Calculating variance of estimated intercept parameter, $\hat\beta_0$

Question

I have the following sample :

$$ \begin{array}{c|lr} X&80&100&120&140&160&180&200&220&240&260\\ \hline Y & 70 &65&90&95&110&115&120&140&155&150 \\ \end{array} $$

Now i have to calculate

$\mathbb V(\hat\beta_0)=\sigma^2[\frac{1}{n}+\frac{\bar X^2}{\sum(X_i-\bar X)^2}]$

here,

X is independent and Y response variable.

$\beta_0$ intercept parameter

$\sigma^2$ population variance

But i have my sample data.

How can i calculate $\sigma^2$ ?

Answer 1

You can't calculate the population variance $\sigma^2$, but you can estimate it with $$\widehat{\sigma^2} = \frac{\Sigma (Y_i - \hat Y_i)^2}{n-2}.$$

Here, $\hat Y_i= \beta_0 + \beta_1 X_i$ denotes the predicted value of $Y$ given $X_i$.