1. This problem uses the United Nations example (data file UN11) to demonstrate many of the properties of added-variable plots. This problem is based on the mean function fertility log (ppgdp) + pctUrban. (a) Examine the scatterplot matrix for (fertility, log (ppgdp), pctUrban), and com- ment on the marginal relationships. (b) Fit the two simple regressions, fertility log (ppgdp) and fertility pctUrban, and verify that the slope coefficients are significantly differ- ent from 0 at any conventional level of significance. (c) Obtain the added-variable plots for both predictors. Based on the added-variable plots, is log (ppgdp) useful after adjusting for pctUrban, and similarly, is pctUrban useful after adjusting for log (ppgdp)? Compute the estimated mean function with both predictors included as regressors, and verify the findings of the added-variable plots. (d) Show that the estimated coefficient for pctUrban is the same as the estimated slope in the added-variable plot for pctUrban after log (ppgdp). This correctly suggests that all the estimates in a multiple linear regression model are adjusted for all the other regressors in the mean function. (e) Show that the residuals in the added-variable plot are identical to the residuals from the mean function with both predictors.