As the effect size increases the power of a hypothesis test also increases, but what happens to alpha?
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The effect size does not change $\alpha$. The significance level $\alpha$ is determined before; usually $\alpha = 0.05$ is chosen. The significance level is the probability of rejecting a true null hypothesis. This does not depend on the effect size, as the effect size says something about what happens when the null hypothesis is not true.
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