We previously used a t test to determine if X contributes significantly to the variation in Y. If H0: β1 = 0 is true (we posit “no linear relationship” between X and Y), then the sole source of variation in Y is the random disturbance term ε since the population regression sum of squares is zero. Now, it can be demonstrated that the statistic
Under H0: β1 = 0, Equation (13.28) becomes
Here the appropriate alternative hypothesis is H1: β1 ≠ 0 so that the critical region is (we have a one-tail alternative using the upper tail of the F distribution). So if we reject H0 in favor of H1, then we can safely conclude that there exists a statistically significant linear relationship between X and Y at the 100 α% level.
Using the information provided in Table 12.4, let us conduct a significance test of the linear relationship between X and Y for α = 0.05. Here we test H0: β1 = 0, against H1: β1 ≠ 0 with . Using Equation (13.28.1) ...