Statistical Inference: A Short Course

12.7 Testing Hypotheses about and

How do we know that X and Y are truly linearly related? Looked at in another fashion, how do we know that X actually contributes significantly to the prediction of Y? Answers to these questions can be garnered by way of testing a particular null hypothesis against an appropriate alternative hypothesis.

The most common type of hypothesis test is that of “no linear relationship between X and Y” or , that is, the conditional mean of Y given X does not depend linearly on X. Hence the implied null hypothesis of “no linear relationship between X and Y” is . Possible alternative hypotheses are:

a. If we are uncertain about the relationship between X and Y (we have no clue as to whether or not these variables vary directly or inversely), then the appropriate alternative hypothesis is . Clearly, this alternative implies a two-tail test with . If we reject H₀ in ...