A manufacturing company uses an acceptance scheme on items from a production line before they are shipped. The plan is a two-stage one. Boxes of 24 items are readied for shipment, and a sample of 3 items is tested for defectives. If any defectives are found, the entire box is sent back for 100% screening. If no defectives are found, the box is shipped. (a) What is the probability that a box containing 2 defectives will be shipped? (b) What is the probability that a box containing only 1 defective will be sent back for screening? (a) The probability that a box containing 2 defectives will be shipped is (Round to four decimal places as needed.) (b) The probability that a box containing only 1 defective will be sent back for screening is (Round to four decimal places as needed.)

MATLAB: An Introduction with Applications
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## Acceptance Sampling in Manufacturing

A manufacturing company employs an acceptance scheme on items produced on a production line before they are shipped. The acceptance plan involves two stages. Boxes containing 24 items each are prepared for shipment. From each box, a sample of 3 items is tested for defects. If any defective items are found within the sample, the entire box is sent back for 100% screening. If no defectives are found, the box is shipped. 

### Problem Statement

**(a)** What is the probability that a box containing 2 defectives will be shipped?  
**(b)** What is the probability that a box containing only 1 defective will be sent back for screening?

### Solution

**(a)** The probability that a box containing 2 defectives will be shipped is:
\[ \text{(Round to four decimal places as needed.)} \]
\[ \_\_\_\_\_ \]

**(b)** The probability that a box containing only 1 defective will be sent back for screening is:
\[ \text{(Round to four decimal places as needed.)} \]
\[ \_\_\_\_\_ \]

---

In this exercise, students are expected to compute the probabilities using appropriate statistical methods, specifically focusing on combinations and the hypergeometric distribution. Understanding the calculation will not only help in similar real-world scenarios in manufacturing but also in scenarios involving quality control and sampling theory.
Transcribed Image Text:## Acceptance Sampling in Manufacturing A manufacturing company employs an acceptance scheme on items produced on a production line before they are shipped. The acceptance plan involves two stages. Boxes containing 24 items each are prepared for shipment. From each box, a sample of 3 items is tested for defects. If any defective items are found within the sample, the entire box is sent back for 100% screening. If no defectives are found, the box is shipped. ### Problem Statement **(a)** What is the probability that a box containing 2 defectives will be shipped? **(b)** What is the probability that a box containing only 1 defective will be sent back for screening? ### Solution **(a)** The probability that a box containing 2 defectives will be shipped is: \[ \text{(Round to four decimal places as needed.)} \] \[ \_\_\_\_\_ \] **(b)** The probability that a box containing only 1 defective will be sent back for screening is: \[ \text{(Round to four decimal places as needed.)} \] \[ \_\_\_\_\_ \] --- In this exercise, students are expected to compute the probabilities using appropriate statistical methods, specifically focusing on combinations and the hypergeometric distribution. Understanding the calculation will not only help in similar real-world scenarios in manufacturing but also in scenarios involving quality control and sampling theory.
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