The article Earthmoving Productivity Estimation Using Linear Regression Techniques (S

In an experiment to determine the factors affecting tensile strength in steel plates, the tensile strength (in kg/mm2), the manganese content (in parts per thousand), and the thickness (in mm) were measured for a sample of 20 plates. The following MINITAB output presents the results of fitting the model \(\text { Tensile strength }=\beta_{0}+\beta_{1} \text { Manganese }+\beta_{2} \text { Thickness}\). a. Predict the strength for a specimen that is 10 mm thick and contains 8.2 ppt manganese. b. If two specimens have the same thickness, and one contains 10 ppt more manganese, by how much would you predict their strengths to differ? c. If two specimens have the same proportion of manganese, and one is 5 mm thicker than the other, by how much would you predict their strengths to differ? Equation Transcription: Text Transcription: Tensile strength=beta_0+beta_1 Manganese+beta_2

Refer to Exercise 1. a. Find a 95% confidence interval for the coefficient of Manganese. b. Find a 99% confidence interval for the coefficient of Thickness. c. Can you conclude that \(\beta_{1}>3\)? Perform the appropriate hypothesis test. d. Can you conclude that \(\beta_{2}<-0.1\)? Perform the appropriate hypothesis test. Equation Transcription: Text Transcription: beta_1>3 beta_2<-0.1

The data used to fit the model in Exercise 1 are presented in the following table, along with the residuals and the fitted values. Plot the residuals versus the fitted values. Does the plot indicate that the linear model is reasonable? Explain.

The article “Application of Analysis of Variance to Wet Clutch Engagement” (M. Mansouri, M. Khonsari, et al., Proceedings of the Institution of Mechanical Engineers, 2002:117–125) presents the following fitted model for predicting clutch engagement time in seconds \((y)\) from engagement starting speed in \(\mathrm{m} / \mathrm{s}\ \left(x_{1}\right)\), maximum drive torque in \(\mathrm{N} \cdot \mathrm{m}\ \left(x_{2}\right)\), system inertia in \(\mathrm{kg} \cdot \mathrm{m}^{2}\ \left(x_{3}\right)\), and applied force rate in \(\mathrm{kN} / \mathrm{s}\ \left(x_{4}\right)\): \(y=-0.83+0.017 x_{1}+0.0895 x_{2}+42.771 x_{3}+0.027 x_{4}-0.0043 x_{2} x_{4}\) The sum of squares for regression was SSR = 1.08613 and the sum of squares for error was SSE = 0.036310. There were 44 degrees of freedom for error. a. Predict the clutch engagement time when the starting speed is 20 m/s, the maximum drive torque is \(17 \mathrm{~N} \cdot \mathrm{m}\), the system inertia is \(0.006 \mathrm{~kg} \cdot \mathrm{m}^{2}\), and the applied force rate is 10 kN/s. b. Is it possible to predict the change in engagement time associated with an increase of 2 m/s in starting speed? If so, find the predicted change. If not, explain why not. c. Is it possible to predict the change in engagement time associated with an increase of \(2 \mathrm{~N} \cdot \mathrm{m}\) in maximum drive torque? If so, find the predicted change. If not, explain why not. d. Compute the coefficient of determination \(R^2\). e. Compute the F statistic for testing the null hypothesis that all the coefficients are equal to 0. Can this hypothesis be rejected? Equation Transcription: Text Transcription: (y) m/s (x_1) N{cdot}m (x_2) kg{cdot}m^2(x_3) kN/s(x_4) y=-0.83+0.017x_1+0.0895x_2+42.771x_3+0.027x_4-0.0043x_2x_4 SSR=1.08613 SSE=0.036310 20 m/s 17 N{cdot}m 0.006 kgm^2 10 kN/s 2 m/s 2 N{cdot}m R^2

