A First Course in Statistics, 12th Edition

9.3 Model Assumptions

In Section 9.2, we assumed that the probabilistic model relating the drug reaction time y to the percentage x of drug in the bloodstream is

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

We also recall that the least squares estimate of the deterministic component of the model, ${\beta }_0+{\beta }_{1}x,$ is

$$\widehat{y}={\widehat{\beta }}_0+{\widehat{\beta }}_{1}x=-.1+.7x$$

Now we turn our attention to the random component $\epsilon$ of the probabilistic model and its relation to the errors in estimating ${\beta }_0$ and ${\beta }_1.$ We will use a probability distribution to characterize the behavior of $\epsilon .$ We will see how the probability distribution of $\epsilon$ determines how well the model describes the relationship between the dependent variable y and the independent variable x.

Step 3 in a regression analysis requires us to specify the probability distribution ...