What test is used for the significance of slope and intercept of linear regression models?

Question

Training a linear model in R, I get something like the following as summary (I think it looks similar in SPSS too).

Call:
lm(formula = df.full$diff.err ~ df.full$diff.emo)

Residuals:
    Min       1Q   Median       3Q      Max 
-0.96323 -0.10255 -0.00002  0.10104  0.94691 

Coefficients:
                  Estimate Std. Error t value Pr(>|t|)    
(Intercept)      1.737e-05  1.647e-03   0.011    0.992    
df.full$diff.emo 8.207e-01  7.924e-03 103.573   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.163 on 10060 degrees of freedom
Multiple R-squared:  0.5161,    Adjusted R-squared:  0.516 
F-statistic: 1.073e+04 on 1 and 10060 DF,  p-value: < 2.2e-16

Under the "Coefficients" section I am given p-values for the intercept and the slope. My question is, what is the name of the test which is (typically) used to obtain these p-values?

I am under the impression, that people online just generally refer to it as "the p-value of the slope/intercept" without discussing where these come from, for example here

Significance of Regression Intercept (R lm model)

http://blog.minitab.com/blog/adventures-in-statistics/how-to-interpret-regression-analysis-results-p-values-and-coefficients

http://blog.minitab.com/blog/adventures-in-statistics/regression-analysis-how-to-interpret-the-constant-y-intercept

Answer 1

As the last line of your output suggests, the p-value comes from the F-statistic (it is the probability of getting as high an F-statistic as you got under the null hypothesis), which in turn comes from the F test. There's a detailed discussion of the procedure of evaluating multiple linear regression slope with an F test here