Question

Design an acceptance sampling plan for which AQL is \(0.05, \mathrm{RQL}\) is \(0.10, \alpha\) is \(0.05\), and \(\beta\) is \(0.05\).

Short answer

This problem requires balancing different parameters. Use a Single Sampling Plan method to get a balance between type I and type II errors, and satisfy the conditions set by the AQL and RQL with the calculated sample size and acceptance number.

Step-by-step solution

Define Resources

From the problem statement, we know the following parameters: AQL=0.05, RQL=0.10, \(\alpha\)=0.05, and \(\beta\)=0.05.

Decide on the Sampling Method

In order to reduce both type I and II errors, and also keep in mind of AQL and RQL, we decide to use a Single Sampling Plan. In a Single Sampling Plan, a sample of items is inspected. If the number of defectives is less than a certain value, the lot is accepted; otherwise, it is rejected.

Calculate Sample Size and Acceptance Number

Calculate the sample size (n) and the acceptance number (c) using appropriate statistical methods or tables. They should satisfy the following conditions simultaneously: The probability of accepting a lot at the AQL should be 1- \(\alpha\), the probability of rejecting a lot at the RQL should be 1-\(\beta\).

Key concepts

AQL (Acceptable Quality Level)

In the domain of quality control, the Acceptable Quality Level (AQL) is a critical threshold that represents the maximum tolerable defect rate in a batch of products. It's the worst quality level that is considered acceptable for the proportion of defective items in a lot. When an AQL of 0.05 is specified, as in the exercise, it means that the customer is willing to accept a lot where up to 5% of items are defective.

During an acceptance sampling inspection, if the defective rate is at or below the AQL, the batch is likely to be accepted. The perspective from a manufacturer's end is that they strive to maintain a quality level at or below the AQL to keep rejection rates low and to satisfy customer quality expectations. It's a delicate balance that ensures that quality is maintained without overburdening resources with over-stringent requirements.

RQL (Rejectable Quality Level)

Conversely, the Rejectable Quality Level (RQL), or Lot Tolerance Percent Defective (LTPD), is the threshold at which a lot's quality is deemed unsatisfactory and hence should be rejected. Setting an RQL of 0.10 means that a lot with a defect rate higher than 10% significantly increases the risk of rejection during quality control.

The choice of RQL reflects the level of quality that customers find unacceptable, leading to a rejection decision. It's a safeguard that triggers an action to either improve production processes or to perform 100% inspection and repair or discard defects, which could be an economically more impactful scenario for manufacturers. Understanding RQL is essential for businesses to set limits that prevent the shipment of low-quality goods while also managing the risks of over inspection and waste.

Type I and II Errors

In acceptance sampling, Type I and II errors represent the risk of incorrect decisions made based on sample inspection. A Type I error, represented by \( \alpha \) in the exercise, occurs when a good lot that meets the quality standards (AQL) is incorrectly rejected, commonly referred to as a producer's risk. In our given problem, a 5% \( \alpha \) signifies that there's a 5% chance of rejecting a lot that should have been accepted.

On the other hand, a Type II error, signified by \( \beta \) in the problem, happens when a bad lot that does not meet quality standards (RQL) is mistakenly accepted, known as a consumer's risk. A \( \beta \) of 5% indicates there’s a 5% risk of accepting a lot that should have been rejected. Minimizing these errors is crucial for both the protection of the consumer and the credibility of the producer. A proper sampling plan aims to strike a balance between these errors while being economically justifiable.

Single Sampling Plan

A Single Sampling Plan is a method used in acceptance sampling that involves the examination of one sample of items from a lot. Based on the number of defective items found, a decision is made to accept or reject the entire lot. The process is straightforward: compare the number of defectives to a predetermined threshold (the acceptance number \( c \) ). If the number of defectives is equal to or less than \( c \), the lot is accepted; if it exceeds \( c \), the lot is rejected.

This type of plan is simpler and quicker compared to multiple or sequential sampling plans, making it well-suited for settings where decisions need to be made swiftly. However, it's crucial to carefully determine the appropriate sample size (n) and acceptance number (c) to manage the risks of Type I and II errors effectively. The aim is to ensure that both the producer and the consumer are protected - the producer is not unduly penalized for good quality lots, and the consumer is shielded from receiving low-quality products.