Hypothesis testing: Parametric test

It is understood that a parametric test is that lies on assumption of population parameters whereas non-parametric test is where few parameter assumptions are required (e.g. ranked observations). But still am unclear on its application. Help on this.

So I am testing whether EPS have the same mean in US and Australian companies (this is as brain dead as most CFAI examples) so I sample 20 companies randomly from each country. Now I can go a few ways: a) Assume the EPS is normally distributed in Australia and also in US. Do parametric test. b) Rank EPS observations and do a non-parametric test (this would be Kruskal-Wallis, I guess). In a) if my assumption is right, I’ve got a much better test than b) because I have imposed all kinds of structure on the problem. My test is more pwerful and is much more likely to show a difference if there is one. However, if my assumption is wrong my test is of unknown quality and it’s power depends in some vague sense about how wrong I am. The power of the test in b) is much less dependent on the distribution of the observations.

Great. So doing any of the two ways will get me to the same answer? (Provided that in a) your assumpion is right)

No - that’s the point. If your assumption is right, a) is more likely to get you to the right answer. If the assumption is wrong, b) might be more likely to get you to the correct answer but it is not clear.