Acceptance Sampling A shipment of 50,000 transistors arrives at a manufacturing plant. The quality control engineer at the plant obtains a random sample of 500 resistors and will reject the entire shipment if 10 or more of the resistors are defective. Suppose that 4% of the resistors in the whole shipment are defective. What is the probability the engineer accepts the shipment? Do you believe the acceptance policy of the engineer is sound? 

Verified step by step guidance

1. Identify the random variable of interest: let X be the number of defective resistors found in the sample of 500 resistors. Since each resistor can be defective or not, X follows a binomial distribution with parameters n = 500 and p = 0.04 (the defective rate).
2. Write down the binomial distribution for X: \(P(X = k) = \binom{500}{k} (0.04)^k (0.96)^{500-k}\), where \(k\) is the number of defective resistors found in the sample.
3. The engineer will accept the shipment if fewer than 10 resistors are defective in the sample, so we want to find \(P(X < 10) = P(X \leq 9) = \sum_{k=0}^9 P(X = k)\).
4. Since n is large and p is small, approximate the binomial distribution with a normal distribution: \(X \sim N(\mu, \sigma^2)\) where \(\mu = np = 500 \times 0.04\) and \(\sigma = \sqrt{np(1-p)} = \sqrt{500 \times 0.04 \times 0.96}\).
5. Use the normal approximation with continuity correction to find \(P(X \leq 9)\) by calculating \(P\left(Z \leq \frac{9.5 - \mu}{\sigma}\right)\), where Z is a standard normal variable. This probability represents the chance the shipment is accepted.

Key Concepts

Acceptance Sampling

Acceptance sampling is a quality control method where a random sample from a batch is inspected to decide whether to accept or reject the entire shipment. It balances inspection cost and risk by using sample results to infer the quality of the whole lot without checking every item.

Binomial Probability Distribution

The binomial distribution models the number of defective items in a fixed-size sample when each item has the same probability of being defective. It helps calculate the probability of observing a certain number of defects, assuming independent and identical defect probabilities.

Operating Characteristic (OC) Curve and Acceptance Criteria

The OC curve shows the probability of accepting shipments with varying defect rates under a given sampling plan. Evaluating the acceptance policy involves analyzing the probability of acceptance at the actual defect rate to determine if the plan effectively controls quality risks.