In the article "Application of Statistical Design in the Leaching Study of Low-Grade Manganese Ore Using Aqueous Sulfur Dioxide" (P. Naik, L. Sukla, and S. Das, Separation Science and Technology, 2002:1375-1389), a fitted model for predicting the extraction of manganese in \(\%\ (y)\) from particle size in mm \((x_1)\), the amount of sulfur dioxide in multiples of the stoichiometric quantity needed for the dissolution of manganese \((x_2)\), and the duration of leaching in minutes \((x_3)\) is given as \(y=56.145-9.046 x_{1}-33.421 x_{2}+0.243 x_{3}-0.5963 x_{1} x_{2}-0.0394 x_{1} x_{3}+0.6022 x_{2} x_{3}+0.6901 x_{1}^{2}+11.7244 x_{2}^{2}-0.0097 x_{3}^{2}\) There were a total of observations, with and a. Predict the extraction percent when the particle size is , the amount of sulfur dioxide is , and the duration of leaching is 20 minutes. b. Is it possible to predict the change in extraction percent when the duration of leaching increases by one minute? If so, find the predicted change. If not, explain why not. c. Compute the coefficient of determination \(R^2\). d. Compute the statistic for testing the null hypothesis that all the coefficients are equal to Can this hypothesis be rejected? Equation Transcription: Text Transcription: % (y) (x_1) (x_2) (x_3) y=56.145-9.046x_1-33.421x_2-0.5963x_1x_2-0.0394x_1x_3+0.6022x_2x_3+0.6901x_1^2+11.7244x_2^2-0.0097x_3^2 n=27 SSE=209.55 SST=6777.5 R^2

The article "Earthmoving Productivity Estimation Using Linear Regression Techniques" (S. Smith, Journal of Construction Engineering and Management, ) presents the following linear model to predict earthmoving productivity (in \(\mathrm m^3\) moved per hour): \(\begin{aligned} \text { Productivity }=&-297.877+84.787 x_{1}+36.806 x_{2}+151.680 x_{3}-0.081 x_{4}-110.517 x_{5} \\ &-0.267 x_{6}-0.016 x_{1} x_{4}+0.107 x_{4} x_{5}+0.0009448 x_{4} x_{6}-0.244 x_{5} x_{6} \end{aligned} \) where \(x_{1}=\) number of trucks \(x_{2}=\) number of buckets per load \(x_{3}=\) bucket volume, in \(\mathrm m^3\) \(x_{4}=\) haul length, in m \(x_{5}=\) match factor (ratio of hauling capacity to loading capacity) \(x_{6}=\) truck travel time, in s a. If the bucket volume increases by \(1\ \mathrm m^3\), while other independent variables are unchanged, can you determine the change in the predicted productivity? If so, determine it. If not, state what other information you would need to determine it. b. If the haul length increases by 1 m, can you determine the change in the predicted productivity? If so, determine it. If not, state what other information you would need to determine it. Equation Transcription: Text Transcription: m^3 Productivity=-297.877+84.787x_1+36.806x_2+151.680x_3-0.081x_4-110.517x_5 -0.267x_6-0.016x1x4+0.107x_4x_5+0.0009448x_4x_6-0.244x_5x_6 x_1= x_2= x_3= m^3 x_4= x_5= x_6= 1 m^3

In a study of the lung function of children, the volume of air exhaled under force in one second is called \(\mathrm{FEV}_{1}\). (\(\mathrm{FEV}_{1}\) stands for forced expiratory volume in one second.) Measurements were made on a group of children each year for two years. A linear model was fit to predict this year’s\(\mathrm{FEV}_{1}\) as a function of last year’s \(\mathrm{FEV}_{1}\) (in liters), the child’s gender (0 = Male, 1 = Female), the child’s height (in m), and the ambient atmospheric pressure (in mm). The following MINITAB output presents the results of fitting the model \(\mathrm{FEV}_{1}=\beta_{0}+\beta_{1}\ \mathrm{Last}\ \mathrm{FEV}_{1}+\beta_{2} \text { Gender }+\beta_{3} \text { Height }+\beta_{4} \text { Pressure }+\varepsilon\) a. Predict the \(\mathrm{FEV}_{1}\) for a boy who is 1.4 m tall, if the measurement was taken at a pressure of 730 mm and last year’s measurement was 2.113 L. b. If two girls differ in height by 5 cm, by how much would you expect their \(\mathrm{FEV}_{1}\) measurements to differ, other things being equal? c. The constant term \(\beta_0\) is estimated to be negative. But \(\mathrm{FEV}_{1}\) must always be positive. Is something wrong? Explain. Equation Transcription: Text Transcription: FEV_1 FEV_1 FEV_1 FEV_1 FEV_1=beta_0+beta_1 Last FEV_1+beta_2 Gender+beta_3 Height+beta_4 Pressure+varepsilon FEV_1 FEV_1 beta_0 FEV_1

