Power paradox: overestimated effect size in low-powered study, but the estimator is unbiased If we have an underpowered study but manage to reject the null hypothesis, anyway, it makes sense to wonder if we have overestimated the effect size.
However, such a concern seems unwarranted if we use an unbiased estimator of the effect size (such as $\bar X_1-\bar X_2$ to estimate $\mu_1-\mu_2$).
At the same time, there is an appeal to speculating that we must have lucked into a particularly large observed effect to have managed to reject despite the low power.
This seems like a paradox. Does it have a resolution?

Possibly the following image might shed some light

Given that the null hypothesis is true, there will always be an $\alpha\%$ chance to reject the null hypothesis, no matter what the power of a test is*.
But the power of the test makes the overall picture a lot more different. Possibly the paradox stems from gazing too much exclusively at the null hypothesis and p-value.

*Or actually the percentage to reject might be a bit higher because the hypothesis test is based on a theoretical model for the error and the reality might be different (sampling errors like outliers or correlation between measurements).
