The probability of Type II error ( β) tends to 0.
The standard practice is to set up your test so that the probability of Type I error ( α ) is 5%. If you follow this practice while increasing your sample size, then α naturally stays at 5%.
This says that if you follow the standard practice, you must care more and more about Type II error versus Type I error as your sample size grows — you have to be really afraid of a Type II error if you prefer having β=0.00000001% and α=5% over something more balanced.
The idea that you should care more about Type II error as the sample size increases seems pretty absurd to me, so this is a good argument against the standard practice of putting a bound on α and minimizing β subject to that.
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