Pläne für messende Prüfung bei Berücksichtigung von „Vorkenntnissen” über die Verteilung der Mittelwerte

Summary

Bayes-sampling-plans for inspection by variables are calculated under the following assumptions:

• (1) A single unit of an inspected lot is characterized by a measured valuex. Within the lotx is normally distributed with an unknown conditional meanE{x/μ}=μ and the known (conditional) variance V{x|μ}=σ² not dependent on μ.

• (2) Between the lots the mean values μ are normally distributed with E{μ}=μ₀ and V|μ}=σ ²₀ (both values known; prior distribution of μ).

• (3) Lots with μ≤a should be accepted, lots with μ>a should be rejected.

• (4) The conditional probabilities of rejecting a good lot and accepting a bad one are given values (α;β).

• (5) A sample ofn units is inspected giving the mean value$\bar x$. For

  

Introducing new variables and expanding the solution as a power series of$\lambda /\sqrt {n[\lambda = \sigma /\sigma _0 }$ the ratio of the standard deviations mentioned above is generally λ≤1] one gets rather simple equations which by iteration may easily be solved for the pair of values (n; b) characterising the inspection plan. The “asymptotic solution” for largen which is valid for n>10λ² has extremely simple properties which are given in detail.