Sampling Plan for a Battery Assembly Line
A plant assembles 2,000 battery packs per day. Historical defect rate is 2%, but a new supplier has raised concern that the rate may be 4%. The quality team can either test 40 randomly selected packs per day with a full electrical test, or use a two-stage plan: test 20 randomly selected packs; if 0 or 1 is defective, accept the day's lot without further testing; if 2 or more are defective, stop the line and test all 2,000 packs. Each full test costs $8; stopping the line and testing all packs costs an additional $12,000 in lost production. The team wants to know how often the two-stage plan would accept a lot that actually has a 4% defect rate, and whether the savings justify the risk. Assume defects are independent and the sample is random. Use the binomial distribution; exact values are not required, but show the calculation setup and approximate probabilities.
1. Under the two-stage plan, what is the approximate probability of accepting a lot with a 4% defect rate? Show the binomial terms.
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Let X be the number of defectives in the 20-pack sample; X ~ Binomial(n=20, p=0.04). The plan accepts if X ≤ 1. P(X=0) = 0.96^20 ≈ 0.442. P(X=1) = C(20,1)(0.04)(0.96^19) ≈ 20 × 0.04 × 0.460 ≈ 0.368. P(accept) ≈ 0.442 + 0.368 = 0.810, or about 81%. So the two-stage plan would pass about 81% of lots that truly have a 4% defect rate.
2. Compare that with testing 40 packs and accepting the lot only if 0 or 1 defect is found. What is the approximate acceptance probability at 4% defects?
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With 40 packs and acceptance if X ≤ 1: P(X=0) = 0.96^40 ≈ 0.195. P(X=1) = 40 × 0.04 × 0.96^39 ≈ 1.6 × 0.203 ≈ 0.325. P(accept) ≈ 0.195 + 0.325 = 0.520, or about 52%. The 40-pack plan is substantially more discriminating: it cuts acceptance of 4%-defect lots from about 81% to about 52%.
3. Which plan should the team choose? Include a cost-based argument and one limitation.
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Choose the full 40-pack test. At the historical 2% defect rate, the two-stage plan rejects only about 6% of the time (P(X≥2) with n=20, p=0.02 ≈ 1 − 0.940 = 0.060). Its expected daily cost is 20 × $8 + 0.060 × ($16,000 + $12,000) ≈ $160 + $1,680 = $1,840. The 40-pack test costs 40 × $8 = $320 per day. Even before counting the cost of defective packs reaching customers, the two-stage plan is far more expensive in expectation and much less protective. Limitation: the 40-pack plan still accepts about 52% of 4%-defect lots, so if the supplier risk is serious, the team should tighten the acceptance rule (for example, accept only at 0 defects) or increase the sample; a sample of 40 cannot reliably distinguish 2% from 4% with a one-defect threshold.