# F-test in multiple linear regression

I'm currently reading Introduction to statistical learning. When trying to prove the collective significance of a regression linear model, we use the F-test with the following formula.

$$F=\frac{(TSS-RSS)/p}{RSS/(n-p-1)}$$

Whre $$TSS=\sum(y_i-/y)²$$, and $$RSS=\sum(y_i-\hat y_i)²$$

What I understood from this formula is that we would want that $$F$$ be larger than $$1$$ in order to reject the null hypothesis, because that means the amount of variability in $$y$$ explained by our model ($$TSS-RSS$$) is larger than the amount that the model couldn't remove ($$RSS$$).

Anyway, I am stuck on proving the result in the following section:

If the linear model assumptions are correct, one can show that: $$E{RSS/(n − p − 1)} = σ²$$ and that, provided $$H0$$ is true, $$E{(TSS − RSS)/p} = σ²$$.

with: $$H_0$$: $$b_1=b_2=...=b_p=0$$

$$b_i$$ is the coefficient for the ith regressor.

So, what I want is a proof or a hint for the quote above. And additional explanations would be appreciated.

• σ² is the variance of the error term. – Youssef Esseddiq Feb 8 at 15:32
• Do you understand how the F-test works in an ANOVA? – gung - Reinstate Monica Feb 8 at 15:36
• No not very much, I know F-test is used to compare models fitted to a certain dataset..it would be very helpful if you could explain it. – Youssef Esseddiq Feb 8 at 15:42
• You may glean some insight from my answer here: How does the standard error work? – gung - Reinstate Monica Feb 8 at 15:45
• It looks like you mis-stated the null hypothesis. – Isabella Ghement Feb 8 at 16:58