9.4.2 - Comparing Correlation and Slope

Some of you may have noticed that the hypothesis test for correlation and slope are very similar. Also, the test statistic for both tests follows the same distribution with the same degrees of freedom, \(n-2\).

This similarity is because the two values are mathematically related. In fact,

\(\hat{\beta}_1=r\dfrac{\sqrt{\sum (y_i-\bar{y})^2}}{\sqrt{\sum(x_i-\bar{x})^2}}\)

Here is a summary of some of the similarities and differences between the sample correlation and the sample slope.

Similarities Section

  • The test for correlation will lead to the same conclusion as the test for slope.
  • The sign of the slope (i.e. negative or positive) will be the same for the correlation. In other words, both values indicate the direction of the relationship

Differences Section

  • The value of the correlation indicates the strength of the linear relationship. The value of the slope does not.
  • The slope interpretation tells you the change in the response for a one-unit increase in the predictor. Correlation does not have this kind of interpretation.