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 "l" 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

Understand the Problem

We are given a situation where a batch of components is accepted if the number of defective ones in a sample of 10 is at most 2. We need to find the probability of this condition being met for different defective proportions (parts a and b). Then, we replicate the analysis with changes in sample sizes and acceptance criteria (parts c and d) and determine the best plan (part e).

02

Define the Binomial Probability

We use the binomial distribution to model the number of defective components in the sample. If there are 10 components and a proportion \( p \) of them is defective, the number of defective components in the sample, \( X \), is binomially distributed: \( X \sim \text{Binomial}(n=10, p) \).

03

Calculate Acceptance Probability for Part a

For each value of \( p \), compute the probability that \( X \leq 2 \) (meaning 0, 1, or 2 defectives in the sample) using the binomial probability formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]Then, calculate:\[ P(X \leq 2) = \sum_{k=0}^{2} \binom{10}{k} p^k (1-p)^{10-k} \] For \( p = 0.01, 0.05, 0.10, 0.20, 0.25 \).

04

Sketch the Operating Characteristic Curve

Using the probabilities calculated in Step 3, plot \( P(X \leq 2) \) against \( p \) to form the operating characteristic curve. This curve visually represents how the acceptance probability changes with the defect proportion \( p \).

05

Update Acceptance Criteria for Part c

Change the acceptance criteria to allow at most 1 defective in the sample of 10. Recalculate:\[ P(X \leq 1) = \sum_{k=0}^{1} \binom{10}{k} p^k (1-p)^{10-k} \]for the same \( p \) values. Plot the new curve.

06

Change Sample Size for Part d

Increase the sample size to 15, keeping the acceptance criterion as at most 2 defectives. Recalculate:\[ P(X \leq 2) = \sum_{k=0}^{2} \binom{15}{k} p^k (1-p)^{15-k} \]for the same \( p \) values. Plot this curve as well.

07

Evaluate the Plans for Part e

Compare the curves from parts a, c, and d. Assess the plans based on how effectively they separate acceptable and non-acceptable batches. The most satisfactory plan minimizes the probability of accepting a bad batch and rejecting a good one, based on the curves produced.

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Key Concepts

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

Binomial Distribution

Binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials of a binary experiment. Each trial can result in one of two possible outcomes, often labeled success or failure.
In the context of acceptance sampling, we use the binomial distribution to model the number of defective items out of a fixed number of samples. Here, we want to determine the probability that a sample of 10 components contains at most 2 defective items. This can be represented as:

• The total number of trials, or components sampled, is 10 (denoted as \( n = 10 \)).
• The probability of finding a defective item in one trial is \( p \), the true proportion of defects in the batch.

The random variable \( X \), the number of defective items in the sample, follows a binomial distribution: \( X \sim \text{Binomial}(n=10, p) \).
The binomial probability formula, \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \), is used to find the likelihood of observing a specific number of defective items in the sample.

Probability Calculation

Probability calculation in this problem involves computing the likelihood of having zero, one, or two defective components in the sample. For various values of defect proportion \( p \), we calculate the cumulative probability \( P(X \leq 2) \).
This involves evaluating the sum:

• \( P(X \leq 2) = \sum_{k=0}^{2} \binom{10}{k} p^k (1-p)^{10-k} \)

For each specified value of \( p \, (0.01, 0.05, 0.10, 0.20, 0.25) \), this computation helps determine the probability of accepting the batch.
By using a binomial distribution calculator or software, this calculation allows us to efficiently determine whether a batch will be accepted, by checking how the actual defect rate \( p \) influences the sample result.

Operating Characteristic Curve

An operating characteristic (OC) curve is used to visually represent the performance of an acceptance sampling plan. It shows the probability of acceptance for a batch as a function of the defect rate \( p \).
The OC curve plots \( P(X \leq 2) \) for different values of \( p \), illustrating how changes in the actual proportion of defects affect the likelihood that a batch will be accepted. This is crucial for understanding the effectiveness of the sampling plan.

• A steep curve indicates a sharper distinction between acceptable and unacceptable batches.
• A flatter curve suggests that even batches with a higher defect rate have a significant chance of being accepted.

The OC curve helps us visualize and choose the best sampling plan by comparing different sampling sizes and criteria. It aids in identifying which plan minimizes errors such as accepting a defective batch or rejecting an acceptable one.

Sample Size Analysis

Sample size analysis examines how changing the sample size affects the probability of accepting a batch. By varying the number of components inspected, we can influence the precision and accuracy of our sampling plan.
For example, increasing the sample size from 10 to 15, while keeping the acceptance threshold at 2 defectives, changes the probability calculations to:

• \( P(X \leq 2) = \sum_{k=0}^{2} \binom{15}{k} p^k (1-p)^{15-k} \)

Here, a larger sample improves the confidence in the acceptance decision by reducing variability. More samples reduce the likelihood of accepting a bad batch due to chance.
However, increasing sample size also means more cost and time. Thus, a balance between resources and quality assurance needs to be considered when determining the ideal sample size for effective batch acceptance. Through the sample size analysis, we find the most satisfactory sampling plan that optimizes this balance without compromising on quality.