Is the following statement true? A UMP test exists for every testing problem

No. H₀: μ = 0 vs H₁: μ ≠ 0 for N(μ, 1) The MP test for μ > 0 rejects for large X̄, for μ < 0 for small X̄; no single test is best against both.

Counterexample

Is this true?

A UMP test exists for every testing problem

No — it is false.

The counterexample

$H_{0}: \mu = 0$ vs $H_{1}: \mu \ne 0$ for $N(\mu, 1)$

Why it fails

The MP test for $\mu > 0$ rejects for large X̄, for $\mu < 0$ for small X̄; no single test is best against both.

Keep practicing

What the inference means

p-values, size, power and error types say specific things. This was not one of them.

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.