 2 of these). This observation may have had a substantial impact on the form of the regression function and subsequent conclusions. Eliminate it and redo part(a). What do you
=+a. Determine the equation of the least squares line for this data and then calculate and interpret the coefficient of determination.
=+analysis. Here is representative data:x: 50 71 55 50 33 58 79 y: 152 1929 48 22 2 5 35 x: 26 69 44 37 70 20 45 49 y: 7 269 38 171 13 43 185 25
=+19. The invasive diatom species Didymosphenia geminata has the potential to inflict substantial ecological and economic damage in rivers. The article“Substrate Characteristics Affect Colonization by the Bloom-Forming Didymosphenia geminata”(Aquatic Ecology, 2010: 33–40) described an
=+18. Suppose data is collected on two quantitative variables, x and y. Let r be the corresponding sample correlation coefficient for (x, y). The x and y values are then transformed as follows: x= 5 a 1 bx, y= 5 c 1 dy wherea, b,c, and d are constants. Let r=be the corresponding sample
=+Can you see why r in part (a) is smaller than r in part (b)? Does this suggest that a correlation coefficient based on averages (called an “ecological” correlation) might be misleading? Explain.
=+c. Construct a scatterplot of the nine (x, y) pairs and another one of the three pairs of averages.
=+b. Let x1 be the average score on the first midterm exam for the 8 a.m. students and y1 be the average score on the second midterm for these students.Denote the two averages for the noon students by x2 and y2, and for the night students by x3 and y3.Calculate r for these three (x, y) pairs.
=+a. Calculate the sample correlation coefficient for the nine (x, y) pairs.
=+17. Nine students currently taking introductory statistics are randomly selected, and both the first midterm exam score (x) and the second midterm score(y) are determined. Three of the students have the class at 8 a.m., another three have it at noon, and the remaining three have a night class.
=+15. A sample of automobiles traversing a certain stretch of highway is selected. Each automobile travels at a roughly constant rate of speed, though speed does vary from auto to auto. Let x 5 speed and y 5 time needed to traverse this segment of highway. Would the sample correlation coefficient
=+14. An employee of an auction house has a list of 25 recently sold paintings. Eight artists were represented in these sales. The sale price of each painting is on the list. Would the correlation coefficient be an appropriate way to summarize the relationship between artist (x) and sale price
=+b. Calculate the value of the sample correlation coefficient. Does it confirm your impression from the scatterplot?
=+a. Construct a scatterplot of the data. Does it seem to be the case that, in general, when the measured load is low (high), the calculated load is also low (high)? For each sample, are the two variables relatively close in value?
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