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I have 2 independent populations, each with N=50 people. I took n=20 samples from each population. For each person, I asked a question with an answer from 1 to 5. I want to know if there is a significant difference between the distributions of the answers of the 2 populations (e.g., mean). I can assume that the variance is the same in the 2 populations.

Which statistical test fits this scenario, and how do I use it in SPSS?

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I will assume your variables are likert and ordinal from your wording in your question. Mann–Whitney U test is a common statistical test for your scenario.

It's also possible to use t-test, but that requires normality assumption. Read the following blog post to decide yourself wherther you like t-test or Mann-Whitney (I prefer Mann-Whitney):

http://blog.minitab.com/blog/adventures-in-statistics-2/best-way-to-analyze-likert-item-data:-two-sample-t-test-versus-mann-whitney

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    $\begingroup$ What are the implications of using a Mann-Whitney test when we have small populations and relatively large sample sizes (as a proportion of the population size)? $\endgroup$
    – Groovy_Worm
    Commented Jan 14, 2017 at 14:21
  • $\begingroup$ Yes shouldn't there be a finite population correction for each population that shoulld be used? It adds information about the mean and reduces sample variance. $\endgroup$
    – Michael R. Chernick
    Commented Jan 14, 2017 at 18:37
  • $\begingroup$ Yes, and our sampling also affects the composition of the remaining unsampled population so that observations are not independent of one another. $\endgroup$
    – Groovy_Worm
    Commented Jan 15, 2017 at 9:14
  • $\begingroup$ Personally, I would start thinking about a simulation based approach using the 40 observations as a best guess at a single parent population. But the question asks for something suitable to use in SPSS. I did start to think of Chi-square/G-test approaches (with exact probabilities from multivariate hypergeometric distribution) but Tomer seems to want to explicitly test for differences in central location. $\endgroup$
    – Groovy_Worm
    Commented Jan 15, 2017 at 9:20

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