Lesson 5 of 9 · All lessons in this module
  1. The regression model and least squares
  2. Interpreting coefficients, fitted values and residuals
  3. The four assumptions and residual plotsVideo lesson
  4. Sums of squares, \(R^2\), the ANOVA table and the SEEVideo lesson
  5. Testing the slope: t-test, correlation test and F-testVideo lesson
  6. Intercept tests, indicator variables and p-values
  7. Forecasts and prediction intervals
  8. Functional forms: log-lin, lin-log and log-logVideo lesson
  9. Integrated cases: reading a full regression printout

Lesson 5 of 9 · 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).

Unlock this lesson free for 7 days

Create a free account to get 7 days free: every lesson, video and flashcard, and the question bank. No card needed.

Start your 7-day free trialLog in