Let X1,...,Xn be a sample from N(, 2). The power of Students t-test is an increasing function

Let X1,...,Xn be a sample from N(ξ, σ2). The power of Student’s t-test is an increasing function of ξ/σ in the one-sided case H : ξ ≤ 0, K : ξ > 0, and of |ξ|/σ in the two-sided case H : ξ = 0, K : ξ = 0.

[If S =

1 1 n − 1

(Xi − X¯)2, the power in the two-sided case is given by 1 − P



−CS

σ −

√nξ

σ ≤

√n(X¯ − ξ)

σ ≤ CS

σ −

√nξ

σ



and the result follows from the fact that it holds conditionally for each fixed value of S/σ.]