Discussion
X and Y are two continuous random variables independent of each other. The combined sample based on two independent samples of size 5 and 4 on X and Y respectively has the following pattern :
\(y_{4} \lt x_{1} \lt x_{5} \lt y_{3} \lt x_{2} \lt y_{1} \lt y_{2} \lt\)
\(x_{4} \lt x_{3}\)
Hence the value of Mann-Whitney-Wilcoxon test statistic is :
Answer: B
The Mann-Whitney statistic is 8.
Count, for each \(y\), the number of \(x\)'s that come before it in the combined ordering \(y_{4} \lt x_{1} \lt x_{5} \lt y_{3} \lt x_{2}\) \(\lt y_{1} \lt y_{2} \lt x_{4} \lt x_{3}\):
- \(y_{4}\): 0
- \(y_{3}\): 2 (\(x_{1}, x_{5}\))
- \(y_{1}\): 3
- \(y_{2}\): 3
The total is \(U = 8\). Counting the other way, the number of \(y\)'s before each \(x\) gives \(1 + 1 + 2 + 4 + 4 = 12\), and \(8 + 12 = 5 \times 4 = 20\) as it must. The test statistic is the smaller count, 8. 12 is the complementary count, and 6 and 10 do not arise from either count.
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