To illustrate the use of this notation, suppose we want to estimate the mean value of quarterly sales for all Armand’s restaurants located near a campus with 10,000 students. Y* = b 0 + b 1x* = the point estimator of E(y*) and the predictor of an individual value of y* when x = x* Y* = the random variable denoting the possible values of the dependent variable y when x = x*Į(y*) = the mean or expected value of the dependent variable y when x = x* X* = the given value of the independent variable x
For this reason, as we discuss in more depth the issues concerning estimation and prediction, the following notation will help clarify matters.
When we are using the estimated regression equation to estimate the mean value of y or to predict an individual value of y, it is clear that the estimate or prediction depends on the given value of x. In this case, we are using y as the predictor of y for a new observation when x = 10. Hence, we would predict quarterly sales of $110,000 for such a new restaurant. For example, to predict quarterly sales for a new restaurant Armand’s is considering building near Talbot College, a campus with 10,000 students, we would compute y = 60 + 5(10) = 110. We can also use the estimated regression equation to predict an individual value of y for a given value of x. In this case we are using y as the point estimator of the mean value of y when x = 10. Thus, a point estimate of the mean quarterly sales for all restaurant locations near campuses with 10,000 students is $110,000. Using the estimated regression equation y = 60 + 5x, we see that for x = 10 (10,000 students), y = 60 + 5(10) = 110. For example, suppose Armand’s managers want to estimate the mean quarterly sales for all restaurants located near college campuses with 10,000 students. If a significant relationship exists between x and y and the coefficient of determination shows that the fit is good, the estimated regression equation should be useful for estimation and prediction.įor the Armand’s Pizza Parlors example, the estimated regression equation is y = 60 + 5x.Īt the end of Section 14.1, we stated that y can be used as a point estimator of E(y), the mean or expected value of y for a given value of x, and as a predictor of an individual value of y. We then use the least squares method to obtain the estimated simple linear regression equation. When using the simple linear regression model, we are making an assumption about the relationship between x and y.
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