Solved – Calculate the error variance in a linear regression model

$(X'X)^{-1}=dfrac{1}{150}.left( begin{array}{cc} 5 & -10 \ -10 & 50 \ end{array} right)$. $hat{beta}=(X'X)^{-1}X'Y=1/150.left( begin{array}{cc} 5 & -10 \ -10 & 50 \ end{array} right).left( begin{array}{cc} 20 \ 10 \ end{array} right)=left( begin{array}{cc} 0 \ 2 \ end{array} right)$
$hat{beta}sim N(beta,sigma^2.(X'X)^{-1})$. You can estimate $sigma^2$ by $s^2=dfrac{1}{n-p-1}(y-Xhat{beta})'(y-Xhat{beta})$. Now if you want to simultaneously test $H_0: beta=beta_0$ vs $H_1:betaneq beta_0$, where $beta_0$ is a $p$-dimensional constant, then you need to use the $F$ test as follow:

$F=dfrac{(hat{beta}-beta_0)'(X'X)^{-1}(hat{beta}-beta_0)}{_ps^2}sim F_{p,n-p}$. Here $hat{beta}-beta_0=left( begin{array}{c} -0.25 \ 1.75 \ end{array} right) $.
And $(hat{beta}-beta_0)'(X'X)^{-1}(hat{beta}-beta_0)=162.1875 $. Here $p=1$ and $n=N$. If we let $_ps^2=sigma^2=1$, then $F$ statistics is 162.1875 and we need to compare it with $F_{1,N-1}$. If $P_r(F_{alpha,1,N-1}geq 162.1875 )geq (1-alpha)$ then $H_0$ cannot be rejected otherwise accept $H_1$. See e.g. page 70 of Linear Regression Analysis: Theory and Computing.