Relationship between Response variable and Predictors How can I know if there's a relationship between my response variable (Y) and the predictors (${X}_{i}$)?. I'm confussed, I don't know if I have to use the ${R}^{2}$ or the F-statistic and its p-value. Or none of these? 
Thanks a lot!
 A: There's a lot to your question. I'll focus on this:

I don't know if I have to use the $R^2$ or the F-statistic and its p-value.

tl;dr The F-statistic and p-value (and ANOVA more generally) are tools to formally test your model fit, and specifically to learn about the fit of certain predictors. $R^2$ is another metric to test model fit, but what constitutes "good" or "bad" is fuzzier.
I'd also recommend googling around, especially on this site, for $R^2$ and ANOVA for linear regression. You'll get a lot of hits that may help answer your questions.
$R^2$
The $R^2$ is a metric that tells you how much of the variation in your model is explained by the variation in the predictors. It's measured as a proportion, so it varies between 0 and 1. 
$$R^2 = 1 - \frac{SSE}{SSTO}$$
$R^2$ close to 1 means that your predictors explain a lot of the variability in your model. If its close to 0, then your predictors don't explain much. So $R^2$ is a very common measure of how well the model fits the data. However, there are a bunch of pitfalls if you blindly apply $R^2$ to selecting your model. See here for a good list. 
For the purposes of comparing against the F-test/p-value note that for $R^2$ there is no cut off that says, "below x% the model fits poorly" or "above y% and the model fits well". Whereas the ANOVA test does have such a cut off.
F-statistic and p-value
The F-statistic and it's p-value is is formal way of determining if a coefficient belongs in the model. Now, there are tons of pitfalls with looking at individual coefficients and their p-values. Feel free to google it. However, putting that aside, the more formal test is a hypothesis test. Usually there are three kinds:


*

*Test if $\beta_i=0$

*Test if all $\beta's=0$

*Test if a subset of the $\beta's =0$


To perform these tests, you'd run an ANOVA and get an F-statistic. 
$$F = \frac{MSR}{MSE}$$
It's a number. I won't go into the details of ANOVA or F-test here (google will turn up tons of results) but for the purposes of your question, the F-statistic quantifies the difference in error between the model assuming your null hypothesis (i.e. $\beta_i=0$) and your fitted model. Then the p-value is the probability of getting that F-statistic assuming the null hypothesis that you're testing, is true. If this number is small (i.e. it's a crazy result that we wouldn't expect to happen if the null were true) then we reject the null (because the probability of occurrence is so small), and say $\beta_i \ne 0$. Otherwise, if the p-value is big, we fail to reject the null. If you're doing a lot of linear regression work, I'd recommend reading more about F-tests and p-values. Of course, selecting what "small" means (usually 5%) is up to you.
