With known probabilities p and q perform either E or F, with X distributed as N(, 1)

With known probabilities p and q perform either E or F, with X distributed as N(θ, 1) under E or N(−θ, 1) under F. For testing H : θ = 0 against

θ > 0 there exist a UMP unconditional and a UMP conditional level-α test. These coincide and do not depend on the value of p.