# Acceptance sampling

Acceptance sampling |
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**Acceptance sampling** is a statistical method used to determine whether a batch of goods or products meets a certain level of quality. It involves taking a random sample of items from the batch and inspecting them to determine if the batch as a whole is acceptable or not. The process is used in industries such as manufacturing, construction, and food production to ensure that the products produced meet certain standards of quality and safety. It is also used in quality control to evaluate the performance of suppliers.

## Usage of acceptance sampling

Acceptance sampling is typically used when there is a need to determine the quality of a batch of goods or products, and it is not feasible or cost-effective to inspect every item in the batch. It is also used when there is a need to evaluate the performance of suppliers or to ensure that products meet certain standards of quality and safety.

Acceptance sampling is commonly used in industries such as manufacturing, construction, food production, and healthcare. For example, in manufacturing, acceptance sampling is used to ensure that products meet certain specifications, such as size, weight, and color. In construction, acceptance sampling is used to ensure that materials, such as concrete and steel, meet certain strength and durability requirements. In food production, acceptance sampling is used to ensure that products meet certain safety and quality standards, such as being free of contaminants.

Acceptance sampling can be especially useful in situations where the cost of inspecting every item in a batch would be prohibitively high, or where there is a high degree of variability in the quality of the products being produced.

## Acceptance sampling plan

An acceptance sampling plan is a set of guidelines and procedures that specify how a random sample of items should be selected and inspected from a batch of goods or products. It includes details such as the sample size, the acceptance and rejection criteria, and the procedures for handling nonconforming items.

There are several types of acceptance sampling plans, such as:

**Single sampling plan**: A single sample of items is taken and inspected, and based on the results of the inspection, the entire batch is either accepted or rejected.**Double sampling plan**: A first sample of items is taken and inspected, and if it fails to meet the acceptance criteria, a second sample is taken and inspected. If the second sample passes, the batch is accepted, and if it fails, the batch is rejected.**Sequential sampling plan**: A sample of items is inspected and the inspection continues until a certain number of nonconformities are found or a certain level of confidence is reached.

The choice of acceptance sampling plan will depend on the characteristics of the product, the cost of inspection, and the desired level of quality.

## Acceptance sampling formulas

There are several different formulas used in acceptance sampling, depending on the type of sampling plan being used. Some of the most common formulas used include:

**Single sampling plan**: The formula used to determine the sample size for a single sampling plan is:- n = (z^2 * p * (1-p)) / e^2
- Where "n" is the sample size, "z" is the standard normal deviate (related to the level of confidence desired), "p" is the proportion of nonconforming items in the population, and "e" is the allowable error or tolerance.

**Double sampling plan**: The formula used to determine the sample size for a double sampling plan is:- n = (z^2 * p * (1-p)) / e^2 + z^2
- Where "n" is the sample size, "z" is the standard normal deviate, "p" is the proportion of nonconforming items in the population, and "e" is the allowable error or tolerance.

**Acceptable quality level (AQL) sampling**: The formula used to determine the sample size for AQL sampling is:- n = (c^2 * L * p) / (e^2 * (C-p))
- Where "n" is the sample size, "c" is the number of nonconformities that can be allowed, "L" is the lot size, "p" is the proportion of nonconforming items in the population, "e" is the allowable error or tolerance, and "C" is the acceptance number of nonconformities.

**Operating characteristic (OC) curve**: The formula used to plot the OC curve is:- P(Accept|p) = P(X<=c|p) = Σ(i=0 to c) [nCi * p^i * (1-p)^(n-i)],
- Where n is sample size and c is the acceptance number of nonconformities.

It's important to note that these are just a few examples, and the specific formula used will depend on the sampling plan and the acceptance criteria being used.

## Acceptance sampling limitations

Acceptance sampling has several limitations that should be considered when deciding whether or not to use it:

**Sampling error**: Acceptance sampling is based on the assumption that a random sample is representative of the entire batch. However, there is always a chance that the sample selected is not representative of the batch, leading to incorrect conclusions about the quality of the batch.**False acceptance**: If the acceptance criteria are not set correctly, it is possible for a batch of nonconforming items to be accepted.**False rejection**: If the acceptance criteria are set too strictly, it is possible for a batch of conforming items to be rejected.**Limited feedback**: Acceptance sampling only provides information on whether or not a batch meets certain acceptance criteria. It does not provide information on how to improve the quality of the batch.**Limited to batch inspection**: Acceptance sampling is typically used for batch inspection, which means that it can only be used when products are produced in batches. It is not suitable for inspecting individual items.- Doesn't provide information about the entire population: Acceptance sampling only provides information about a sample of items, not the entire population of items. Therefore, it may not be able to detect certain issues that only occur in small or specific parts of the population.

It's important to note that acceptance sampling should be used in conjunction with other quality control methods to ensure that a comprehensive quality control program is in place.

## References

- Wetherill, G. B., & Chiu, W. K. (1975). A review of acceptance sampling schemes with emphasis on the economic aspect. International Statistical Review/Revue Internationale de Statistique, 191-210.
- Rosaiah, K., & Kantam, R. R. L. (2005). Acceptance sampling based on the inverse Rayleigh distribution.
- Schilling, E. G., & Neubauer, D. V. (2009). Acceptance sampling in quality control. Chapman and Hall/CRC.