Regression Inference: Slope Test
Is the slope really nonzero? t-test for β₁, standard error, and a 95% CI for the slope.
Tests H₀: β₁ = 0 (no linear relationship) vs Hₐ: β₁ ≠ 0.
① Answer
② Steps
③ Why it works
The least-squares slope b₁ estimates the true slope β₁, but it wobbles from sample to sample with standard error SE(b₁) = s/√Sxx, where s measures scatter around the line. The t-statistic b₁/SE(b₁) tests whether the slope is distinguishable from zero — a flat line means x tells you nothing about y.