Question: Because the coefficient of determination R2 always increases when a

Because the coefficient of determination, R2, always increases when a new independent variable is added to a model, it is tempting to include many variables in the model in order to force R2 to be near 1. However, doing so reduces the number of degrees of freedom available for estimating σ2, which adversely affects our ability to make reliable inferences. Suppose you want to use 18 psychological and sociological factors to predict a student’s Scholastic Assessment Test (SAT) score. You fit the model
y = β0 + β1x1 + β2x2 + ∙∙∙ + β17x17 + β18x18 + ε
Where y = SAT score and x1, x2, ∙∙∙,x18 are the psychological and sociological factors. Only 20 years of data (n = 20) are used to fit the model, and you obtain R2 = .95. Test to see whether this impressive-looking R2 is large enough for you to infer that the model is useful—that is, that at least one term in the model is important in predicting SAT scores. Use α = .05.

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