GATE 2026 ST – Question 48

Regression Analysis · Regression Inference and Quadratic Forms · 2 marks · Multiple choice

In Y_i=β₀+β₁x_i1+β₂x_i2+β₃x_i3+ε_i,i=1,…,31, the errors are iid N(0,1). The 5% overall F-test rejects all three slopes zero. Upper-tail F critical values are F_(.05;3,27)=2.96,F_(.05;3,30)=2.92,F_(.025;3,27)=4.01,F_(.025;3,30)=3.91. Which R² is impossible?

0.50
0.80
0.30
0.20

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Correct answer: (D) 0.20

Explanation

There are 3 regression and 31−4=27 residual degrees of freedom. The statistic is (R²/3)/[(1−R²)/27]=9R²/(1−R²), which must exceed 2.96. For the listed values it is 9,36,27/7≈3.857, and 2.25 respectively. Only **0.20, D**, fails to reject.

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