A problem I've been thinking of and want to make sure I'm on the right track. It's somewhat similar to this question I think: Binomial Probability Question. But I was curious if I could approach it from a different direction.
Say I'm picking between two basketball players for my team. I want to evaluate them based only on their make percentage from $n$ free throws taken during the tryout. How many shots should I have them take to have some level of certainty in my selection of who is the better shooter?
My initial thought is that I could answer this via power analysis. If I assume a probability of Type I error, $\alpha=0.05$, probability of Type II error, $\beta=0.2$, and assume that make% of player 1 $P_1 = 0.5$ and make% of player 2 $P_2= 0.6$, then what does my sample size $n$ need to be?
I plug this information into G*Power (Proportions: Inequality, two independent groups (Fisher's exact test), two-tailed) and get a sample size of $n=404$ shots.
Is this a valid way to approach this problem? Can you think of a better or more generalizable way?