Refer to Exercise 7. a. Find a 95% confidence interval for the coefficient of Last FEV. b. Find a 98% confidence interval for the coefficient of Height. c. Can you conclude that \(\beta_{2}<-0.08\)? Perform the appropriate hypothesis test. d. Can you conclude that \(\beta_{3}>0.5\)? Perform the appropriate hypothesis test. Equation Transcription: Text Transcription: beta_2<-0.08 beta_3>0.5

The article “Drying of Pulps in Sprouted Bed: Effect of Composition on Dryer Performance” (M. Medeiros, S. Rocha, et al., Drying Technology, 2002:865–881) presents measurements of pH, viscosity (in \(\mathrm{kg} / \mathrm{m} \cdot \mathrm{s}\)), density (in \(\mathrm{g} / \mathrm{cm}^{3}\)), and BRIX (in percent). The following MINITAB output presents the results of fitting the model \(\mathrm{pH}=\beta_{0}+\beta_{1} \text { Viscosity }+\beta_{2} \text { Density }+\beta_{3} \mathrm{BRIX}+\varepsilon\) a. Predict the pH for a pulp with a viscosity of \(1500 \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}\), a density of \(1.04 \mathrm{~g} / \mathrm{cm}^{3}\), and a BRIX of 17.5%. b. If two pulps differ in density by \(0.01 \mathrm{~g} / \mathrm{cm}^{3}\), by how much would you expect them to differ in pH, other things being equal? c. The constant term \(\beta_0\) is estimated to be negative. But pulp pH must always be positive. Is something wrong? Explain. d. Find a 95% confidence interval for the mean pH of pulps with viscosity \(1200 \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}\), density \(1.08 \mathrm{~g} / \mathrm{cm}^{3}\), and BRIX 18.0%. e. Find a 95% prediction interval for the pH of a pulp with viscosity \(1000 \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}\), density \(1.05 \mathrm{~g} / \mathrm{cm}^{3}\), and BRIX 19.0%. f. Pulp A has viscosity 2000, density 1.03, and BRIX 20.0. Pulp B has viscosity 1000, density 1.05, and BRIX 19.0. Which pulp will have its pH predicted with greater precision? Explain. Equation Transcription: Text Transcription: kg/m{cdot}s g/cm^3 pH=beta_0+beta_1 Viscosity+beta_2 Destiny+beta_3 BRIX+varepsilon 1500 kg/m{cdot}s 1.04 g/cm^3 0.01 g/cm^3 beta_0 1200 kg/m{cdot}s 1.08 g/cm^3 1000 kg/m{cdot}s 1.05 g/cm^3

A scientist has measured quantities \(y, x_{1}\), and \( x_{2}\). She believes that is related to \( x_{1}\) and \( x_{2}\) through the equation \(y=\alpha e^{\beta_{1} x_{1}+\beta_{2} x_{2}} \delta\), where \(\delta\) is a random error that is always positive. Find a transformation of the data that will enable her to use a linear model to estimate \(\beta_1\) and \(\beta_2\). Equation Transcription: Text Transcription: y,x_1 x_2 x_1 x_2 y={alpha}e^{beta_1x1+beta_2x2}delta delta beta_1 beta_2

The following MINITAB output is for a multiple regression. Something went wrong with the printer, so some of the numbers are missing. Fill in the missing numbers.

The following MINITAB output is for a multiple regression. Some of the numbers got smudged and are illegible. Fill in the missing numbers.

