The t-test for the true slope β₁ = 0 is identical to the t-test for the true correlation ρ = 0.

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The t-test for the true slope β₁ = 0 is identical to the t-test for the true correlation ρ = 0 in a simple linear regression model. This can be shown mathematically using the formula for the t-test statistic for the slope (t = β/SE) and the formula for the correlation coefficient (r = cov(X,Y)/sd(X)*sd(Y)). In a multiple regression model, however, the t-test for the slope may be different from the t-test for the correlation, as the relationship between variables can be more complex.

The T-Test for the True Slope β1 = 0 Is

The T-Test for the True Slope β₁ = 0 Is