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applied statistics and probability for engineers

Questions and Answers of Applied Statistics And Probability For Engineers

=+Variation in Forming of High Strength Steels” (J. of Manuf. Sci. and Engr., 2008: 1–9) included data on y 5 springback from the wall opening angle and x 5 blank holder pressure (BHP). Three
=+43. The use of high-strength steels (HSS) rather than aluminum and magnesium alloys in automotive body structures reduces vehicle weight. However, HSS use is still problematic because of
=+b. If we include in the model only the predictors x1, x2, and x3, the corresponding SSResid 5 11.428.Carry out a test at significance level .01 to decide whether at least one of the second-order
=+a. The authors fit the complete second-order model with predictors x1, x2, x3, x1 2, x2 2, x3 2, x1x2, x1x3, and x2x3, which resulted in SSResid 5 .215 and SSTo 5 16.798. Determine the
=+42. Soluble dietary fiber (SDF) can provide health benefits by lowering blood cholesterol and glucose levels. The article “Effects of Twin-Screw Extrusion on Soluble Dietary Fiber and
=+2, and x1x2 resulted in SSResid 5 390.64. Carry out a test at significance level .01 to decide whether at least one of the second-order predictors provides useful information about shear strength.
=+41. The article “The Undrained Strength of Some Thawed Permafrost Soils” (Canadian Geotechnical J., 1979: 420–427) reported the following data on undrained shear strength of sandy soil (y, in
=+by the first-order predictors? State and test the appropriate hypotheses using a significance level of .05.
=+c. Fitting the model with all 4 predictors as well as all second-order interactions gave R2 5 .960(this model was also fit by the investigators).Does it appear that at least one of the interaction
=+b. Fitting the model with predictors x3, x4, and the interaction x3x4 gave R2 5 .834. Does this model appear to be useful? Can an F test be used to compare this model to the model of part (a)?
=+a. Does there appear to be a useful relationship between power and at least one of the predictors?Carry out a formal test of hypotheses.
=+ Here is the Minitab output from fitting the model with the aforementioned independent variables as predictors (also fit by the authors of the cited article):Predictor Coef SE Coef T p Constant
=+40. The article “Sensitivity Analysis of a 2.5 kW Proton Exchange Membrane Fuel Cell Stack by Statistical Method” (J. of Fuel Cell Sci. and Tech., 2009: 1–6)used regression methodology to
=+e. When x 5 150, the estimated standard deviation of yn is syn 5 .1410. Calculate a 99% confidence interval for true average power when frequency is 150, and also a 99% prediction interval for a
=+d. Carry out a test of hypotheses to decide whether the quadratic predictor should be retained in the model.
=+c. Carry out a test of hypotheses to decide whether the quadratic regression model is useful.
=+b. What proportion of observed variation in output power can be attributed to the model relationship between power and frequency?
=+a. Why is b2 negative rather than positive?
=+ Fitting a quadratic regression model to this data yielded the following summary quantities:a521.5127, b15.391902, b252.00163141, SSResid 5 .29, SSTo 5 202.87, and sb2 5 .00003391.
=+39. The accompanying data on x 5 frequency (MHz)and y 5 power (W) for a certain laser configuration was read from a graph in the article “Frequency Dependence in RF Discharge Excited Waveguide
=+d. The estimated standard deviation of a prediction for repair time when elapsed time is 6 months and the repair is electrical is .192. Predict repair time under these circumstances by calculating
=+c. Calculate and interpret a 95% confidence interval for 2.
=+b. Given that elapsed time since the last service remains in the model, does type of repair provide useful information about repair time? State and test the appropriate hypotheses using a
=+a. Does there appear to be a useful linear relationship between repair time and the two model predictors? Carry out a test of the appropriate hypotheses using a significance level of .05.
=+38. A regression analysis carried out to relate y 5 repair time for a water filtration system (hr) to x1 5 elapsed time since the previous service (months) and x2 5 type of repair (1 if electrical
=+c. Calculate a 99% confidence interval for true mean yarn tenacity when yarn count is 16.5, yarn contains 50% polyester, first nozzle pressure is 3, and second nozzle pressure is 5 if the estimated
=+b. Again using n 5 25, calculate the value of adjusted R2.
=+a. Assuming that the sample size was n 5 25, state and test the appropriate hypotheses to decide whether the fitted model specifies a useful linear relationship between the dependent variable and
=+). The estimate of the constant term in the corresponding multiple regression equation was 6.121. The estimated coefficients for the four predictors were 2.082, .113, .256, and 2.219,
=+37. The article “Analysis of the Modeling Methodologies for Predicting the Strength of Air-Jet Spun Yarns” (Textile Res. J., 1997: 39–44) reported on a study carried out to relate yarn
=+c. Fitting the model with predictors x1 and x2 gave SSResid 5 27,454, whereas fitting with x1, x2, and x3 5 x1x2 resulted in SSResid 5 20519.Using 5 .01, can we conclude that the x1x2 term adds
=+b. The estimated standard deviation of yn when x1 is 20,000 and x2 is .002 is syn 5 21.7. Calculate a 95% confidence interval for the mean value of deposition under these circumstances.
