Exact Tests for Correlation and for the Slope in Simple Linear Regressions Without Making Assumptions

We present an exact test for whether two random variables that have known bounds on their support are negatively correlated. The alternative hypothesis is that they are not negatively correlated. No assumptions are made on the underlying distributions. We show by example that the Spearman rank correlation test as the competing exact test of correlation in nonparametric settings rests on an additional assumption on the data generating process without which it is not valid as a test for correlation. We then show how to test for the signi.cance of the slope in a linear regression analysis that invovles a single independent variable and where outcomes of the dependent variable belong to a known bounded set