5. (a) Give two circumstances where it may be more appropriate to use Spearman's rank correlation coefficient rather than Pearson's product moment correlation coefficient.
(b) A farmer needs a new tractor. The tractor salesman selects 6 tractors at random to show the farmer. The farmer ranks these tractors, in order of preference, according to their ability to meet his needs on the farm. The tractor salesman makes a note of the price and power take-off (PTO) of the tractors.
| Tractor | Farmer's rank | PTO (horsepower) | Price ( \(\boldsymbol { \pounds } \mathbf { 1 0 0 0 s }\) ) |
| A | 1 | 77.5 | 80 |
| B | 6 | 87.9 | 45 |
| C | 5 | \(53 \cdot 0\) | 47 |
| D | 4 | \(41 \cdot 0\) | 53 |
| E | 2 | \(112 \cdot 0\) | 60 |
| F | 3 | \(90 \cdot 0\) | 61 |
Spearman's rank correlation coefficient between the farmer's ranks and the price is 0.9429 .
- Test at the \(5 \%\) significance level whether there is an association between the price of a tractor and the farmer's judgement of the ability of the tractor to meet his needs on the farm.
- Calculate Spearman's rank correlation coefficient between the farmer's rank and PTO.
- How should the tractor salesman interpret the results in (i) and (ii)?