CSP passes but evenness and IPI fail: a sampling and acceptance decision
Buyer specification for a 40s combed yarn is CSP at least 2,400, U% at most 12.0, IPI at most 200 per km and count CV at most 2.5%. The mill tests 20 lea samples drawn from 10 of 60 bales. Results: mean CSP 2,465 with SD 85; mean U% 12.4 with SD 0.5; mean IPI 235 with SD 30; mean count 39.6 Ne with CV 2.8%. The mill QC manager argues that the mean CSP is above the specification, so the lot should pass. The buyer rejects the lot. The shipment is waiting, and re-testing will take two days. The contract says the lot is judged on the mean of a random sample with a 95% lower confidence limit for CSP. The mill has a calibrated lea tester and a capacitive evenness tester, but no Classimat. The buyer's inspector selected bales from the outer rows of the godown; the mill's samples came from the first 10 bales opened.
1. Compute the one-sided 95% lower confidence limit for mean CSP and decide whether CSP meets the contract. Explain the difference between a mean result and an individual test result.
Read the worked answer
Standard error of the mean = SD / √n = 85 / √20 = 85 / 4.472 = 19.0 g.
For n = 20, df = 19, the one-sided 95% t value is 1.729. Margin = 1.729 × 19.0 = 32.9 g.
Lower confidence limit = 2,465 − 32.9 = 2,432 g, which is above the contractual minimum of 2,400. On CSP alone, the lot meets the contract.
A mean result is an estimate of the lot average, not a guarantee that every lea is above 2,400. Individual lea results will scatter around 2,465 with an SD of 85; some leas can be below 2,400 even when the mean passes. The contract is written on the mean with a confidence limit, so the correct test is the lower confidence limit, not a single low lea. Equally, the QC manager cannot use the CSP pass to ignore the other three specifications.
2. Evaluate the U%, IPI and count CV results. What is the statistical error in the QC manager's argument, and can the lot be accepted?
Read the worked answer
U% mean = 12.4 against a maximum of 12.0. The standard error is 0.5/√20 = 0.112, so 12.4 is about 3.6 standard errors above the limit. This is not sampling noise; the lot is genuinely above the evenness limit.
IPI mean = 235 against a maximum of 200. Standard error = 30/√20 = 6.7, so the mean is about 5.2 standard errors above the limit. Count CV = 2.8% against 2.5%; even without a stated SD, the mean count is already off and the CV is outside specification.
The QC manager's error is selective acceptance: he tests four specifications but argues from the one that passes. A lot must satisfy all agreed specifications. A high CSP with poor U% and high IPI is consistent with a yarn that has good average strength but many weak, thick and thin places; such yarn is risky in knitting and weaving even though the lea strength looks good. The lot cannot be accepted as first quality on these results.
3. Calculate the number of lea samples needed to estimate CSP within ±25 g at 95% confidence. What sampling change would you insist on before re-testing, and what commercial disposition do you recommend?
Read the worked answer
Use n = (z × SD / E)² = (1.96 × 85 / 25)² = (6.664)² = 44.4, so 45 lea samples are needed for a ±25 g estimate at 95% confidence.
Sampling change: insist on a random sample across all 60 bales, not the first 10 bales opened and not the outer rows only. The outer bales may have different moisture and handling history. Draw bales from different layers and positions, condition them, and record the bale numbers. For evenness and IPI, test at least five randomly selected bobbins from different bales and positions.
Commercial disposition: do not ship as first quality merely because CSP passed. Re-test with 45 lea samples for CSP and a proper evenness sample. If U%, IPI and count CV remain outside specification, reject or offer the lot as second quality with a price concession, or divert it to a fabric where evenness and IPI are less critical. The two-day re-test is cheaper than a customer claim or a returned shipment.