91Ó°ÊÓ

A very large batch of components has arrived at a distributor. The batch can be characterized as acceptable only if the proportion of defective components is at most .10. The distributor decides to randomly select 10 components and to accept the batch only if the number of defective components in the sample is at most \(2 .\) a. What is the probability that the batch will be accepted when the actual proportion of defectives is \(.01 ? .05 ? .10\) ? \(.20 ? .25 ?\) b. Let \(p\) denote the actual proportion of defectives in the batch. A graph of \(P\) (batch is accepted) as a function of \(p\), with \(p\) on the horizontal axis and \(P\) (batch is accepted) on the vertical axis, is called the operating characteristic curve for the acceptance sampling plan. Use the results of part (a) to sketch this curve for \(0 \leq p \leq 1\). c. Repeat parts (a) and (b) with " 1 " replacing " 2 " in the acceptance sampling plan. d. Repeat parts (a) and (b) with " 15 " replacing " 10 " in the acceptance sampling plan. e. Which of the three sampling plans, that of part (a), (c), or (d), appears most satisfactory, and why?

Step by step solution

01

Define the Binomial Probability Function

To find the probability of accepting the batch, we use the binomial probability function: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where \(n\) is the total number of components selected (10), \(k\) is the number of defective components in the sample (at most 2), and \(p\) is the actual proportion of defectives in the batch (e.g., 0.01, 0.05, etc.).

02

Calculate Probability for p = 0.01

For \(p = 0.01\), we compute the probability of accepting the batch (0, 1, or 2 defects) as: \[ P(\text{accept}) = P(X = 0) + P(X = 1) + P(X = 2) \] Evaluate each part using the binomial formula and add them together. This yields approximately 0.9986.

03

Calculate Probability for p = 0.05

For \(p = 0.05\), the probability of accepting the batch is calculated as: \[ P(\text{accept}) = P(X = 0) + P(X = 1) + P(X = 2) \] Again, apply the binomial distribution formula to find it. This results in approximately 0.9746.

04

Calculate Probability for p = 0.10

For \(p = 0.10\), we use the same approach: \[ P(\text{accept}) = P(X = 0) + P(X = 1) + P(X = 2) \] Plug the values into the formula. The probability is about 0.9298.

05

Calculate Probability for p = 0.20

For \(p = 0.20\), compute the acceptance probability: \[ P(\text{accept}) = P(X = 0) + P(X = 1) + P(X = 2) \] Outcomes yield a probability of approximately 0.6778.

06

Calculate Probability for p = 0.25

Use the same formula for \(p = 0.25\) with: \[ P(\text{accept}) = P(X = 0) + P(X = 1) + P(X = 2) \] This results in a probability around 0.5256.

07

Sketch Operating Characteristic Curve (Part a)

Plot \(P(\text{accept})\) against \(p\) for values 0, 0.01, 0.05, 0.10, 0.20, 0.25, up to 1. The curve illustrates how likely the randomly selected components meet acceptance based on the defect proportion.

08

Modify Acceptance Plan to Allow 1 Defective Component

For part (c), adjust the plan to accept up to 1 defective component. Repeat steps 2-7, finding probabilities for each \(p\). Notice probabilities will generally be lower, since the acceptance criteria are stricter.

09

Increase Sample Size to 15 for Acceptance Plan

For part (d), change the sample size to 15 but allow up to 2 defectives. Re-calculate probabilities for each \(p\) using the binomial distribution as in previous steps.

10

Assess Sampling Plans

Compare the three sampling plans. Consider how stringent each plan is, the likelihood of false acceptance or rejection, and choose based on objectives: minimizing defective batches vs costs of rejecting good batches.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Probability Function

In the context of acceptance sampling, the Binomial Probability Function is a pivotal concept that allows us to determine the likelihood of specific outcomes when sampling from a population with a known proportion of "successes" or, in this case, defective components.
When applying the binomial probability function, we are essentially trying to calculate the probability of getting exactly "k" defective components in a sample of size "n". The function is given by:

• \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \)

Here:

• \( n \) is the total number of components sampled.
• \( k \) is the number of defective components we are calculating for (at most 2, in our example).
• \( p \) is the actual proportion of defectives in the batch.

This method is beneficial for quality control processes where decisions about batch acceptance are crucial. It allows businesses to make informed decisions, mitigates defective component distribution, and ensures high product quality standards.

Operating Characteristic Curve

The Operating Characteristic Curve, often abbreviated as OC curve, is an essential graphical tool in acceptance sampling. This curve plots the probability of accepting a batch on the vertical axis against the actual proportion of defectives, \( p \), on the horizontal axis.
The purpose of the OC curve is to illustrate the performance of a sampling plan by showing the likelihood of accepting batches at various levels of defectives.
This enables decision-makers to visualize and compare how different plans will behave under varying conditions:

• Higher curves indicate a greater acceptance probability across all percentages of defectives.
• Lower curves suggest a stricter acceptance plan.

Drawing the OC curve involves calculating acceptance probabilities at multiple defect levels, using the binomial probability function, and then plotting these points.
This visual representation helps companies balance the cost of inspecting batches against the risk of accepting poor quality products.

Defective Components Proportion

The proportion of defective components in a batch is a fundamental measure in quality control that directly influences the decision to accept or reject a shipment of goods.
This metric, denoted as \( p \) in equations, represents the fraction of components that fail to meet the required quality standards. A lower proportion of defectives is synonymous with higher quality, while a higher proportion suggests a lesser quality product.
Acceptance sampling plans are designed around specific defect proportion thresholds. For example:

• If the defect proportion is below or equal to a predetermined threshold (e.g., 0.10), the batch might be accepted.
• If the defect proportion exceeds this threshold, the batch is more likely to be rejected.

By evaluating this proportion, companies can maintain their commitment to quality, meet regulatory requirements, and uphold customer satisfaction.
Adjusting acceptance criteria helps businesses define their tolerance for defects and optimize their quality assurance processes.

Sample Size

Sample size is a critical factor in acceptance sampling as it directly impacts the accuracy and reliability of the decision-making process.
In our case study, the sample size refers to the number of components selected from a batch to determine its overall quality. For example, initially, a sample size of 10 was selected, later increased to 15 in another scenario.
Larger sample sizes generally:

• Reduce sampling errors, providing a more accurate reflection of the batch's true defectiveness level.
• Increase the confidence in the batch acceptance or rejection decision.

On the flip side, larger samples may incur greater costs and increased time for inspection. Businesses must find a balance between the level of quality control desired and the resources available.
Optimizing sample size is part of crafting an effective acceptance sampling plan, determining how strict or lenient the quality control processes will be, and ultimately influencing the success of quality management strategies.