Problem 1.1. State the model for the one-way layout, define every symbol, and give the two side conditions under which \(\mu \) and the \(\tau _j\) become individually determined. Say which quantities are unaffected by the choice.

Problem 1.2. An experiment compares \(k=5\) diets on \(N=40\) animals. Write down the degrees of freedom for treatments, error and total, and state the distribution of the \(F\) ratio under the null hypothesis.

Problem 1.3. Show that \(\sum _{j}\sum _{i}\left (Y_{ij}-\overline {Y}_{\cdot \cdot }\right )^2 =\sum _{j}n_j\left (\overline {Y}_{\cdot j}-\overline {Y}_{\cdot \cdot }\right )^2 +\sum _{j}\sum _{i}\left (Y_{ij}-\overline {Y}_{\cdot j}\right )^2\), and identify the step at which the cross-product term vanishes. Where to start: add and subtract \(\overline {Y}_{\cdot j}\) inside the bracket.

Problem 1.4. Explain the difference between a fixed effects and a random effects model, using the example of the personnel officers in Section 1.1. In which case is the conclusion about the five officers who were sampled, and in which case about officers in general?

Problem 1.5. A researcher compares four fertilisers on plots lying in three fields of visibly different fertility, and analyses the data as a one-way layout. Explain what this does to the error mean square and to the power of the test, and give the design that should have been used.

Problem 1.6. For a randomised block design with \(b=5\) blocks and \(k=4\) treatments, complete the degrees of freedom column of the analysis of variance table. If the block sum of squares is large, what does that say about the decision to block — and what would it have cost to ignore it?

Problem 1.7. Randomisation is described in Section 1.1 as the corner stone of experimental design. Explain what it protects against that blocking does not, and why both are needed.