Problem 3.1. Show that in the one-way layout \(\mu +\overline {\tau }\), where \(\overline {\tau }=\tfrac {1}{k}\sum _j\tau _j\), is estimable, and give a linear function of the observations that estimates it unbiasedly.

Problem 3.2. For the randomised block model \(Y_{ij}=\mu +\beta _i+\tau _j+\varepsilon _{ij}\), determine which of \(\mu \), \(\tau _1\), \(\tau _1-\tau _2\), \(\beta _1-\beta _2\) and \(\mu +\beta _1+\tau _1\) are estimable. Where to start: write down a typical row of \(X\) and apply Theorem 3.4.4.

Problem 3.3. Verify Definition 3.4.1 for \(A=\begin {pmatrix}1&1\\1&1\end {pmatrix}\) by exhibiting two different generalised inverses, and confirm that \(\underline {\lambda }^t=(1,-1)\) gives the same value of \(\underline {\lambda }^tA^{-}\underline {\lambda }\) for both while \(\underline {\lambda }^t=(1,0)\) does not.

Problem 3.4. With \(k=4\) treatments, write down three mutually orthogonal contrasts and verify their orthogonality. Explain what each one tests if the treatments are four doses of a fertiliser, \(0,1,2,3\) units.

Problem 3.5. Two statistical packages fit the same one-way analysis of variance and report different values of \(\widehat {\tau }_1\) but identical values of \(\widehat {\tau }_1-\widehat {\tau }_2\) and identical analysis of variance tables. Explain, and say which of the reported quantities you would be willing to put in a report.

Problem 3.6. Show that the hypothesis \(\beta _2=\beta _3=0\) in a regression on \(X_1,X_2,X_3,X_4\) can be written as \(CB=\underline {0}\), give \(C\), and state the degrees of freedom of the resulting \(F\) test.