Equivalence and Noninferiority Tests for Quality, Manufacturing and Test Engineers

179

Appendix II

Theorem

For any interval on the real line, say, A, and a critical region χ , such that:

sup Pr {x ∈ χ|γ = inf A} = 1 − β

there is always an interval B, such that

inf A > sup B

and

sup Pr {x ∈ χ|δ = sup B} = α < 1 − β.

Proof: Let

f (θ) = Pr {x ∈ χ|θ}

be a continuous, monotonically decreasing function of θ. Then

f (γ) = Pr {x ∈ χ|γ} = 1 − β

and there is a value δ < γ such that

f (δ) = Pr {x ∈ χ|δ} = α < 1 − β

by continuity and monotonicity. Let A be any interval such that γ = inf A.

Since δ < γ, dene the interval B such that δ = sup B.