12.105 Regression with an error term4 An alternative to the regression formula my = a + bx expresses each y value, rather than the mean of the y values, in terms of x. This approach models an observation on y as y = mean + error = a + bx + e, where the mean my = a + bx and the error = e.

The error term denoted by e (the Greek letter epsilon)

represents the deviation of the observation from the mean, that is, e = error = y - mean.

a. If an observation has e 7 0, explain why the observation falls above the mean.

b. What does e equal when the observation falls exactly at the mean? The e term represents the error that results from using the mean value 1a + bx2 of y at a certain value of x for the prediction of the individual observation on y.

c. For the sample data and their prediction equation yn = a + bx, explain why an analogous equation to the population equation y = a + bx + e is y = a + bx +

e, where e is the residual, e = y - yn.

(The residual e estimates

e. We can interpret e as a sample residual and e as a population residual.)

d. Explain why it does not make sense to use the simpler model, y = a + bx, which does not have the error term.