Problem 6.1. Explain why the differences must be taken before, not after, computing the covariance matrix in a paired design, and what goes wrong if the \(2n\) observations are treated as two independent samples.

Problem 6.2. Show that Theorem 6.5 reduces to the square of the ordinary two-sample \(t\) statistic when \(p=1\), and check that the degrees of freedom agree.

Problem 6.3. Two independent samples of sizes \(n_1=25\) and \(n_2=30\) are taken on \(p=3\) variables. State the multiplier \(c^{2}\) for simultaneous \(T^{2}\) intervals at \(\alpha =0.05\), and the corresponding Bonferroni multiplier for the three component comparisons. Which is larger, and why must it be?

Problem 6.4. Using the additivity of the Wishart, prove that \(E\left (S_{\text {pooled}}\right ) = \Sigma \). Where to start: Theorem 5.17 parts (i) and (ii).

Problem 6.5. A two-sample \(T^{2}\) is significant at the \(1\%\) level, yet no individual variable differs significantly between the groups. Explain how this arises and what should be reported.

Problem 6.6. Discuss what happens to the level of the two-sample test when \(\Sigma _1\neq \Sigma _2\), distinguishing the cases \(n_1=n_2\) and \(n_1\ll n_2\).