Dropping a slope
Five points have x=(1,2,3,4,5) and y=(1,1,2,3,3). The least-squares line is \widehat y=0.2+0.6x, with residual sum of squares 0.4. The sample mean of y is 2, and the total sum of squares about the mean is 4.0.
Test H_0:\beta_1=0 at level 0.05. Use F_{1,3;0.95}=10.13.
Worked answer
The intercept-only residual sum of squares is the total sum of squares, 4.0. The line has residual sum of squares 0.4 on 5-2=3 degrees of freedom. One restriction is being tested, so
F=\frac{(4.0-0.4)/1}{0.4/3}=\frac{3.6}{0.1333}=27.
Since 27>10.13, reject H_0:\beta_1=0 at level 0.05. The five points are not compatible with a flat mean.