4 An agricultural company conducts a trial of five fertilisers (A, B, C, D, E) in an experimental field at its research station. The fertilisers are applied to plots of the field according to a completely randomised design. The yields of the crop from the plots, measured in a standard unit, are analysed by the one-way analysis of variance, from which it appears that there are no real differences among the effects of the fertilisers.
A statistician notes that the residual mean square in the analysis of variance is considerably larger than had been anticipated from knowledge of the general behaviour of the crop, and therefore suspects that there is some inadequacy in the design of the trial.
- Explain briefly why the statistician should be suspicious of the design.
- Explain briefly why an inflated residual leads to difficulty in interpreting the results of the analysis of variance, in particular that the null hypothesis is more likely to be accepted erroneously.
Further investigation indicates that the soil at the west side of the experimental field is naturally more fertile than that at the east side, with a consistent 'fertility gradient' from west to east.
- What experimental design can accommodate this feature? Provide a simple diagram of the experimental field indicating a suitable layout.
The company decides to conduct a new trial in its glasshouse, where experimental conditions can be controlled so that a completely randomised design is appropriate. The yields are as follows.
| Fertiliser A | Fertiliser B | Fertiliser C | Fertiliser D | Fertiliser E |
| 23.6 | 26.0 | 18.8 | 29.0 | 17.7 |
| 18.2 | 35.3 | 16.7 | 37.2 | 16.5 |
| 32.4 | 30.5 | 23.0 | 32.6 | 12.8 |
| 20.8 | 31.4 | 28.3 | 31.4 | 20.4 |
- Construct the usual one-way analysis of variance table. Carry out the appropriate test, using a \(5 \%\) significance level. Report briefly on your conclusions.
- State the assumptions about the distribution of the experimental error that underlie your analysis in part (iv).