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Show that the mean and variance of S (in Prob. 17) under the hypothesis of independence are 0 and 1/(n – 1), respectively.
Argue that the distribution of S in Prob. 16 is independent of the form of the distributions of X and Y provided that X and Yare continuous and independently distributed random variables. Hence S can be used as a test statistic in a nonparametric test of the null hypothesis of independence.
A common measure of association for random variables X and Y is the rank correlation, or Spearman`s correlation. The X values are ranked, and the observations are replaced by their ranks; similarly the Y observations are replaced by their ranks. For example, for a sample of size 5 the observations
are replaced by
Let r(Xt) denote the rank of XI and r( Yt) the rank of Yt. Using these paired ranks, the ordinary sample correlation is computed:
(b) Compute the ordinary correlation and Spearman`s correlation for the above data.