Manuscript Psy04-A@1 · Psy04-A-v1.tex · proposition · d5800e2a

Proposition: Prospective power certificate

Finite-sample exclusion certificates › Power and planned sample size · lines 1323–1355 · proof included or linked

Psy04-A@1#prop:power, Psy04-A@1#eq:samplesize

Suppose an alternative has conditional means

μ_{i} \geq c + g, g > 0.

For the fixed-sample rule (eq:rejection), if $r_{N} (δ) < g$ , its failure-to-reject probability is at most

exp [- 2 N (g - r_{N} (δ))^{2}] .

A sufficient sample size for Type-I error at most $δ$ and failure-to-reject probability at most $β$ is

N \geq \frac{( lo g ( 1/ δ ) + lo g ( 1/ β ) ) ^{2}}{2 g ^{2}} .

Proof. If rejection fails,

i \sum (S_{i} - μ_{i}) \leq N (r_{N} - g) .

Apply the lower-tail bound. The stated sample size ensures

g \geq r_{N} (δ) + r_{N} (β) .

Displayed formulas in this passage

unlabeled · line 1326
$μ_{i} \geq c + g, g > 0.$
unlabeled · line 1331
$exp [- 2 N (g - r_{N} (δ))^{2}] .$
Psy04-A@1#eq:samplesize · line 1336
$N \geq \frac{( lo g ( 1/ δ ) + lo g ( 1/ β ) ) ^{2}}{2 g ^{2}} .$
unlabeled · line 1347
$i \sum (S_{i} - μ_{i}) \leq N (r_{N} - g) .$
unlabeled · line 1352
$g \geq r_{N} (δ) + r_{N} (β) .$