=+a. Does there appear to be a useful linear relationship between y and at least one of the predictors?
=+36. Exercise 37 of Section 3.5 gave R output for a regression of y 5 deposition over a specified time period on two complex predictors x1 and x2 defined in terms of PAH air concentrations for
=+d. Calculate a 95% prediction interval for the deposition rate resulting from a single experimental run with x1 5 11.5 and x2 5 40.
=+c. When x1 5 11.5 and x2 5 40, the estimated standard deviation of yn is syn 5 .02438. Calculate a 95% confidence interval for true average deposition rate for the given values of x1 and x2.
=+b. Calculate and interpret a 95% confidence interval for 2, the population regression coefficient of x2
=+a. Carry out the model utility test.
=+WeldSpd 0.002775 0.001121 2.47 0.023 s = 0.0448530 R-sq = 99.3% R-sq(adj) = 99.2%Analysis of Variance SOURCE DF SS MS F p Regression 2 5.0726 2.5363 1260.71 0.000 Error 19 0.0382 0.0020 Total 21
=+35. Exercise 35 of Section 3.5 gave data on x1 5 wire feed rate, x2 5 welding speed, and y 5 deposition rate of a welding process. Minitab output from fitting the multiple regression model with x1
=+d. Using SSResid 5 30.1033 and SSTo 5 102.3922, what proportion of observed variation in VO2max can be attributed to the model relationship?
=+c. Suppose that an observation made on a male whose weight was 170 lb, walk time was 11 min, and heart rate was 140 beats/min resulted in VO2max 5 3.15. What would you have predicted for VO2max in
=+b. How would you interpret the estimated coefficient b1 5 .6566?
=+a. How would you interpret the estimated coefficient b3 5 2.0996?
=+34. The article “Validation of the Rockport Fitness Walking Test in College Males and Females” (Research Quarterly for Exercise and Sport, 1994: 152–158)recommended the following estimated
=+b. What is the mean value of sales for an out-let without a drive-up window that has 3 competing outlets and 5000 people within a 1-mile radius?
=+a. What is the mean value of sales when the number of competing outlets is 2, there are 8000 people within a 1-mile radius, and the outlet has a drive-up window?
=+33. Let y 5 sales at a fast-food outlet (1000s of $), x1 5 number of competing outlets within a 1-mile radius, x2 5 population within a 1-mile radius (1000s of people), and x3 be an indicator
=+been programmed into all of the most popular statistical computer packages. When using any particular package, it is necessary only to enter the data, make an appropriate request, and know how to
=+As described in Section 3.5, the minimization requires taking k 1 1 partial derivatives, equating these to zero to obtain a system of linear equations (the normal equations), and solving this
=+discussed in Section 11.4. Estimation of model parameters and other inferences are based on a sample of n observations, each one consisting of k 1 1 numbers: a value of x1, a value of x2, . . . , a
=+11.5 Inferences in Multiple Regression We now assume that a dependent or response variable y is related to k independent, predictor, or explanatory variables x1, . . . , xk via the general additive
=+b. When viscosity is 30, what is the change in mean life associated with an increase of 1 in load? When viscosity is 40, what is the change
=+a. What is the mean value of life when viscosity is 40 and load is 1100?
=+32. Let y 5 wear life of a bearing, x1 5 oil viscosity, and x2 5 load. Suppose that the multiple regression model relating life to viscosity and load is y 5 125.0 1 7.750x1 1 .0950x2 2 .0090x1x2 1 e
=+c. What is the change in mean free flow percentage when the viscosity increases from 450 to 460? From 460 to 470?
=+b. Would mean free flow percentage be higher for a viscosity value of 450 or 470?
=+a. Graph the true regression function y 5 2296 1 2.20x – .003x 2for x values between 350 and 485.
=+Self-Flow Alumina Castables” (The Amer. Ceramic Soc. Bull., 1998: 60–66) proposed a quadratic regression model to describe the relationship between x 5 viscosity (MPa • sec) and y 5 free
=+31. High-alumina refractory castables have been extensively investigated in recent years because of their significant advantages over other refractory brick of the same class: lower production and
=+b. What is the mean yield when x1 5 40, x2 5 5, x3 5 230, and x4 5 360?
=+a. Interpret the population regression coefficients1 and 3.
=+30. Consider the regression model y 5 26.50 1.250x1 1.600x2 2 .150x3 1 .160x4 1e, where y 5 gasoline yield(% of crude oil), x1 5 crude oil gravity ( API), x2 5 crude oil vapor pressure (PSIA), x35
=+b. How would you interpret 1 5 .060, the coefficient of the predictor x1? What is the interpretation of 2 5 .900?c. If 5 .5 hour, what is the probability that travel time will be at most 6
=+. What is the mean value of travel time when distance traveled is 50 miles and three deliveries are made?
=+29. A trucking company considered a multiple regression model for relating the dependent variable y 5 total daily travel time for one of its drivers (hours)to the predictors x1 5 distance traveled
=+28. Obtain an expression for sa, the estimated standard deviation of the intercept of the least squares line.Then use the fact that t 5 (a 2 )ysa has a t distribution with n 2 2 df to test H0:
=+c. Calculate a 99% prediction interval for the protein from a single cow whose production is 30 kg/day.
