Fitted Values. Let (left.widehat{y}_{i}=pileft(mathbf{x}_{i}^{prime} mathbf{b}_{M L E} ight) ight)) denote the (i) th fitted value for the

Fitted Values. Let \(\left.\widehat{y}_{i}=\pi\left(\mathbf{x}_{i}^{\prime} \mathbf{b}_{M L E}\right)\right)\) denote the \(i\) th fitted value for the logit function. Assume that an intercept is used in the model so that one of the explanatory variables \(x\) is a constant equal to one. Show that the average response is equal to the average fitted value; that is, show \(\bar{y}=n^{-1} \sum_{i=1}^{n} \widehat{y}_{i}\).