Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 218 / 3 / d / Solution

The printed values appear to favor model 2 because $-12.529<336.381$. They are not directly comparable: model 2 reports the Gaussian likelihood of $Z_i=\log T_i$, whereas model 3 reports a density for $T_i$. The change-of-variables formula for a probability density givesso the transformed model's AIC on the original response scale isBecause the standardized predictors have zero sample means and the ordinary-least-squares residuals sum to zero,Thereforewhich is slightly worse than model 3's $336.381$.