# F-statistic and Linear Regression formula for slope?

One of the derivations of the Linear Regression Method leads to the following two formulas, for the slope and the intercept:

$$\begin{equation} m = \frac{Cov(x, y)}{Var(x)} \label{eq:slope} \\ b = \bar{y} - m \bar{x} \end{equation}$$

for the model: $y = mx + b$

The above equation for the slope, $m$, is seeming (at least conceptually) similar to the equation for F-test statistic:

$$\begin{equation} F = \frac{Between-group-variance}{Within-group-variance} = \frac{\frac{\sum_{i=1}^K n_i(\bar{Y_i} - \bar{Y} )^2}{K-1}}{\frac{\sum_{i=1}^K\sum_{j=1}^{n_i} (Y_{ij} - \bar{Y_i})^2}{N-K}} \end{equation}$$

Is there any relation between these two concepts?

• Yes. What you are calling $\bar{Y}_i$ is constructed by $b + m x_i$. The null hypothesis for the F test is that $m = 0$. The "within group" variance is proportional to the squared distance of the observations about the line-of-best-fit constructed by $y=b + mx$ – AdamO Apr 12 '18 at 16:26