 Solution

Use a bivariate correlation to describe the direction and strength of association between two measured variables, such as study time and test score. The Pearson correlation coefficient describes linear association between two quantitative variables: its sign gives direction, and its magnitude reflects linear association on a standardized scale. A confidence interval describes estimation uncertainty; a corresponding hypothesis test can assess a specified null association under its sampling assumptions.

Inspect a scatter plot for curvature, clusters, influential observations and restriction of range. A coefficient near zero does not exclude a strong nonlinear relationship. For ordered or markedly nonnormal measurements, a Spearman rank correlation can describe monotone association, using ranks; it does not require a linear relation in the original units. Ordinary inferential procedures require independent observational units and appropriate assumptions for their chosen method. Correlation alone does not identify a causal direction or exclude confounding.