# Which coeffcients are used for a F-statistic

I have an F-statistic, $$F(4,10)$$, my constant and 4 Coefficients $$\beta_2 , \beta_3 , \beta_4$$ and $$\beta_5$$

I already know that the (in this case) 10 reflects the number of obsverations. But what exactly does the 4 tell me? Is it the number of coefficients to be tested? Also what does the outcome - $$F(4,10) = 9.59$$ tell me?

Thank you in advance.

With $$F$$-test in multiple linear regression, you test the $$H_0$$ that a "trivial" model $$y = \beta_1 + u$$ (i.e. dependent variable as a function of intercept and random error only) is equally good in predicting variability in the dependent variable as your "main" model $$y = \beta_1 + \beta_2 x_2 + \beta_3 x_3 + \beta_4 x_4 + \beta_5 x_5 + u$$.
Basically, you test the significance of the model as a whole. In $$F(k, (n-k-1))$$, parameter $$k$$ is the number of parameters restricted (4), and $$n-k-1$$ is the degrees of freedom in multiple linear regression - i.e. you are estimating your model with 15 observations only, which is far too low for 5 parameter-estimation.
Also, the critical value for $$F(4,10)=2.6$$ (at 5% sig. level), so you are fine (reject $$H_0$$ in favour of significant model). However, let me repeat that estimating 5 coeffs using 15 observations is generally not a good idea.