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I'm trying to solve the above exercise, but my question begins with which test should I choose: Tukey test or Fisher's LSD?

I read several things and still can not tell the difference between the two when it is preferable to use each other?

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I would typically default to Tukey. If you're just learning stats do it both ways and see how they differ. Tukey was really designed to allow one to make all of the pairwise comparisons, which you plan to do here. Tukey will be more conservative, but not as conservative as Scheffe. I've also read that one shouldn't do LSD for more than three groups.

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