ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Known-blank normal limits at five-percent error and ten-percent CV

Example

Assume AC and $σ_{0} > 0$ . In the known-blank model $X \sim N (c, σ_{0}^{2})$ , take $α = β = 0.05$ and $k_{Q} = 0.10$ . Then $L_{C} = z_{0.95} σ_{0}, L_{D} = 2 z_{0.95} σ_{0}, L_{Q} = 10 σ_{0} .$ The NIST SP 958 table reports the corresponding known-blank values rounded to $1.64 σ_{0}$ , $3.29 σ_{0}$ , and $10 σ_{0}$ . These are conditional on the equal-known-variance normal law and the stated error and CV targets.

Facts & Assumptions

Given: AC; $σ_{0} > 0$ ; the known-blank experiment and definitions in Exact limits in the known-blank equal-variance normal model and Critical, detection, and quantification levels, with $α = β = 0.05$ and $k_{Q} = 0.10$ .

[A1]

AC is assumed by the normal-limit theorem and the standard normal quantile lemma. The Axiom of Choice Exact limits in the known-blank equal-variance normal model The standard normal CDF has unique symmetric quantiles

[F1]

Under the known-blank equal-variance normal model, $L_{C} = z_{1 - α} σ_{0}$ , $L_{D} = (z_{1 - α} + z_{1 - β}) σ_{0}$ , and $L_{Q} = σ_{0} / k_{Q}$ . Exact limits in the known-blank equal-variance normal model

[F2]

The standard normal $q$ -quantile is the unique $z_{q}$ with $Φ (z_{q}) = q$ , and $z_{1 - q} = - z_{q}$ . The standard normal CDF has unique symmetric quantiles

Verification

Given: AC; $σ_{0} > 0$ ; $α = β = 0.05 = 1 / 20$ ; $k_{Q} = 0.10 = 1 / 10$ ; the known-blank model; and the suppliers [F1]–[F2].

1.1A1F1F2algebra

Since $1 - α = 1 - β = 0.95$ , [A1] and [F1] give $L_{C} = z_{0.95} σ_{0}$ and $L_{D} = (z_{0.95} + z_{0.95}) σ_{0} = 2 z_{0.95} σ_{0}$ , with $z_{0.95}$ the unique quantile from [F2].

2.1F1step 1.1algebra∎

By [F1], the stated CV criterion gives $L_{Q} = σ_{0} / k_{Q} = σ_{0} / (1 / 10) = 10 σ_{0}$ . Together with step 1.1, this gives all three exact limits.

Source notes

The exact expressions follow from [F1]. The NIST table reports the known-blank row as $1.64$ , $3.29$ , and $10$ times the blank standard deviation for the same targets; those table entries are rounded values, not exact replacements for $z_{0.95}$ .

Depends on

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Sources

A Century of Excellence in Measurements, Standards, and Technology: A Chronicle of Selected NBS/NIST Publications 1901–2000, NIST SP 958, pp. 177–179 (printed pp. 164–166) (standard reference, not scraped)
H. Cantwell (ed.), Eurachem Guide: The Fitness for Purpose of Analytical Methods, 3rd ed. (2025), Annex B (standard reference, not scraped)