=+b. Estimate true average protein for all cows whose production is 30 kg/day; use a confidence interval with a confidence level of 99%. Does the resulting interval suggest that this mean value has
=+a. Does the simple linear regression model specify a useful relationship between production and protein?
=+27. Milk is an important source of protein. How does the amount of protein in milk from a cow vary with milk production? The article “Metabolites of Nucleic Acids in Bovine Milk” (J. of Dairy
=+d. Determine a 90% prediction interval for the threshold stress of a single steel specimen whose yield strength is 800 MPa.
=+c. Determine a 90% confidence interval for the true average threshold stress of all similar steel specimens whose yield strength is 800 MPa.
=+b. What proportion of observed variation in stress can be attributed to the approximate linear relationship between the two variables?
=+a. Does a scatterplot support the use of the simple linear regression model for relating y to x?
=+composition of a standard grade of steel was analyzed. The following data on y 5 threshold stress(% SMYS) and x 5 yield strength (MPa) was read from a graph in the article (which also included the
=+26. During oil drilling operations, components of the drilling assembly may suffer from sulfide stress cracking. The article “Composition Optimization of High-Strength Steels for Sulfide
=+(ii) a 95% PI for age when x 5 35; (iii) a 95% CI for mean age when x 5 42; (iv) a 95% PI for age when x 5 42; (v) a 99% CI for mean age when x 5 42; (vi) a 99% PI for age when x 5 42. Without
=+on tooth characteristics. A single observation on y 5 age (yr) was made for each of the following values of x 5 % of root with transparent dentine:15, 19, 31, 39, 41, 44, 47, 48, 55, 64. Consider
=+25. The article “Root Dentine Transparency: Age Determination of Human Teeth Using Computerized Densitometric Analysis” (Amer. J. of Physical Anthro., 1991: 25–30) reported on an
=+2 5 .975, and se 5 5.24. Use the fact that syn 5 1.44 when rainfall volume is 40 m3 to predict runoff in a way that conveys information about reliability and precision. Does the resulting interval
=+24. The simple linear regression model provides a very good fit to the data on rainfall and runoff volume given in Exercise 4 of Section 11.1. The equation of the least squares line is yn 5 21.128
=+b. Would a prediction interval for diffusivity when temperature is 1200°F using the same prediction level as in part (a) be wider or narrower than the interval of part (a)? Answer without
=+23. Refer to Exercise 6 of Section 11.1.a. Predict oxygen diffusivity for a single observation to be made when temperature is 1500°F, and do so in a way that conveys information about reliability
=+e. If a 95% CI is calculated for true average DNP sorbed concentration when equilibrium concentration is .2, what will be the simultaneous confidence level for both this interval and the interval
=+d. Calculate a prediction interval with a prediction level of 95% for the DNP sorbed concentration of a single river sediment specimen using an equilibrium concentration of .4.
=+c. Calculate a confidence interval with a confidence level of 95% for the true average DNP sorbed concentration of all river sediment specimens using an equilibrium concentration of .4.
=+.4. Explain why syn is larger when x 5 .2 than when x 5 .4.
=+b. Using the simple linear regression model fit to this data, confirm that yn 5 3.404, syn 5 .107 when x 5 .2, and yn 5 6.616, syn 5 .088 when x 5
=+a. Calculate point estimates of the slope and intercept of the population regression line.
=+data on y 5 sorbed concentration (g/g) and x 5 equilibrium concentration (g/mL) of 2, 4-Dinitrophenol (DNP) in a particular natural river sediment was read from a graph in the article.x: 0.11
=+Phenolic Compounds on the Surface and Sediment of Rivers” (J. of Envir. Engr., 2011: 1114–1121), the authors examined the sorption characteristics of three selected phenolic compounds. The
=+herbicide manufacturing, and fiberglass manufacturing. These compounds are toxic, carcinogenic, and have contributed over the past decades to environmental pollution of aquatic environments.In one
=+22. Phenolic compounds are found in the effluents of coal conversion processes, petroleum refineries,
=+Mist Generation from Metal Removal Fluids”(Lubrication Engr., 2002: 10–17) gave the accompanying data on x 5 fluid flow velocity for a 5%soluble oil (cm/sec) and y 5 the extent of mist
=+21. 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
=+d. The researchers concluded that the freshwater and seawater leaching agents yield similar nitrate extraction efficiencies. Using the regression models from part (b), calculate a point estimate
=+c. Does the simple linear regression model appear to specify a useful relationship between either dependent variable and x in part (b)? State and test the relevant hypotheses.
=+b. In Section 3.4, we described how a power transformation can be applied to create a linear pattern in the transformed data. Using the transformation x 5 1yx, construct scatterplots of yfw
=+a. Construct scatterplots of yfw versus x and ysw versus x. Note the nonlinearity of the plots. Would it be reasonable to describe the patterns in both plots as curved and monotonic?

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