Suppose we obtain model estimates that yield predicted conditional mean \(\widehat{E}[y \mid x]=\exp (1+0.01 x) /[1+\exp (1+0.01 x)]\). Suppose the sample is of size 100 and \(x\) takes integer values \(1,2, \ldots, 100\). Obtain the following estimates of the estimated marginal effect \(\partial \widehat{\mathrm{E}}[y \mid x] / \partial x\).

(a) The average marginal effect over all observations.

(b) The marginal effect of the average observation.

(c) The marginal effect when \(x=90\).

(d) The marginal effect of a one-unit change when \(x=90\), computed using the finite-difference method.