Consider a small, hypothetical dataset with two groups: A : 10, 15, 50 B : 10, 1
ID: 3224443 • Letter: C
Question
Consider a small, hypothetical dataset with two groups:
A : 10, 15, 50
B : 10, 17, 19
(a) How many possible pairings of values (Xi, Yj) are there?
(b) List all possible pairings (Xi, Yj).
(c) Calculate the Mann-Whitney Statistic, U.
(d) Calculate the associated p-value for (c) if it is a two-sided test.
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Answer the following questions with TRUE or FALSE.
(a) For a resampling permutation test, as the value Rincreases, the error for the true p-value decreases.
(b) The Wilcoxon Rank-Sum test is not highly eected by outliers.
(c) For the Mann-Whitney test, if all values Xi < Yj,then the one-sided p-value will be 1.
(d) The permutation test can be used with the mean ortotal and the result will not change.
(e) If F1(x) F2(x), then µ1 > µ2.
Explanation / Answer
a.
There are 3 elements in A and 3 elements in B. So, 3*3=9 possible pair is there.
b.
The possible pairings are................
(10,10),(10,17),(10,19),(15,10),(15,17),(15,19),(50,10),(50,17),(50,19)
c.
To calculate Mann-Whitney Statistics let us first rank all the treatments in increasing order...........
Rank Treatment
1 10
2 10
3 15
4 17
5 19
6 50
Now the 10 has duplicate values with rank 1 and 2. We will take the average rank (1+2)/2=1.5 for it. Now let us assign those ranks in corresponding groups and calculate the sum.
A rank B rank
10 1.5 10 1.5
15 3 17 4
50 6 19 5
sum(rankA)=1.5+3+6=10.5, sum(rankB)=1.5+4+5=10.5
length(A)=no of elements in A=3, length(B)=no of elements in B=3
U(A)=10.5-3*(3+1)/2=10.5-6=4.5
U(B)=10.5-3*(3+1)/2=10.5-6=4.5
U=min(U(A),U(B))=4.5
d.
For two sample test with n1=3 and n2=3, we find the corresponding p-value for U is almost 1.00
d