Question

Suppose that X = (Xi, X2, . . . , Xn) and Y = (y,Y2, . . . ,Yn) are random samples from continuous distributions F and G, respectively. Wilcoxon's two-sample test statistic W = W(X,Y) is defined to be Σ-ngi Ri where Ri is the rank of in the combined sample.

Unexposed: 8,11,12,14,20,43,111

Exposed: 35,56,83,92,128,150,176,208

Answer 1

Steps:

1. Let the sample X(sample size n) be the exposed sample and sample Y(sample size m) will the unexposed sample.

2. Now, combine both the samples and rank them in increasing order, keeping track of which observation belongs to which sample.

3. Compute W(N)= Sum of ranks associated with X sample values in the combined ranking scheme.

here, n=8 and m=7 , N=m+n=15

Ho: Both the sample means are equal

H1: mean of exposed sample is 25 units higher than mean of unexposed sample.

After combining and ranking, the data will look like: 8,11,12,14,20,35,43,56,83,92,111,128,150,176,208

and the sample from which each obs belongs is: Y,Y,Y,Y,Y,X,Y,X,X,X,Y,X,X,X,X

following above steps we get W(N)= 87

From table at alpha = 5% , critical value= 16

Since, W(n)> W(c) we reject the null hypothesis.