Suppose that you wish to test H 0 : = 0 versus H 1 :

Suppose that you wish to test H₀: μ = μ₀ versus H₁: μ ≠ μ₀, where the population is normal with known σ. Let 0 < ε < α, and define the critical region so that you will reject H₀ if z₀ > z_ε or if z₀ < −z_α−ε where z₀ is the value of the usual test statistic for these hypotheses.

(a) Show that the probability of type I error for this test is α.

(b) Suppose that the true mean is μ₁ = μ₀ + Δ. Derive an expression for β for the above test.