Question 6 of 6: Acceptance Sampling, MIL-STD-105E Features, and a Single Sampling Plan

Question 6: Acceptance Sampling, MIL-STD-105E Features, and a Single Sampling Plan (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a) Acceptance sampling vs. 100% inspection; when sequential sampling is recommended

Acceptance sampling's advantages are lower inspection cost and time (only a fraction of the lot is examined), reduced handling damage (fewer units are touched, tested, or subjected to destructive testing), applicability to destructive tests (100% inspection is impossible when the test destroys the unit), and less inspector fatigue (which paradoxically often gives sampling better real-world detection performance per unit inspected than a bored 100%-inspection operator). Its disadvantages are sampling risk (a genuinely bad lot can be accepted — consumer's risk — and a genuinely good lot can be rejected — producer's risk), less information about any individual unit's quality, and the administrative overhead of designing, maintaining, and periodically switching among sampling plans.

100% inspection's advantage is, in principle, that every unit is checked — when inspection is perfectly reliable, no defective unit should reach the customer and no good unit should be needlessly rejected. Its disadvantages are high cost and time, inspector fatigue and monotony on large lots (well documented to reduce inspection effectiveness, sometimes badly enough that 100% inspection finds well under 100% of the defects present), the impossibility of applying it to destructive tests, and cumulative handling damage from testing/handling every unit.

Sequential sampling (items inspected one at a time, or in small groups, with accept/reject/continue decision boundaries recomputed after each) is recommended when the cost of inspecting each individual unit is high (so minimizing the average number inspected has real economic value) and lots are typically clearly good or clearly bad (letting the sequential test terminate quickly, often with a much smaller average sample number than a single or double sampling plan of comparable protection) — it is the most sample-size-efficient of the classical plans precisely because it can stop as soon as the evidence is decisive, rather than committing to a fixed sample size up front.

(b) Attributes vs. variables sampling plans; MIL-STD-105E features

An attributes sampling plan classifies each inspected unit simply as conforming or nonconforming (pass/fail) and bases the accept/reject decision on the count of nonconforming units found against an acceptance number $Ac$ — it requires only a go/no-go measurement, works for any characteristic (including purely qualitative ones), and is simple to administer, but needs relatively large sample sizes to give strong protection because each inspected unit yields only one bit of information. A variables sampling plan instead measures the actual numerical value of a continuous quality characteristic on each sampled unit and bases the decision on a statistic computed from those measurements (typically the sample mean and standard deviation, compared against a critical distance to the specification limit) — because a measurement carries far more information than a pass/fail classification, a variables plan gives equivalent protection with a substantially smaller sample size, at the cost of requiring the characteristic to actually be measurable on a continuous scale, more sophisticated statistical assumptions (commonly normality), and a plan and calculation procedure per characteristic rather than one general procedure for many attributes at once.

MIL-STD-105E (the standard this question uses) is a widely adopted attributes acceptance-sampling system with these main features: plans are indexed by lot size and a chosen inspection level (I, II, or III general; S-1–S-4 special) to a sample-size code letter (Table I); each code letter, together with the buyer/producer-agreed Acceptable Quality Level (AQL), indexes into master tables (Table II-A/B/C) giving the sample size and acceptance/rejection numbers ($Ac$/$Re$) for single, double, or multiple sampling; the standard provides normal, tightened, and reduced inspection, with formal switching rules that move a supplier to tightened inspection after a run of rejected lots (raising the bar) and permit reduced inspection after a run of good history (lowering inspection cost when a supplier has demonstrated consistent quality) — a built-in incentive structure that rewards sustained good performance and penalizes deteriorating performance automatically.

(c) Single sampling plan for $N=1{,}500$, $AQL=1\%$, and the risks at $LQL=3\%$

Given. Lot size $N=1{,}500$; required $AQL=1\%$; normal inspection, general inspection level II; consumer's-side quality of interest $LQL=3\%$.

Find. The MIL-STD-105E single sampling plan ($n$, $Ac$, $Re$); the producer's risk $\alpha$ (probability of rejecting a lot at $p=AQL$); and the consumer's risk $\beta$ (probability of accepting a lot at $p=LQL$).

Approach. Look up the sample-size code letter from Table I using the lot-size range and inspection level, read the normal-inspection single sampling plan ($n$, $Ac$, $Re$) for that code letter and the required AQL from Table II-A, then use the Poisson approximation to the binomial (appropriate here since $n$ is a small fraction of $N$ and $p$ is small) to evaluate the OC curve at $p=AQL$ and $p=LQL$.

1. Sample-size code letter (Table I). A lot size of 1,500 falls in the $1{,}201$–$3{,}200$ range; under general inspection level II this indexes to code letter K.
2. Sampling plan (Table II-A, normal inspection). Code letter K carries a sample size of $n=125$; at $AQL=1.0\%$ the master table gives acceptance number $Ac=3$ and rejection number $Re=4$: $$\boxed{n=125,\ \ Ac=3,\ \ Re=4}\qquad(\text{accept the lot if }\leq3\text{ nonconforming units are found in the sample of 125}).$$
3. Producer's risk at $p=AQL=1\%$. Using the Poisson approximation with $\lambda=np=125(0.01)=1.25$: $$P_a(AQL)=\sum_{d=0}^{3}\frac{e^{-1.25}1.25^d}{d!}=0.9617,\qquad \alpha=1-P_a(AQL)=\boxed{0.038\ (3.8\%)}.$$
4. Consumer's risk at $p=LQL=3\%$. With $\lambda=np=125(0.03)=3.75$: $$\beta=P_a(LQL)=\sum_{d=0}^{3}\frac{e^{-3.75}3.75^d}{d!}=\boxed{0.484\ (48.4\%)}.$$