The article ??Permanent Deformation Characterization of Subgrade Soils from RLT Test?? (A. Puppala, L. Mohammad, et al., Journal of Materials in Civil Engineering, 1999:274??282) presents measurements of plastic strains (in percent) on soils at various confining and deviatoric stresses in kPa. The following table presents the plastic strains (y), the confining stress (x₁), and the deviatoric stress (x₂) for tests on a sandy soil.

a. Fit the model y = β₀ + β₁x₁ + β₂x₂ + ε. Plot the residuals versus the fitted values. Does the model seem appropriate?

b. Fit the model ln y = β₀ + β₁ ln x₁ + β₂ ln x₂ + ε. Plot the residuals versus the fitted values. Does the model seem appropriate?

c. Use the more appropriate of the models from (a) and (b) to predict y when x₁ = 50 and x₂ = 100.

d. Is the model you used in part (c) improved by including an interaction term? Explain.

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y X2 y X1 X2 0.01 21 21 0.01 70 140 0.02 21 35 0.07 70 210 0.05 21 52.5 0.002 105 70 0.09 21 70 0.0003 105 105 0.003 35 35 0.0009 105 140 0.006 35 70 0.01 105 210 0.05 35 105 0.001 140 70 0.23 35 140 0.0003 0.0005 140 105 0.003 70 35 140 210 0.0008 70 70 0.03 140 280