One mistake is treating 95% as the chance of the study indicating two-tailed coins, given that they were two-tailed coins. More likely it was meant as the chance of the study not indicating two-tailed coins, given that they were not two-tailed coins.

Try this:

You want to test if a coin is biased towards heads. You flip it 5 times, and consider 5 heads as a positive result, 4 heads or fewer as negative. You're aiming for 95% confidence but have to get 31/32 = 96.875%. Treating 4 heads as a possible result wouldn't work either, as that would get you less than 95% confidence.