A Statistician collected the data of 78 values with two independent variable X₁ and X₂, and considered the four models: (i) Y = B₀ + e;

(ii) Y = B₀ + B₁X₁ + e;

(iii) Y = B₀ + B₁ X₁ + e and

(iv) Y = B₀ + B₁ X₁ + B₂ X₂ + e.

The results obtained are: σ²=0.91, SS(B₀) = 652.42, SS(B₀,B₁) = 679.34, SS(B₀,B₂) = 654.00,SS(B₀,B₁,B₂) = 687.79 . Find the additional contribution of (i) X₂ over X₁ and (ii) X₁ over X₂. Test whether their inclusion in the model is justified.

We can use the extra sum of squares method to test whether including the independent variable X2 in the model (iv) is justified and whether including X1 in the model (iii) is justified.

(i) To test the additional contribution of X2 over X1, we compare model (iii) with model (iv). The extra sum of squares due to X2 is given by:

ESS(X2) = SS(model (iii)) - SS(model (iv), including both X1 and X2)
= SS(B0, B1) - SS(B0, B1, B2)

The degrees of freedom for ESS(X2) is 1, since we are comparing two models that differ by one additional independent variable.

The null hypothesis is that X2 does not significantly improve the model ___ _____ _____ _________ ___ ____ _________ ____.
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