SOLUTION: A shipment of 10 items has two defective and eight non defective units. In the inspection of the shipment , a sample of units will be selected and tested. If a defective unit is fo
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Question 1193442: A shipment of 10 items has two defective and eight non defective units. In the inspection of the shipment , a sample of units will be selected and tested. If a defective unit is found , the shipment of 10 units will be rejected.If management would like a .90 probability of rejecting a shipment with two defective and eight non defective units , how large a sample would you recommend ?
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
**1. Define the Probability of Rejection:**
* The probability of rejecting a shipment is the probability of finding at least one defective unit in the sample.
**2. Calculate the Probability of Not Rejecting:**
* The probability of not rejecting a shipment is the probability of finding no defective units in the sample.
**3. Determine the Sample Size:**
* We need to find the sample size (n) that satisfies the following condition:
Probability of rejecting the shipment ≥ 0.90
1 - Probability of not rejecting the shipment ≥ 0.90
Probability of not rejecting the shipment ≤ 0.10
**4. Calculate the Probability of Not Rejecting (for a given sample size n):**
* The probability of not rejecting a shipment is the probability of selecting only non-defective units in the sample.
* This can be calculated using the hypergeometric distribution:
Probability of not rejecting = (Number of ways to choose n non-defective units from 8 non-defective units) / (Total number of ways to choose n units from 10 total units)
Probability of not rejecting = (8Cn) / (10Cn)
**5. Find the Sample Size (n) by Trial and Error:**
* Start with a small sample size (e.g., n = 1) and calculate the probability of not rejecting.
* Increase the sample size gradually and recalculate the probability until the probability of not rejecting is less than or equal to 0.10.
**Using a Spreadsheet or Statistical Software:**
* You can use a spreadsheet program (like Excel) or statistical software (like R or Python) to efficiently calculate the hypergeometric probabilities for different sample sizes. This will help you quickly find the minimum sample size that meets the desired rejection probability.
**Example (Illustrative):**
* **n = 1:** Probability of not rejecting = (8C1) / (10C1) = 0.8
* **n = 2:** Probability of not rejecting = (8C2) / (10C2) = 0.56
* **n = 3:** Probability of not rejecting = (8C3) / (10C3) = 0.3
* **n = 4:** Probability of not rejecting = (8C4) / (10C4) = 0.1429
* **n = 5:** Probability of not rejecting = (8C5) / (10C5) = 0.051
In this example, a sample size of **n = 5** would likely satisfy the requirement, as the probability of not rejecting is less than 0.10.
**Note:**
* This is a simplified example. In real-world scenarios, you might need to consider factors like the cost of inspection, the potential consequences of rejecting a good shipment, and the desired level of consumer confidence.
* Statistical software or tables can provide more accurate and efficient calculations for the hypergeometric distribution.
I hope this helps! Let me know if you have any further questions.
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