> library(faraway)
> data(savings)
> fit <- lm(sr ~ pop15 + pop75 + dpi + ddpi, savings)
> fit.sum <- summary(fit)
> fit.sum
Call:
lm(formula = sr ~ pop15 + pop75 + dpi + ddpi, data = savings)


Residuals:
    Min      1Q  Median      3Q     Max
-8.2422 -2.6857 -0.2488  2.4280  9.7509


Coefficients:
              Estimate Std. Error t value Pr(>|t|)
(Intercept) 28.5660865  7.3545161   3.884 0.000334 ***
pop15       -0.4611931  0.1446422  -3.189 0.002603 **
pop75       -1.6914977  1.0835989  -1.561 0.125530
dpi         -0.0003369  0.0009311  -0.362 0.719173
ddpi         0.4096949  0.1961971   2.088 0.042471 *
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 3.803 on 45 degrees of freedom
Multiple R-squared: 0.3385,     Adjusted R-squared: 0.2797
F-statistic: 5.756 on 4 and 45 DF,  p-value: 0.0007904

What does p value mean ? Why should be reject that parameter if the p value is < 0.05?

> parameter.mean <- coef(fit.sum)[3, 1]
> parameter.sigma <- coef(fit.sum)[3, 2]
> parameter.prob <- coef(fit.sum)[3, 4]
> tstat <- qt(0.975, 45)
> bands <- parameter.mean + c(-1, 1) * parameter.sigma * tstat
> bands
[1] -3.8739780  0.4909826

One can see that the pop75 variable confidence bands contain 0 and hence pop75 can be ignored. Since we are checking for 0.975 value, which is 5 percent prob, the display prob is more than 5 percent Hence null is accepted that parameter value pop75 = 0