A First Course in Statistics, 12th Edition

9.4 Assessing the Utility of the Model: Making Inferences about the Slope ${\beta }_1$

Now that we have specified the probability distribution of $\epsilon$ and found an estimate of the variance ${\sigma }^2,$ we are ready to make statistical inferences about the linear model’s usefulness in predicting the response y. This is step 4 in our regression modeling procedure.

Refer again to the data of Table 9.1, and suppose the reaction times are completely unrelated to the percentage of drug in the bloodstream. What could then be said about the values of ${\beta }_0$ and ${\beta }_1$ in the hypothesized probabilistic model

$$y={\beta }_0+{\beta }_{1}x+\epsilon$$

if x contributes no information for the prediction of y? The implication is that the mean of y—that is, the deterministic part of the model $E(y)={\beta }_0+{\beta }_{1}x$—does not change ...