An environmental study measured the health of a river system at 7 different locations. At each location, the researchers measured the water temperature, TTT (∘ ^\circ\,∘C), and the dissolved oxygen content, DDD (mg/L). Additionally, a subjective visual pollution rank, VVV, was recorded for each location, where Rank 1 indicates the clearest water and Rank 10 indicates the most debris.
| Location | A | B | C | D | E | F | G |
|---|---|---|---|---|---|---|---|
| TTT | 12 | 15 | 18 | 21 | 24 | 27 | 30 |
| DDD | 10.2 | 9.4 | 8.1 | 7.3 | 6.4 | 5.5 | 4.2 |
| VVV | 2 | 4 | 5 | 7 | 6 | 9 | 10 |
The following summary statistics are provided: Stt=252S_{tt} = 252Stt=252, Sdd=27.12S_{dd} = 27.12Sdd=27.12, Std=−82.5S_{td} = -82.5Std=−82.5.
Calculate the product moment correlation coefficient (PMCC) between water temperature and dissolved oxygen content.
Use your value from part (a) to test, at the 1% significance level, whether there is evidence of a negative correlation between temperature and dissolved oxygen. State your hypotheses clearly.
State an assumption about the data required for the test in part (b) to be valid.
Calculate the Spearman's rank correlation coefficient between dissolved oxygen content and the visual pollution rank.
Using a 5% level of significance and a two-tailed test, determine whether there is evidence of a correlation between dissolved oxygen and visual pollution rank.