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: Degrees of freedom.
\(\text{df}_{\text{reg}}=3\), \(\text{df}_{\text{res}}=31-3-1=27\). Use \(F_{0.05;3,27}=2.96\).

Step 2: F in terms of R^2.
\[ F=\frac{R^2/3}{(1-R^2)/27}=\frac{9R^2}{1-R^2}. \]

Step 3: Rejection requires F>2.96.

Step 4: Test each value.
\(R^2=0.50\): \(F=9\), OK. \(R^2=0.80\): \(F=36\), OK. \(R^2=0.30\): \(F\approx3.86\), OK. \(R^2=0.20\): \(F=2.25<2.96\), FAILS.

Final Answer: \[ \boxed{0.20} \]