Question

Consider the multiple linear regression model
\[ Y_i = \beta_0+\beta_1 x_{i1}+\beta_2 x_{i2}+\beta_3 x_{i3}+\epsilon_i, \quad i=1,2,\ldots,31, \]
where \(\epsilon_i\) are iid \(N(0,1)\) variables. The \(F\)-test for testing significance of regression rejects \(H_0: \beta_1=\beta_2=\beta_3=0\) at \(5\%\) level. Given \[ F_{0.05;3,27}=2.96,\quad F_{0.05;3,30}=2.92,\quad F_{0.025;3,27}=4.01,\quad F_{0.025;3,30}=3.91. \] Then the value of \(R^2\) cannot be equal to

• A. \(0.50\)
• B. \(0.80\)
• C. \(0.30\)
• D. \(0.20\)

Hint:

Write F in terms of R^2 using the correct residual degrees of freedom n-p-1=27, then check which option gives F below the critical value 2.96.

Answer 1

The Correct Option is D

Step 1: Get the two degrees of freedom right.
Numerator df $=3$, denominator df $=31-4=27$. Use $F_{0.05;3,27}=2.96$.

Step 2: Find boundary R^2.
\[ 9R^2=2.96(1-R^2) \Rightarrow R^2\approx0.2475 \]

Step 3: Compare each option.
$0.50,0.80,0.30$ all above $0.2475$; only $0.20$ below.

Step 4: Confirm.
At $R^2=0.20$: $F=2.25<2.96$.
\[ \boxed{0.20} \]