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**Abstract**

This practice presents methodology for the setting of an upper confidence bound regarding an unknown fraction or quantity non-conforming, or a rate of occurrence for nonconformities, in cases where the method of attributes is used and there is a zero response in a sample. Three cases are considered. In Case 1, the sample is selected from a process or a very large population of interest. In Case 2, a sample of n items is selected at random from a finite lot of N items. In Case 3, there is a process, but the output is a continuum, such as area (for example, a roll of paper or other material, a field of crop), volume (for example, a volume of liquid or gas), or time (for example, hours, days, quarterly, etc.) The sample size is defined as that portion of the “continuum” sampled, and the defined attribute may occur any number of times over the sampled portion.

In this practice, allowance is made for misclassification error but only when misclassification rates are well understood or known and can be approximated numerically. It is possible to impose the language of classical acceptance sampling theory on this method. Terms such as lot tolerance percent defective, acceptable quality level, and consumer quality level are not used in this practice.

*This abstract is a brief summary of the referenced standard. It is informational only and not an official part of the standard; the full text of the standard itself must be referred to for its use and application. ASTM does not give any warranty express or implied or make any representation that the contents of this abstract are accurate, complete or up to date. *

**Significance and Use**

4.1 In Case 1, the sample is selected from a process or a very large population of interest. The population is essentially unlimited, and each item either has or has not the defined attribute. The population (process) has an unknown fraction of items p (long run average process non-conforming) having the attribute. The sample is a group of n discrete items selected at random from the process or population under consideration, and the attribute is not exhibited in the sample. The objective is to determine an upper confidence bound, p_{u}, for the unknown fraction p whereby one can claim that p ≤ p_{u} with some confidence coefficient (probability) C. The binomial distribution is the sampling distribution in this case.

4.2 In Case 2, a sample of n items is selected at random from a finite lot of N items. Like Case 1, each item either has or has not the defined attribute, and the population has an unknown number, D, of items having the attribute. The sample does not exhibit the attribute. The objective is to determine an upper confidence bound, D_{u}, for the unknown number D, whereby one can claim that D ≤ D_{u} with some confidence coefficient (probability) C. The hypergeometric distribution is the sampling distribution in this case.

4.3 In Case 3, there is a process, but the output is a continuum, such as area (for example, a roll of paper or other material, a field of crop), volume (for example, a volume of liquid or gas), or time (for example, hours, days, quarterly, etc.) The sample size is defined as that portion of the “continuum” sampled, and the defined attribute may occur any number of times over the sampled portion. There is an unknown average rate of occurrence, λ, for the defined attribute over the sampled interval of the continuum that is of interest. The sample does not exhibit the attribute. For a roll of paper, this might be blemishes per 100 ft^{2}; for a volume of liquid, microbes per cubic litre; for a field of crop, spores per acre; for a time interval, calls per hour, customers per day or accidents per quarter. The rate, λ, is proportional to the size of the interval of interest. Thus, if λ = 12 blemishes per 100 ft^{2} of paper, this is equivalent to 1.2 blemishes per 10 ft^{2} or 30 blemishes per 250 ft^{2}. It is important to keep in mind the size of the interval in the analysis and interpretation. The objective is to determine an upper confidence bound, λ_{u}, for the unknown occurrence rate λ, whereby one can claim that λ ≤ λ_{u} with some confidence coefficient (probability) C. The Poisson distribution is the sampling distribution in this case.

4.4 A variation on Case 3 is the situation where the sampled “interval” is really a group of discrete items, and the defined attribute may occur any number of times within an item. This might be the case where the continuum is a process producing discrete items such as metal parts, and the attribute is defined as a scratch. Any number of scratches could occur on any single item. In such a case, the occurrence rate, λ, might be defined as scratches per 1000 parts or some similar metric.

4.5 In each case, a sample of items or a portion of a continuum is examined for the presence of a defined attribute, and the attribute is not observed (that is, a zero response). The objective is to determine an upper confidence bound for either an unknown proportion, p (Case 1), an unknown quantity, D (Case 2), or an unknown rate of occurrence, λ (Case 3). In this practice, confidence means the probability that the unknown parameter is not more than the upper bound. More generally, these methods determine a relationship among sample size, confidence and the upper confidence bound. They can be used to determine the sample size required to demonstrate a specific p, D, or λ with some degree of confidence. They can also be used to determine the degree of confidence achieved in demonstrating a specified p, D, or λ.

4.6 In this practice, allowance is made for misclassification error but only when misclassification rates are well understood or known, and can be approximated numerically.

4.7 It is possible to impose the language of classical acceptance sampling theory on this method. Terms such as lot tolerance percent defective, acceptable quality level, and consumer quality level are not used in this practice. For more information on these terms, see Practice E1994.

**1. Scope**

1.1 This practice presents methodology for the setting of an upper confidence bound regarding a unknown fraction or quantity non-conforming, or a rate of occurrence for nonconformities, in cases where the method of attributes is used and there is a zero response in a sample. Three cases are considered.

1.1.1 The sample is selected from a process or a very large population of discrete items, and the number of non-conforming items in the sample is zero.

1.1.2 A sample of items is selected at random from a finite lot of discrete items, and the number of non-conforming items in the sample is zero.

1.1.3 The sample is a portion of a continuum (time, space, volume, area, etc.) and the number of non-conformities in the sample is zero.

1.2 Allowance is made for misclassification error in this practice, but only when misclassification rates are well understood or known and can be approximated numerically.

1.3 The values stated in inch-pound units are to be regarded as standard. No other units of measurement are included in this standard.

1.4 *This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.*

**2. Referenced Documents** *(purchase separately)* The documents listed below are referenced within the subject standard but are not provided as part of the standard.

**ISO Standards**

**ASTM Standards**

E141 Practice for Acceptance of Evidence Based on the Results of Probability Sampling

E456 Terminology Relating to Quality and Statistics

E1402 Guide for Sampling Design

E1994 Practice for Use of Process Oriented AOQL and LTPD Sampling Plans

E2586 Practice for Calculating and Using Basic Statistics

**ICS Code**

ICS Number Code 03.120.30 (Application of statistical methods)

**UNSPSC Code**

UNSPSC Code 81130000(Statistics)

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**DOI:** 10.1520/E2334-09R18

**Citation Format**

ASTM E2334-09(2018), Standard Practice for Setting an Upper Confidence Bound for a Fraction or Number of Non-Conforming items, or a Rate of Occurrence for Non-Conformities, Using Attribute Data, When There is a Zero Response in the Sample, ASTM International, West Conshohocken, PA, 2018, www.astm.org

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