Problem 3.7. Four treatments are compared with \(n=8\) observations each and \(MSE=12.4\). Compute the half-width of a \(95\%\) confidence interval for a single difference \(\tau _i-\tau _j\) by (a) the ordinary \(t\) interval, (b) Bonferroni for all six pairwise differences, and (c) Tukey. Rank the three and explain the ordering.

Problem 3.8. Explain in one sentence each what Bonferroni, Tukey and Scheffé guarantee, and state which of the three is the only one you may use for a contrast you decided to look at after seeing the data.

Problem 3.9. The studentised range distribution is used rather than the \(t\) distribution in Tukey’s method. Explain what quantity it describes and why that is the right reference for the largest of several differences.

Problem 3.10. In a two-factor study with factors \(A\) at 3 levels and \(B\) at 4 levels and \(n=2\) replicates per cell, give the degrees of freedom for \(A\), \(B\), the interaction, error and total.

Problem 3.11. A two-factor experiment gives a significant interaction. A colleague reports the main effect of \(A\) averaged over the levels of \(B\). Explain what is wrong with that summary and what should be reported instead.

Problem 3.12. Explain why a two-factor design with one observation per cell cannot test the interaction, and say what is being assumed when such a design is analysed anyway.

Problem 3.13. A Latin square with \(k=5\) eliminates two nuisance factors using only \(25\) runs, where a full three-factor design would need \(125\). State the assumption that buys this economy, and what happens to the analysis if it is false.

Problem 3.14. Write down the model for a \(4\times 4\) Latin square, give the degrees of freedom for rows, columns, treatments and error, and comment on whether the error degrees of freedom are adequate.