Explain the advantages of randomisation and replication in a statistically designed experiment.
The usual statistical model underlying the one-way analysis of variance is given, in the usual notation, by
$$x _ { i j } = \mu + \alpha _ { i } + e _ { i j }$$
where \(x _ { i j }\) denotes the \(j\) th observation on the \(i\) th treatment. Define carefully all the terms in this model and state the properties of the term that represents experimental error.
A trial of five fertilisers is carried out at an agricultural research station according to a completely randomised design in which each fertiliser is applied to four experimental plots of a crop (so that there are 20 experimental units altogether). The sums of squares in a one-way analysis of variance of the resulting data on yields of the crop are as follows.
Source of variation
Sum of squares
Between fertilisers
219.2
Residual
304.5
Total
523.7
State the customary null and alternative hypotheses that are tested. Provide the degrees of freedom for each sum of squares. Hence copy and complete the analysis of variance table and carry out the test at the 5\% level.