3. A cafe owner wishes to know whether the price of strawberry jam is related to the taste of the jam. He finds a website that lists the price per 100 grams and a mark for the taste, out of 100, awarded by a judge, for 9 different strawberry jams \(A , B , C , D , E , F , G , H\) and \(I\). He then ranks the marks for taste and the prices.
The ranks are shown in the table below.
| Rank | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| Price | \(A\) | \(B\) | \(E\) | \(C\) | \(D\) | \(F\) | \(G\) | \(H\) | \(I\) |
| Taste | \(A\) | \(B\) | \(F\) | \(E\) | \(H\) | \(G\) | \(I\) | \(C\) | \(D\) |
- Calculate Spearman's rank correlation coefficient for these data.
- Test, at the \(5 \%\) level of significance, whether or not there is a relationship between the price and the taste of these strawberry jams. State your hypotheses clearly.
A friend suggests that it would be better to use the price per 100 grams, \(c\), and the mark for the taste, \(m\), for each strawberry jam rather than rank them.
Given that
$$\mathrm { S } _ { c c } = 2.0455 \quad \mathrm {~S} _ { m m } = 243.5556 \quad \mathrm {~S} _ { c m } = 16.4943$$
- calculate the product moment correlation coefficient between the price and the mark for taste of these strawberry jams, giving your answer correct to 3 decimal places.
- Use your value of the product moment correlation coefficient to test, at the \(5 \%\) level of significance, whether or not there is evidence of a positive correlation between the price and the mark for taste of these 9 strawberry jams. State your hypotheses clearly.
- State which of the tests in parts (b) and (d) is more appropriate for the cafe owner to use. Give a reason for your answer.