explanation of the summary of linear model

0: residuals distribution

1: coefficients

2. stdandard variance of coefficients

3: using t-test to test the significant of coefficients.  p-value of Hypothesis: coef=0. The p-value is the accept region of coef=0. The less the p-value is, the more significant the coefficient is.

4. (Pearson) Correlation coefficient r. R-sqared value is square of r. The 'Adjusted R-squared' is more demanding as it takes into account the number of parameters of the regression model. The greater the r is , the more fit the model is. 

Information about r from wiki:

formular:

     for a population:

 for a sample:

graph:

    

5. p-value of hypothesis: coef_1=coef_2=...=coef_n=0. Usually, if the model fails this test (e.g, with a p-value that is considered too high, for example, higher than 0.1), it makes no sense to look at the t-tests on the individual coefficients.

ps: The explanation of anova result is similar.