The article “Evaluating Vent Manifold Inerting Requirements: Flash Point Modeling for Organic Acid-Water Mixtures” (R. Garland and M. Malcolm, Process Safety Progress, 2002:254–260) presents a model to predict the flash point (in \({ }^{\circ} \mathrm{F}\)) of a mixture of water, acetic acid, propionic acid, and butyric acid from the concentrations (in weight %) of the three acids. The results are as follows. The variable “Butyric Acid ? Acetic Acid” is the interaction between butyric acid concentration and acetic acid concentration. a. Predict the flash point for a mixture that is 30% acetic acid, 35% propionic acid, and 30% butyric acid. (Note: In the model, 30% is represented by 30, not by 0.30.) b. Someone asks by how much the predicted flash point will change if the concentration of acetic acid is increased by 10% while the other concentrations are kept constant. Is it possible to answer this question? If so, answer it. If not, explain why not. c. Someone asks by how much the predicted flash point will change if the concentration of propionic acid is increased by 10% while the other concentrations are kept constant. Is it possible to answer this question? If so, answer it. If not, explain why not. Equation Transcription: Text Transcription: ^{o}F

In the article "Low-Temperature Heat Capacity and Thermodynamic Properties of -trifluoro-2, 2-dichloroethane" (R. Varushchenko and A. Druzhinina, Fluid Phase Equilibria, 2002:109-119), the relationship between vapor pressure \((p)\) and heat capacity \((t)\) is given as \(p=t^{\beta_{3}} \cdot e^{\beta_{0}+\beta_{1} t+\beta_{2} / t} \delta\), where \(\delta\) is a random error that is always positive. Express this relationship as a linear model by using an appropriate transformation. Equation Transcription: Text Transcription: (p) (t) p=t^{beta_3}{cdot}e^{beta_0+beta_1t+beta_2/t}{delta} delta

The following data were collected in an experiment to study the relationship between extrusion pressure (in KPa) and wear (in mg). 150 175 200 225 250 275 The least-squares quadratic model is \(y=-32.445714+0.43154286 x-0.000982857 x^{2}\). a. Using this equation, compute the residuals. b. Compute the error sum of squares SSE and the total sum of squares SST. c. Compute the error variance estimate \(s^2\). d. Compute the coefficient of determination \(R^2\). e. Compute the value of the statistic for the hypothesis \(H_{0}: \beta_{1}=\beta_{2}=0\). How many degrees of freedom does this statistic have? f. Can the hypothesis \(H_{0}: \beta_{1}=\beta_{2}=0\) be rejected at the level? Explain. Equation Transcription: Text Transcription: y=-32.445714+0.43154286x-0.000982857x^2 SSE SST s^2 R^2 H_0:beta_1=beta_2=0 H_0:beta_1=beta_2=0

The following data were collected in an experiment to study the relationship between the speed of a cutting tool in m/s \((x)\) and the lifetime of the tool in hours \((y)\). The least-squares quadratic model is \(y=101.4000+3.371429 x-5.142857 x^{2}\). a. Using this equation, compute the residuals. b. Compute the error sum of squares SSE and the total sum of squares SST. c. Compute the error variance estimate \(s^{2}\). d. Compute the coefficient of determination \(R^{2}\). e. Compute the value of the F statistic for the hypothesis \(H_{0}: \beta_{1}=\beta_{2}=0\). How many degrees of freedom does this statistic have? f. Can the hypothesis \(H_{0}: \beta_{1}=\beta_{2}=0\) be rejected at the 5% level? Explain. Equation Transcription: Text Transcription: m/s (x) (y) y=101.4000+3.371429x-5.142857x^2 s^2 R^2 H_0:beta_1=beta_2=0 H_0:beta_1=beta_2=0

