Suppose n = 5 and we observe the data (X1, Y1) = (3.7, 9.2),(X2, Y2) = (4.3, 9.4),(X3, Y3) = (5.0, 9.2),(X4, Y4) = (6.2, 10.4), and X5, Y5) = (5.3, 9.2). What is the conditional probability distribution of rs under H0 (8.1) when average ranks are used to break ties among the Y's? How extreme is the observed value of rs in this conditional null distribution? Compare this fact with that obtained by taking the observed value of rs to the (incorrect) unconditional null distribution of rs. (See also Problem 48.)
Problem 48
For the case of n = 5 untied bivariate (X, Y) observations, obtain the form of the exact null (H0) distribution of rs. (See Comment 41)