Whats the relationship between $R^2$ and F-Test? I was wondering if there is a relationship between $R^2$ and a F-Test.
Usually $$R^2=\frac {\sum (\hat Y_t - \bar Y)^2 / T-1} {\sum( Y_t - \bar Y)^2 / T-1}$$ and it measures the strength of the linear relationship in the regression.
An F-Test just proves a hypothesis.
Is there a relationship between $R^2$ and a F-Test?
 A: Intuitively, I like to think that the result of the F-ratio first gives a yes-no response to the the question, 'can I reject $H_0$?' (this is determined if the ratio is much larger than 1, or the p-value < $\alpha$). 
Then if I determine I can reject $H_0$, $R^2$ then indicates the strength of the relationship between. 
In other words, a large F-ratio indicates that there is a relationship. High $R^2$ then indicates how strong that relationship is.
A: If all the assumptions hold and you have the correct form for $R^2$ then the usual F statistic can be computed as $F = \frac{ R^2 }{ 1- R^2} \times \frac{ \text{df}_2 }{ \text{df}_1 }$.  This value can then be compared to the appropriate F distribution to do an F test.  This can be derived/confirmed with basic algebra.
A: Recall that in a regression setting, the F statistic is expressed in the following way.
$$
F = \frac{(TSS - RSS)/(p-1)}{RSS/(n-p)}
$$
where TSS = total sum of squares and RSS = residual sum of squares, $p$ is the number of predictors (including the constant) and $n$ is the number of observations. This statistic has an $F$ distribution with degrees of freedom $p-1$ and $n-p$.
Also recall that 
$$
R^2 = 1 - \frac{RSS}{TSS} = \frac{TSS - RSS}{TSS}
$$
simple algebra will tell you that
$$
R^2 = 1 - (1 + F \cdot \frac{p-1}{n-p})^{-1}
$$
where F is the F statistic from above.
This is the theoretical relationship between the F statistic (or the F test) and $R^2$.
The practical interpretation is that a bigger $R^2$ lead to high values of F, so if $R^2$ is big (which means that a linear model fits the data well), then the corresponding F statistic should be large, which means that that there should be strong evidence that at least some of the coefficients are non-zero.
A: Also, quickly:
R2 = F / (F + n-p/p-1)
Eg, The R2 of a 1df F test = 2.53 with sample size 21, would be:
R2 = 2.53 / (2.53+19)
R2 = .1175
