Lesson 5 of 8 · 15 min

Testing the slope: t-test, correlation test and F-test

To test a slope, divide (estimate − hypothesised value) by its standard error and compare with a t critical value with n − 2 degrees of freedom; the F-test and the correlation test give the same verdict on a zero slope.

In short

  • Slope t-test: t=(b^1−B1)/sb^1t = (\hat b_1 - B_1)/s_{\hat b_1}, df = n − 2. B1B_1 can be 0, 1 or any value; tests can be two- or one-sided.
  • sb^1=se/∑(Xi−Xˉ)2s_{\hat b_1} = s_e/\sqrt{\sum (X_i-\bar X)^2}: more spread in X → smaller standard error → larger t.
  • Confidence interval for the slope: b^1±tc sb^1\hat b_1 \pm t_c\, s_{\hat b_1} with n − 2 df; reject a two-sided H0H_0 if the hypothesised value lies outside it.
  • Correlation test: t=rn−2/1−r2t = r\sqrt{n-2}/\sqrt{1-r^2}. In simple regression it equals the t for H0:b1=0H_0: b_1 = 0.
  • F-test of fit: F = MSR ÷ MSE with 1 and n − 2 df, one-tailed (right). H0:b1=0H_0: b_1 = 0. In simple regression F=t2F = t^2.
  • Reject H0H_0 when the test statistic falls outside the critical value(s).

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Testing the slope: t-test, correlation test and F-test · Simple Linear Regression