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Both tails determine power for the known-variance two-sided normal z-test

Statement

Assume AC. Let $X_{1}, \dots, X_{n}$ be mutually independent with common law $N (μ, σ^{2})$ , where $n \geq 1$ and $σ > 0$ . For the simple null $H_{0} : μ = μ_{0}$ against the two-sided alternative $H_{1} : μ \neq μ_{0}$ , let $Z_{n} = \sqrt{n} ({\overset{ˉ}{X}}_{n} - μ_{0}) / σ$ , let $c_{α} = z_{1 - α / 2}$ for $α \in (0, 1)$ , and reject when $∣ Z_{n} ∣ > c_{α}$ . This test has size $α$ . At $μ = μ_{0} + δ$ , where $δ \in R$ , its rejection probability (the power function evaluated at this parameter) is $1 - Φ (c_{α} - λ) + Φ (- c_{α} - λ), λ = \frac{\sqrt{n} δ}{σ} .$ For $δ \neq 0$ this is power at the stated alternative; for $δ = 0$ it is the null rejection probability. The first term is the upper-tail rejection probability and the second is the lower-tail rejection probability.

Facts & Assumptions

Given: AC; the independent normal sample and parameters in the Statement; the simple null, the two-sided alternative, and the stated symmetric rejection region.

[A1]

Assume AC, as required by the Gaussian-law convention. It also supplies countable choice: apply AC to the family of values of any sequence of nonempty sets and compose the resulting choice function with that sequence. This discharges the countable-choice premise in [F5]. The Axiom of Choice The Axiom of Countable Choice ( ${A C}_{ω}$ ) The Axiom of Choice implies countable choice

[F1]

The standardized sample mean has law $N (λ, 1)$ at $μ = μ_{0} + δ$ , and $N (0, 1)$ at the null; these normal laws are affine shifts of the standard normal. The mean of a finite independent normal sample is normal Standard normal and normal laws

[F2]

$Φ$ has unique quantiles, is strictly increasing, is symmetric about zero, and has no atoms. Standard-normal CDF, tail, and quantile identities for power calculations

[F3]

The sample-mean observation with its parameter-indexed probability laws is a statistical experiment. Statistical experiments

[F4]

A randomized test has a measurable rejection-probability function $φ$ ; here it is the indicator of the two-sided rejection region. Randomized tests

[F5]

The affine statistic is continuous, its two-tail region is Borel, and a measurable rejection function defines a randomized test. Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function Continuous functions on Euclidean spaces are Borel measurable A measurable function between measurable spaces Every measurable rejection function defines a randomized test

[F6]

Size is the null rejection probability for this singleton null, and the power function is the rejection probability under each parameter. Test size and power

Proof

1.1A1F1F2F3F4F5givenalgebra

By [A1], countable choice is available for the Borel-measurability supplier in [F5]. The affine map $T (x) = \sqrt{n} (x - μ_{0}) / σ$ is continuous, so the sample-mean rejection region $R = {x \in R : ∣ T (x) ∣ > c_{α}} = T^{- 1} ((- \infty, - c_{α}) \cup (c_{α}, \infty))$ is Borel. Its indicator $φ = 1_{R}$ is measurable: preimages of Borel sets in $[0, 1]$ are among $\emptyset, R, R^{c}, R$ . By [F5], it defines a randomized test in the experiment [F3], and $φ ({\overset{ˉ}{X}}_{n}) = 1_{{∣ Z_{n} ∣ > c_{α}}}$ . By [F1], under $H_{0}$ the statistic $Z_{n}$ has law $N (0, 1)$ . Since $1 - α / 2 > 1 / 2$ , [F2] gives $c_{α} > 0$ , and the two rejection tails are disjoint.

2.1F2F6step 1.1algebra

Under the null, the probability of the upper tail is $1 - Φ (c_{α}) = α / 2$ , and the probability of the lower tail is $Φ (- c_{α}) = α / 2$ by symmetry and the quantile identity. Their sum is $α$ , proving the test size.

3.1F1F2step 1.1algebra∎

Under $μ = μ_{0} + δ$ , [F1] gives $Z_{n} \sim N (λ, 1)$ . Thus $P (Z_{n} > c_{α}) = 1 - Φ (c_{α} - λ)$ , while $P (Z_{n} < - c_{α}) = Φ (- c_{α} - λ)$ ; the no-atom clause in [F2] justifies the strict lower tail. The two events are disjoint by step 1.1, so their probabilities add to the stated power-function value.

Depends on

Used by

A two-sided normal z-test retains its lower-tail power Example

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Sources

Gary W. Oehlert, A First Course in Design and Analysis of Experiments, Chapter 7, §§7.1–7.4, printed pp. 149–159 (standard reference, not scraped)
NIST/SEMATECH e-Handbook of Statistical Methods, §7.2.2.2, PDF pp. 2019–2021 (standard reference, not scraped)