Introduction to Tests for Statistical Equality of Two or More Populations: Two random variables X and Y having distribution F X (x) and F Y (y) respectively, are said to be equivalent, or equal in rule, or equal in distribution, if and only if they have the same distribution function. That is, F X (z) = F Y (z), for all z, There are different tests depending on the intended applications. The widely used tests for statistical equality of populations are as follow: 1. Equality of Two Normal Populations: One may use the Z-test and F-test to check the equality of the means, and the equality of variances, respectively. 2. Testing a Shift in Normal Populations: Often we are interested in testing for a given shift in a given population distribution, that is testing if a random variable Y is equal in distribution to another X + c for some constant c. In other words, the distribution of Y is the distribution of X shifted. In testing any shift in distribution one needs to test for normality first, and then testing the difference in
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