Problem 4.9. Prove that \(\sum _{i=1}^{n}h_{ii}=k\) for a full rank \(n\times k\) design matrix, and deduce the average leverage. Where to start: \(tr(H)\) and the cyclic property of the trace.

Problem 4.10. Show that \(\text {Var}\left (\widehat {e}_i\right )=\sigma ^2\left (1-h_{ii}\right )\) and explain why raw residuals should not be compared with one another directly.

Problem 4.11. An observation has \(h_{ii}=0.9\) and a residual close to zero. Is it influential? Is it well fitted? Should it be a concern? Answer all three and explain how they can hold together.

Problem 4.12. Using \(D_i=\dfrac {r_i^2}{k}\cdot \dfrac {h_{ii}}{1-h_{ii}}\), compute Cook’s distance for an observation with \(r_i=2.5\), \(h_{ii}=0.4\) in a model with \(k=4\). Comment.

Problem 4.13. For the cement data of Section 4.4, the full model has \(s^2=5.983\) with \(n=13\). The model in \(X_1\) and \(X_4\) has \(SSE=57.90\) and \(p=3\). Compute \(C_p\) and comment on it in relation to \(p\).

Problem 4.14. Explain why \(R^2\) cannot be used to choose between models with different numbers of regressors, and show algebraically that \(R^2_a\) can decrease when a variable is added.

Problem 4.15. A colleague reports a regression obtained by stepwise selection from 40 candidate variables on 50 observations, with all retained coefficients significant at the \(1\%\) level and \(R^2=0.86\). What questions would you ask, and what would you ask to be done before the model is used?