Spot the fallacy!
- The power of a test is the probability of correctly rejecting the null hypothesis. Write it as 1 – β.
- So, the probability of incorrectly rejecting the null hypothesis is β.
- But the probability of incorrectly rejecting the null is α (the type 1 error probability).
So α = β.
I’ve actually seen this, and variants on it [i].
 Although they didn’t go so far as to reach the final, shocking, deduction.