Identify two conditions under which it would be more appropriate to use Spearman’s rank correlation coefficient rather than the product moment correlation coefficient (PMCC) to analyze the relationship between two variables.

A conservationist monitors the density of a rare orchid species, $D$ (plants per $m^2$), and the local soil pH, $p$, across $n=10$ distinct plots in a meadow.

The Spearman’s rank correlation coefficient for the recorded data was found to be $r_s=0.655$.

Investigate, at the 5% level of significance, whether there is statistical evidence of any correlation between the density of orchids and soil pH. Explicitly state your null and alternative hypotheses and the critical value used.

The product moment correlation coefficient (PMCC) for the same dataset was calculated as $r=0.515$.

Determine, at the 5% level of significance, if there is evidence of a positive correlation between orchid density and soil pH. State your hypotheses clearly and include the relevant critical value.