7.3 Testing Individual Coefficients
Let’s test \(H_0: \beta_1 = 0\) versus \(H_a: \beta_1 \neq 0\). Recall that if the null hypothesis holds, then \[\hat{\beta}_1/se({\beta_1}) \sim t_{n-2}\] where \(t_{n-2}\) denotes t-distribution with \(n-2\) degrees of freedom.
Recall from above, the summary() output reports a t-value for \(\beta_1\) (Y2012) of 21.84.
summary(lin.model)##
## Call:
## lm(formula = Y2016 ~ Y2012, data = election)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.079300 -0.022713 0.000465 0.019404 0.061641
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.02235 0.02148 -1.041 0.303
## Y2012 0.95295 0.04364 21.838 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.03152 on 48 degrees of freedom
## Multiple R-squared: 0.9086, Adjusted R-squared: 0.9066
## F-statistic: 476.9 on 1 and 48 DF, p-value: < 2.2e-16Not surprisingly, the p-value for this t-statistic is extremely small.