Calculate the error variance in a linear regression model I am trying to calculate the error variance for the following question but I don't have clue where to start. Could anyone please help?


 A: $(X'X)^{-1}=\dfrac{1}{150}.\left(
  \begin{array}{cc}
    5 & -10 \\
    -10 & 50 \\
  \end{array}
\right)$. $\hat{\beta}=(X'X)^{-1}X'Y=1/150.\left(
  \begin{array}{cc}
    5 & -10 \\
    -10 & 50 \\
  \end{array}
\right).\left(
  \begin{array}{cc}
    20 \\
    10 \\
  \end{array}
\right)=\left(
  \begin{array}{cc}
    0 \\
    2 \\
  \end{array}
\right)$
$\hat{\beta}\sim N(\beta,\sigma^2.(X'X)^{-1})$. You can estimate $\sigma^2$ by $s^2=\dfrac{1}{n-p-1}(y-X\hat{\beta})'(y-X\hat{\beta})$. Now if you want to simultaneously test $H_0: \beta=\beta_0$ vs $H_1:\beta\neq \beta_0$, 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.
