Mann-Whitney-Wilcoxon Test vs. Z-test, for non-normal samples

Control group includes 150 Poisson (=2.3) cases;
Experiment group includes 200 cases, with fixed effect +1 which is unknown but interested in.

Researcher A chose Z test, considering the big sample sizes.
Researcher B chose Mann-Whitney-Wilcoxon Test, considering the non-normal distributions.
Whose confidence interval will beat?

Researcher A might complain: "Your CI width is actually from Poisson distribution. Most time it is actually zero. Let's try . It would be fairer for Z-Test to use a continual distribution."

Mann-Whitney-Wilcoxon plays better than his expectation even in the continual case.

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