# 13.4: Hypothesis Testing in Regression

Regression, like all other analyses, will test a null hypothesis in our data. In regression, we are interested in predicting $$Y$$ scores and explaining variance using a line, the slope of which is what allows us to get closer to our observed scores than the mean of $$Y$$ can. Thus, our hypotheses concern the slope of the line, which is estimated in the prediction equation by $$b$$. Specifically, we want to test that the slope is not zero:

$\begin{array}{c}{\mathrm{H}_{0}: \text { There is no explanatory relation between our variables }} \\ {\mathrm{H}_{0}: \beta=0}\end{array} \nonumber$

$\begin{array}{c}{\mathrm{H}_{\mathrm{A}}: \text {There is an explanatory relation between our variables}} \\ {\mathrm{H}_{\mathrm{A}}: \beta>0} \\ {\mathrm{H}_{\mathrm{A}}: \beta<0} \\ {\mathrm{H}_{\mathrm{A}}: \beta \neq 0}\end{array} \nonumber$

A non-zero slope indicates that we can explain values in $$Y$$ based on $$X$$ and therefore predict future values of $$Y$$ based on $$X$$. Our alternative hypotheses are analogous to those in correlation: positive relations have values above zero, negative relations have values below zero, and two-tailed tests are possible. Just like ANOVA, we will test the significance of this relation using the $$F$$ statistic calculated in our ANOVA table compared to a critical value from the $$F$$ distribution table. Let’s take a look at an example and regression in action.

## Contributors

• Foster et al. (University of Missouri-St. Louis, Rice University, & University of Houston, Downtown Campus)