Consider a shipment of 1000 items into a factory. Suppose...

Consider a shipment of 1000 items into a factory. Suppose the factory can tolerate about $5 \%$ defective items. Let $X$ be the number of defective items in a sample without replacement of size $n=10 .$ Suppose the factory returns the shipment if $X \geq 2$. (a) Obtain the probability that the factory returns a shipment of items that has $5 \%$ defective items. (b) Suppose the shipment has $10 \%$ defective items. Obtain the probability that the factory returns such a shipment. (c) Obtain approximations to the probabilities in parts (a) and (b) using appropriate binomial distributions. Note: If you do not have access to a computer package with a hypergeometric command, obtain the answer to (c) only. This is what would have been done in practice 20 years ago. If you have access to $\mathrm{R}$, then the command dhyper $(\mathrm{x}, \mathrm{D}, \mathrm{N}-\mathrm{D}, \mathrm{n})$ returns the probability in expression (3.1.7).

Consider a shipment of 1000 items into a factory. Suppose the factory can tolerate about $5 \%$ defective items. Let $X$ be the number of defective items in a sample without replacement of size $n=10 .$ Suppose the factory returns the shipment if $X \geq 2$.
(a) Obtain the probability that the factory returns a shipment of items that has $5 \%$ defective items.
(b) Suppose the shipment has $10 \%$ defective items. Obtain the probability that the factory returns such a shipment.
(c) Obtain approximations to the probabilities in parts (a) and (b) using appropriate binomial distributions.
Note: If you do not have access to a computer package with a hypergeometric command, obtain the answer to (c) only. This is what would have been done in practice 20 years ago. If you have access to $\mathrm{R}$, then the command dhyper $(\mathrm{x}, \mathrm{D}, \mathrm{N}-\mathrm{D}, \mathrm{n})$ returns the probability in expression (3.1.7).

Key Concepts

Hypergeometric Distribution

The hypergeometric distribution describes the probability of obtaining a given number of successes in a sample drawn without replacement from a finite population that contains a fixed number of successes. This concept is crucial when the sampling affects the remaining population, making the probabilities change after each draw. It is commonly used in quality control and acceptance sampling, where the items tested are not returned to the population, and each draw changes the composition of the lot.

Binomial Distribution Approximation

The binomial distribution is used to model the number of successes in a series of independent trials with a fixed probability of success for each trial. When the sample size is much smaller than the overall population, the dependency in the hypergeometric setting can be approximated by the independence assumption of the binomial distribution. This approximation simplifies calculations and is particularly useful in practical applications where the exact hypergeometric probability is computationally intensive or unnecessary.

Acceptance Sampling

Acceptance sampling is a quality control procedure used to decide whether to accept or reject a lot of products based on a sample inspection. The decision rule typically involves pre-defined criteria, such as a maximum allowable number of defective items in the sample. This process leverages probability distributions to estimate the likelihood that a shipment meets the quality standards, making it a critical tool for manufacturers and suppliers in managing product quality and process control.

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