3. Recall that adjusted-R² for multiple linear regression with model

Y = Bo + B₁x₁ + + Bpxp+e

R²dj = 1

(M1) Y = Po+Pixi+…

+Bpxp+e

(M2) Y=B₁ + B₁x₁ + ··· + Bp-kXp-k + e,

is defined as

Compare two models

and

where (M2) contains k fewer predictor variables than (M1).

Then the adjusted-R² values for the two models are given by

RSSM₁/(n-p-1)

adj,M1

SST/(n − 1)

and

RSS/(n-p-1)

SST/(n-1)

RSSM2/(n-p-1+k)

1

SST/(n − 1)

Show that, Radj.M1 > Radj.M2 if and only if the F-ratio for the test

Ho: Pp-k+1==

H₁ : not Ho

is greater than 1. Explain how the situation R²

tion where the above test is statistically significant.

R²

adj,M2

· =Bp=0₂

Radj.M1 > Rªdj.M2 is different from the situa-