Victoria -

Both R, and even much more so, "significance" can be misleading.  If you want to know how well your regression model predicts, you can test it.  You may want to research the term "model assessment."

You can do a graphical check of the residuals, but to avoid overfitting, you should also save out some data when you select your model and estimate coefficients, so that you can then see, with the data you did not use, how well you would have predicted it. 

Parts of the following may be of some use to you:

 https://www.researchgate.net/publication/313925533_Comparing_Model_Performances_Graphically

Cheers - Jim

Dear Victoria

James went deep in the logics and probability. I agree with that. Statistics is a combination of mathematics, probability, and above all, logic. The logic is a key component of all procedures. Hence, whenever you want to assess the strength of a relationship, this is the logic that says 0.05 p value or 0.01 is significant. Sometimes you might reach such significant level, but the strength is considered low from your perspective, and vice versa.

Simply, if you want to test the strength of a model (considering the model doesn't violate the assumptions and you are familiar with the procedures) you can average the absolute value of all residuals. If the average was higher than your tolerance, then the model is not strong enough, and if it was lower, you can consider it strong, even if it was significant at 0.1 level. This is logic.

Regards

Ehsan

Yes, tolerable p-values depend upon sample size, standard deviation, and what the 'test' is supposedly measuring.

Regarding bias, yes it is good to have sum of estimated residuals be approximately zero, indicating perhaps no omitted variable bias, but even unbiased prediction can have huge variance (seen by summing squares of estimated residuals).