Sep 25, 2017 term paper 2

# SHOW THAT THE MEAN AND VARIANCE OF S (IN PROB. 17) UNDER THE HYPOTHESIS OF INDEPENDENCE ARE 0 AND…

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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.

Prob. 17

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.

Prob. 16

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.

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