- 1. Statistics for goods income tests with zero response Zero response If n tested items of the LOT show a zero response, i.e. zero non-conforming items are found, a probability exists that non-conforming items are still in the LOT. For n/M ≤ 0.05, i.e. large LOT size compared to the sample size, the binomial distribution can be used to calculate the upper limit of non-conforming items for a given conﬁdence level C: [1] P(k) = n k pk (1 − p)n−k (1) with n the sample size, k the number of non-conforming items and p the fraction of non-conforming items in a LOT. For k = 0: P(0) = (1 − p)n . (2) 0.5 0.6 0.7 0.8 0.9 1.0 0.00.10.20.30.4 n = 11 Confidence level Fractionofnon−conformingitems Figure 1: Conﬁdence level versus upper fraction of non-conforming items for n = 11. Title: Zero response Revision: 1 Effective: 2015-10-25 Page 1 of 3 ID-Number: Author: Dr. Peter Drechsler Developed: 2015-10-25 Phone: +49 711 46 92 73 40 Mobile: +49 173 2462720 peter.drechsler@qtec-group.com qtec group R qtec services GmbH Humboldtstr. 30 70771 Leinfelden-Echterdingen Germany
- 2. Statistics for goods income tests with zero response Table 1: Maximal fraction of non-conforming items in a LOT in percent as a function of samples n taken and zero non-compliance items k are found for conﬁdence intervals between 75% and 99%. Samples Conﬁdence limit n 0.75 0.90 0.95 0.975 0.99 1 75.00 90.00 95.00 97.50 99.00 3 37.00 53.58 63.16 70.76 78.46 5 24.21 36.90 45.07 52.18 60.19 7 17.97 28.03 34.82 40.96 48.21 9 14.28 22.57 28.31 33.63 40.05 11 11.84 18.89 23.84 28.49 34.21 17 7.83 12.67 16.16 19.51 23.73 21 6.39 10.38 13.29 16.11 19.69 25 5.39 8.80 11.29 13.72 16.82 50 2.73 4.50 5.82 7.11 8.80 100 1.38 2.28 2.95 3.62 4.50 150 0.92 1.52 1.98 2.43 3.02 200 0.69 1.14 1.49 1.83 2.28 250 0.55 0.92 1.19 1.46 1.83 300 0.46 0.76 0.99 1.22 1.52 400 0.35 0.57 0.75 0.92 1.14 500 0.28 0.46 0.60 0.74 0.92 1000 0.14 0.23 0.30 0.37 0.46 and a upper fraction of non-conforming items 1 − C. This leads to Cn + C − 1 = 0. (3) Solved for n: n = ln(1 − C) ln(C) . (4) One can ask for the largest p the makes P(0) reasonably small. Therefore, we set P(0) = 1 − C, where C is the conﬁdence level for the upper limit of non-conforming items pu: 1 − C = (1 − pu)n (5) or pu = 1 − n √ 1 − C (6) Title: Zero response Revision: 1 Effective: 2015-10-25 Page 2 of 3 ID-Number: Author: Dr. Peter Drechsler Developed: 2015-10-25 Phone: +49 711 46 92 73 40 Mobile: +49 173 2462720 peter.drechsler@qtec-group.com qtec group R qtec services GmbH Humboldtstr. 30 70771 Leinfelden-Echterdingen Germany
- 3. Statistics for goods income tests with zero response Table 2: Sample size n for a given conﬁdence level C. C/% pu n 99.99 0.0001 92099 99.90 0.0010 6904 99.5 0.0050 1057 99 0.0100 458 97 0.0300 115 95 0.0500 58 93 0.0700 37 90 0.1000 22 70 0.3000 4 50 0.5000 1 Sometimes it’s very convienient to calculate the sample size n for a given conﬁdence level C References [1] ASTM E 2334-09. Standard practice for setting an upper conﬁdence bound for a fraction or number of non-conforming items, or a rate of occurence for non-conformities, using attribute data, when there is a zero response in the sample, 2009. Title: Zero response Revision: 1 Effective: 2015-10-25 Page 3 of 3 ID-Number: Author: Dr. Peter Drechsler Developed: 2015-10-25 Phone: +49 711 46 92 73 40 Mobile: +49 173 2462720 peter.drechsler@qtec-group.com qtec group R qtec services GmbH Humboldtstr. 30 70771 Leinfelden-Echterdingen Germany