Finding the estimated variance of b2

Problem

Given: the sample variance of \(x\) -data is 2.16, \(RSS=44.2\) and \(n=42\) . Find

\[\widehat{Var}\left( b_2 \right)= \frac{s^2}{\sum_{i=1}^{n}{ {\left( x_i-\bar{x} \right)}^2 }}\]

Solution

\(\sum_{i=1}^{n}{ {\left( x_i-\bar{x} \right)}^2 }\) is equal to \(\left( n-1 \right)\) times sample variance or \(41×2.16=88.56\) .

\(s^2=RSS/(n-2) \) or \(44.2/40=1.105\) .

\[\widehat{Var}\left( b_2 \right)= \frac{1.105}{88.56}=0.0125\]

Also, \(SE\left( b_2 \right)=\sqrt{0.0125}=0.112\) .