8. The heights, in metres, and weights, in kilograms, of a random sample of 9 men are shown in the table below
| Man | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) | \(I\) |
| Height \(( x )\) | 1.68 | 1.74 | 1.75 | 1.76 | 1.78 | 1.82 | 1.84 | 1.88 | 1.98 |
| Weight \(( y )\) | 75 | 76 | 100 | 77 | 90 | 95 | 110 | 96 | 120 |
- Given that \(\mathrm { S } _ { x x } = 0.0632 , \mathrm {~S} _ { y y } = 1957.5556\) and \(\mathrm { S } _ { x y } = 9.3433\) calculate, to 3 decimal places, the product moment correlation coefficient between height and weight for these men.
- Use your value of the product moment correlation coefficient to test whether or not there is evidence of a positive correlation between the height and weight of men. Use a \(5 \%\) significance level. State your hypotheses clearly.
Peter does not know the heights or weights of the 9 men. He is given photographs of them and asked to put them in order of increasing weight. He puts them in the order
$$A C E B G D I F H$$
- Find, to 3 decimal places, Spearman's rank correlation coefficient between Peter's order and the actual order.
- Use your value of Spearman’s rank correlation coefficient to test for evidence of Peter's ability to correctly order men, by their weight, from their photographs. Use a 5\% significance level and state your hypotheses clearly.