The November 24, 2001, issue of The Economist published economic data for 15 industrialized nations. Included were the percent changes in gross domestic product (GDP), industrial production (IP), consumer prices (CP), and producer prices (PP) from Fall 2000 to Fall 2001, and the unemployment rate in Fall 2001 (UNEMP). An economist wants to construct a model to predict GDP from the other variables. A fit of the model \(\mathrm{GDP}=\beta_{0}+\beta_{1} \mathrm{IP}+\beta_{2} \mathrm{UNEMP}+\beta_{3} \mathrm{CP}+\beta_{4} \mathrm{PP}+\varepsilon\) yields the following output: a. Predict the percent change in GDP for a country with IP = 0.5, UNEMP = 5.7, CP = 3.0, and PP = 4.1. b. If two countries differ in unemployment rate by 1%, by how much would you predict their percent changes in GDP to differ, other things being equal? c. CP and PP are both measures of the inflation rate. Which one is more useful in predicting GDP? Explain. d. The producer price index for Sweden in September 2000 was 4.0, and for Austria it was 6.0. Other things being equal, for which country would you expect the percent change in GDP to be larger? Explain. Equation Transcription: Text Transcription: GDP=beta_0+beta_1IP+beta_2UNEMP+beta_3CP+beta_4PP+varepsilon IP=0.5 UNEMP=5.7 CP=3.0 PP=4.1

The article “Multiple Linear Regression for Lake Ice and Lake Temperature Characteristics” (S. Gao and H. Stefan, Journal of Cold Regions Engineering, 1999:59–77) presents data on maximum ice thickness in mm \((y)\), average number of days per year of ice cover \(\left(x_{1}\right)\), average number of days the bottom temperature is lower than \(8^{\circ} \mathrm{C}\) \(\left(x_{2}\right)\), and the average snow depth in mm \(\left(x_{3}\right)\) for 13 lakes in Minnesota. The data are presented in the following table. y \(\left(x_{1}\right)\) \(\left(x_{2}\right)\) \(\left(x_{3}\right)\) y \(\left(x_{1}\right)\) \(\left(x_{2}\right)\) \(\left(x_{3}\right)\) 730 152 198 91 730 157 204 90 760 173 201 81 650 136 172 47 850 166 202 69 850 142 218 59 84 161 202 72 740 151 207 88 720 152 198 91 720 145 209 60 730 153 205 91 710 147 190 63 840 166 204 70 a. Fit the model \(y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3}+\varepsilon\). For each coefficient, find the P-value for testing the null hypothesis that the coefficient is equal to 0. b. If two lakes differ by 2 in the average number of days per year of ice cover, with other variables being equal, by how much would you expect their maximum ice thicknesses to differ? c. Do lakes with greater average snow depth tend to have greater or lesser maximum ice thickness? Explain. Equation Transcription: Text Transcription: (y) (x_1) 8^{o}C (x_2) (x_3) (x_1) (x_2) (x_3) (x_1) (x_2) (x_3) y=beta_0+beta_1x_1+beta_2x_2+beta_3x_3+varepsilon

In an experiment to estimate the acceleration of an object down an inclined plane, the object is released and its distance in meters \((y)\) from the top of the plane is measured every 0.1 second from time \(t=0.1\) to \(t=1.0\). The data are presented in the following table. The data follow the quadratic model \(y=\beta_{0}+\beta_{1} t+\beta_{2} t^{2}+\varepsilon\), where \(\beta_0\) represents the initial position of the object, \(\beta_1\) represents the initial velocity of the object, and \(\beta_2=a/2\), where a is the acceleration of the object, assumed to be constant. In a perfect experiment, both the position and velocity of the object would be zero at time 0. However, due to experimental error, it is possible that the position and velocity at \(t=0\) are nonzero. a. Fit the quadratic model \(y=\beta_{0}+\beta_{1} t+\beta_{2} t^{2}+\varepsilon\). b. Find a 95% confidence interval for \(\beta_2\). c. Find a 95% confidence interval for the acceleration \(a\). d. Compute the P-value for each coefficient. e. Can you conclude that the initial position was not zero? Explain. f. Can you conclude that the initial velocity was not zero? Explain. Equation Transcription: Text Transcription: (y) t=0.1 1=1.0 y=beta_0+beta_1t+beta_2t^2+varepsilon beta_0 beta_1 beta_2=a/2 t=0 y=beta_0+beta_1t+beta_2t^2+varepsilon beta_2 a