Type I error means you claimed an effect that is not there. Type II error means you missed an effect that is really there. The significance level sets your Type I rate, and power controls the Type II rate.
Costs differ by context, and that asymmetry should drive your threshold. Shipping a redesign that does nothing burns engineering time and pollutes future baselines. Killing a feature that actually worked costs revenue forever, quietly, because nobody measures the road not taken.
A medical screening test makes it concrete. A false positive sends a healthy person for an expensive biopsy. A false negative sends a sick person home untreated. With a fixed sample you cannot minimize both at once, since tightening one loosens the other. The only honest way to shrink both is to collect more data.
Rewriting in plainer words…
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