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The Mann-Whitney U Test is a nonparametric version of the independent samples t-test. The test primarily deals with two independent samples that contain ordinal data. The following example explained the comparison between the nonparametric versus parametric tests.
1. Select the best choice for the analysis and explain the reason.
2. Remember the importance of the p-value.
Results S.S. Bayesian Independent Samples T-Test
Since the prompt explicitly states the Mann-Whitney U Test is for "two independent samples that contain ordinal data," the selection would be:
If the actual data is confirmed to be ordinal (ranked) or strongly non-normal, the Mann-Whitney U Test is the best choice.
If the data is continuous and satisfies the assumption of normality, the Bayesian Independent Samples T-Test is the best choice (as it is typically more robust and informative than the standard $t$-test by providing evidence for or against the null hypothesis).
The $p$-value (P stands for probability) is critical in classical (frequentist) hypothesis testing, such as the Mann-Whitney U test or the standard independent samples $t$-test.
Definition: The $p$-value is the probability of observing a test statistic (or a more extreme one) if the null hypothesis ($H_0$) were true.
Decision Rule: Researchers typically set a significance level ($\alpha$), often $\alpha = 0.05$.
If $p < 0.05$: The probability of seeing the data by chance (if $H_0$ were true) is low. You reject the null hypothesis and conclude there is a statistically significant difference between the two groups.
The choice between a nonparametric test (like the Mann-Whitney U test) and a parametric test (like the Independent Samples $t$-test or its Bayesian version) depends entirely on the level of measurement of the dependent variable and the distribution characteristics of the data.