Solution. Treat last year’s yields as the unit values \(u\) of Section 1, and suppose the varieties do not differ, so that any estimated difference is error.

Design 3 is a completely randomised design. By Theorem 1.3.2, with the six unit values having \(S^2 = 2.667\), the estimate \(\bar y_A - \bar y_B\) is unbiased with variance \(2S^2/3 = 1.78\); there are \(4\) error degrees of freedom.

Design 2 is a randomised complete block design with locations as blocks. Within each location the difference between the two plots is \(d = 2\), \(-3\), \(-3\), and the estimate is the average of \(\pm d_j\) with random signs, unbiased with variance \(\sum d_j^2/9 = 22/9 = 2.44\); there are \(2\) error degrees of freedom.

Design 1 has only two possible allocations, and the variety difference is confounded with the difference between the left and right plots. Last year the right plots out-yielded the left ones by \(38.00 - 36.67 = 1.33\) on average, a systematic difference that would be reported as a variety effect; and since every comparison is between sides, there is no valid estimate of error at all. It should not be used.

These figures show something worth noticing: here the locations differ little (their means are \(38.0\), \(36.5\) and \(37.5\)) while the plots within a location differ a good deal, so blocking on location does not pay, and the completely randomised design is both more precise and has more error degrees of freedom. Blocking helps only when the blocks are more homogeneous than the units as a whole. One year’s data on six plots is thin evidence, however, and locations usually differ in soil and climate. If the experimenter expects the location differences of last year to be typical, Design 3 is preferable; if not, Design 2 is the safe recommendation, since it protects against location differences at the cost of two error degrees of freedom. The left–right pattern also suggests that the plots should be examined for a fertility gradient, which, if real, is the nuisance factor to block on.

Solution. The totals are \(66\), \(86\), \(76\) for the detergents and \(46\), \(71\), \(52\), \(59\) for the machines, with \(y_{\cdot \cdot } = 228\) and \(\sum \sum y^2 = 4500\). The correction factor is \(228^2/12 = 4332\), so \(SST = 168\), \(SS_{\mathrm {Detergents}} = (66^2 + 86^2 + 76^2)/4 - 4332 = 50\), \(SS_{\mathrm {Machines}} = (46^2 + 71^2 + 52^2 + 59^2)/3 - 4332 = 115.33\), and \(SSE = 2.67\) on \(6\) degrees of freedom.

\(F_0 = 56.25 > F_{0.05,2,6} = 5.14\) (\(p = 0.0001\)). The means are \(16.5\), \(21.5\) and \(19.0\), and \(HSD = 4.339\sqrt {0.444/4} = 1.45\); all three differences (\(5.0\), \(2.5\), \(2.5\)) exceed it, so detergent \(2\) is best and \(1\) worst. The relative efficiency is \[ RE = \frac {3(38.444) + 4(2)(0.444)}{11(0.444)} = 24.3 : \] the machines differ so much that without blocking about \(24\) times as many loads would have been needed.

Solution. \(y_{\cdot j} = t\mu + t\beta _j + \sum _i\tau _i + \sum _i\varepsilon _{ij} = t(\mu + \beta _j) + \varepsilon _{\cdot j}\), since \(\sum \tau _i = 0\). So \(E(y_{\cdot j}^2) = t^2(\mu + \beta _j)^2 + t\sigma ^2\), and similarly \(E(y_{\cdot \cdot }^2) = (tb\mu )^2 + tb\sigma ^2\). Then \begin {align*} E(SS_{\mathrm {Blocks}}) &= \frac 1t\sum _jE(y_{\cdot j}^2) - \frac {E(y_{\cdot \cdot }^2)}{tb} = t\sum _j(\mu + \beta _j)^2 + b\sigma ^2 - tb\mu ^2 - \sigma ^2 \\ &= t\sum _j\beta _j^2 + (b - 1)\sigma ^2 , \end {align*}

using \(\sum \beta _j = 0\). Dividing by \(b - 1\) gives the result.

Solution. With \(t = 2\), \(d_j = y_{1j} - y_{2j}\), and \(\bar y_{1\cdot } - \bar y_{2\cdot } = \bar d\). Then \(SS_{\mathrm {Treatments}} = b\sum _i(\bar y_{i\cdot } - \bar y_{\cdot \cdot })^2 = b\bar d^2/2\). The residuals are \(e_{1j} = -e_{2j} = (d_j - \bar d)/2\), so \(SSE = \sum _j(d_j - \bar d)^2/2 = (b - 1)s_d^2/2\) on \(b - 1\) degrees of freedom, and \(MSE = s_d^2/2\). Hence \[ F_0 = \frac {b\bar d^2/2}{s_d^2/2} = \Bigl (\frac {\bar d}{s_d/\sqrt b}\Bigr )^2 = t_0^2 . \] For the tyre data the RCBD analysis gives \(F_0 = 31.96 = 5.653^2\).

Solution. The RCBD spends \(b - 1\) degrees of freedom on estimating block effects, leaving \((t - 1)(b - 1)\) for error against \(t(b - 1)\) in the completely randomised design. Fewer error degrees of freedom mean a larger critical value \(t_{\alpha /2}\) or \(F_\alpha \) and a less precise estimate of \(\sigma ^2\). With \(t = 2\) and \(b = 3\), for example, the error degrees of freedom fall from \(4\) to \(2\) and \(t_{0.025}\) rises from \(2.78\) to \(4.30\): blocking must reduce the error variance by more than a factor of \((4.30/2.78)^2 \approx 2.4\) just to break even. With \(20\) or more error degrees of freedom the loss is negligible, and blocking on any nuisance factor that is likely to matter is almost always worthwhile.