Lately, I've been reading a lot about power calculation, effect size and sample size of an experiment, but recently this question arose in my head and I do not know what to do of it and would really like your thoughts about it.
Suppose I have a simple two-sample t-test I'd like to carry out and I want to detect a big effect, say Cohen's d = 0.8, and the minimum sample size in each group is 26 (alpha=0.05,beta=0.2 and using R's pwr package), so I use 52 experimental subjects in total as prescribed. By doing so, I can be sure that I have enough statistical power as to detect a difference in means that is "big" (Cohen's d >= 0.8).
The question then is: what should I think about getting a significant result (p < 0.05) in a difference in means smaller than d=0.8 ? How should I/would you interpret this finding and report/discuss it in a paper?