In standard least squares regression, we find constants $\beta_1$ and $\beta_2$ such that the square of the average error, $\epsilon = y_i - (\beta_1 + \beta_2x_i)$, is minimized, and so the 'line of best fit' is given as:
$$y_{\textrm{avg}}(x)=\beta_1 + \beta_2x$$
Now, my question is, if you had to put associated error terms for each of the obtained terms (i.e. $\Delta\beta_1$ & $\Delta\beta_2$), what would they be? My guess is that for $\beta_1$, the 'error term' that I'm looking for would be the standard deviation $\sigma$, and so for the 'error term' for $\beta_2$ we would get the upper and lower bounds for the slope of the line within the 'wiggle room' defined by $\sigma$. [See picture]
Please give me some direction on this problem of mine if you can.
