The article "Groundwater Electromagnetic Imaging in Complex Geological and Topographical Regions: A Case Study of a Tectonic Boundary in the French Alps" (S. Houtot, P. Tarits, et al., Geophysics, 2002:1048-1060) presents measurements of concentrations of several chemicals (in mmol/L) and electrical conductivity (in 10ˆ’2 S/m) for several water samples in various locations near Gittaz Lake in the French Alps. The results for magnesium and calcium are presented in the following table. Two outliers have been deleted.

a. To predict conductivity (y) from the concentrations of magnesium (x1) and calcium (x2), fit the full quadratic model y = β0 + β1x1 + β2x2 + β3x21 + β4x22 + β5x1x2 + ε. Compute the analysis of variance table.
b. Use the F test to investigate some submodels of the full quadratic model. State which model you prefer and why.
c. Use a best subsets routine to find the submodels with the maximum adjusted R2 and the minimum Mallows Cp. Are they the same model? Comment on the appropriateness of this (these) model(s).

Conductivity Magnesium Calcium Conductivity Magnesium Calcium 2.77 3.03 3.09 3.29 3.37 0.88 0.77 0.97 0.037 0.041 0.215 0.166 0.100 0.031 0.012 0.017 1.342 1.500 1.332 1.609 1.627 0.382 0.364 0467 1.10 1.11 2.57 3.27 2.28 3.32 3.93 4.26 0.027 0.039 0.168 0.172 0.044 0.069 0.188 0.211 0.487 0.497 1.093 1.480 1.093 1.754 1.974 2. 103