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 \(" 10^{\circ}\) 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 Acceptance Probability for Binomial Distribution

To find the probability that a batch gets accepted, we use the binomial distribution. Consider a sample size, , with the probability of selecting a defective item as \(p\). Define (number of defective items in the sample) to be binomially distributed with parameters \(n\) and \(p\). The acceptance rule says should be at most , so we calculate \( P(X \leq 2) \) for several values of \(p\).

02

Calculate Acceptance Probability for Different Defect Rates

For a batch to be accepted:- When \( p = 0.01 \): \( P(X \leq 2) = \sum_{x=0}^{2} \binom{10}{x} (0.01)^x (0.99)^{10-x}\)- When \( p = 0.05 \): \( P(X \leq 2) = \sum_{x=0}^{2} \binom{10}{x} (0.05)^x (0.95)^{10-x}\)- When \( p = 0.10 \): \( P(X \leq 2) = \sum_{x=0}^{2} \binom{10}{x} (0.10)^x (0.90)^{10-x}\)- When \( p = 0.20 \): \( P(X \leq 2) = \sum_{x=0}^{2} \binom{10}{x} (0.20)^x (0.80)^{10-x}\)- When \( p = 0.25 \): \( P(X \leq 2) = \sum_{x=0}^{2} \binom{10}{x} (0.25)^x (0.75)^{10-x}\)

03

Calculate and Sketch the Operating Characteristic Curve

For part (b), plot \( P(X \leq 2) \) as a function of \(p\) for \(0 \leq p \leq 1\). This curve depicts the likelihood of batch acceptance across a range of defect probabilities \(p\).

04

Update Sampling Plan Acceptance Rule

Change the acceptance rule to accept the batch if \( X \leq 1 \) and repeat probability calculations:- Find \( P(X \leq 1) \) and use the same values of \(p\) for calculation as in Step 2.

05

Calculate Acceptance Probability for Larger Sample

Increase sample size to 15 as suggested in part (d) with the same acceptance criteria (\( X \leq 2 \)).- Re-calculate \( P(X \leq 2) \) using \( n = 15\).

06

Analyze and Compare Different Sampling Plans

Compare the retention probabilities from each sampling plan. Analyze which method offers the closest alignment to assuring less than 10% defectives are accepted. For the decision-making in part (e), provide insights into balancing inspection rigor with batch acceptance.

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Key Concepts

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

Binomial Distribution

The Binomial Distribution is a widely used probability distribution in statistics. It applies when there are fixed numbers of independent trials, each with two possible outcomes - success or failure. In the case of acceptance sampling, the success might represent finding a defective item, and failure might indicate a non-defective item.

Key characteristics are:

• Each trial is independent.
• The probability of success, denoted by 'p', is the same for each trial.
• There are a specified number of trials, 'n'.

For our exercise, we assume that selecting a defective item is a success. Given this, the random variable, represented by 'X', will follow a binomial distribution with parameters 'n' (the sample size) and 'p' (probability of a defect in each trial). Accepting the batch is equivalent to finding the probability that the number of defects, X, is at most a certain number. For instance, we calculate \( P(X \leq 2) \) for different values of 'p'.

This approach provides a structured method to decide if a lot is acceptable based on the sample.

Operating Characteristic Curve

The Operating Characteristic (OC) Curve provides a visual representation of the performance of an acceptance sampling plan. It displays the probability that a given batch is accepted based on various defective rates. On an OC curve, the horizontal axis represents the proportion of defectives in the batch (denoted as 'p'), while the vertical axis shows the probability of accepting the batch. In this exercise:

• Each point on the curve corresponds to a specific probability of acceptance for a given defect rate 'p'.
• This curve is sketched using calculations from different proportions of defectives, ranging from 0 to 1.

The OC curve guides us in understanding how stringent or lenient a particular sampling plan is. By plotting and interpreting this curve, we can evaluate how different strategies might affect the acceptance ratio of batches based on their quality metrics.

Probability Calculation

Probability Calculation in acceptance sampling involves compute probabilities that a particular sample will meet the acceptance criteria. Here, we're interested in probabilities like \( P(X \leq 2) \) where 'X' is a binomially distributed random variable.To calculate:

• Use the formula: \( P(X = x) = \binom{n}{x} p^x (1-p)^{n-x} \), where \(\binom{n}{x}\) is the binomial coefficient.
• Add these probabilities for X = 0, 1, and 2 for our rejection criteria \( X \leq 2 \).

Suppose p = 0.05, the stepwise calculation is:

• \(P(X=0) = \binom{10}{0}(0.05)^0(0.95)^{10}\)
• \(P(X=1) = \binom{10}{1}(0.05)^1(0.95)^9\)
• \(P(X=2) = \binom{10}{2}(0.05)^2(0.95)^8\)
• Sum these probabilities for the total \(P(X \leq 2)\).

This process is repeated for other values of 'p' to evaluate batch acceptance under different defect scenarios.

Sampling Plans

Sampling Plans define how many items are inspected from a lot to determine its quality. It sets the criteria for accepting or rejecting a batch based on the results from the sample.Key elements in a sampling plan:

• Sample Size (n): Number of items to test.
• Acceptance Number (c): Maximum number of defects allowed for the batch to be accepted.

Our exercise evaluates three different sampling plans:

• Original plan: Sample 10 items, accept batch if \(X \leq 2\).
• Alternative plan 1: Sample 10 items, accept batch if \(X \leq 1\).
• Alternative plan 2: Sample 15 items, accept batch if \(X \leq 2\).

The choice of plan affects the balance between inspection effort and risk of accepting defective batches. A rigorous plan reduces the chance of accepting poor-quality lots but increases inspection costs. The decision depends on the acceptable risk level and the cost-benefit analysis of quality control processes.