The Brockingham Carpet Company prides itself on high quality carpets. At the end of each​ day, the company quality managers select 3 square yards for inspection. The quality standard requires an average of no more than 2.3 defects per square yard. Last​ night, the inspector found 8 defects in the sample of 3 square yards. The chance of finding 8 or more defects in the sample is 0.9975.
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Part 1

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False

The problem states that the **quality standard** requires an average of **no more than 2.3 defects per square yard**, and the inspector found **8 defects** in a sample of **3 square yards**. To check if the inspector's findings meet the standard, we can compare this scenario against the given chance (0.9975) of finding 8 or more defects.

### Analysis:
- Expected number of defects in 3 square yards (based on standard):  
  \( 2.3 \, \text{defects/yard} \times 3 \, \text{yards} = 6.9 \, \text{defects} \)
  
- The inspector found **8 defects**, which is slightly higher than the expected number of 6.9.  
  However, the chance of finding **8 or more defects** is **0.9975** (or 99.75%).

This means it is very likely to find 8 or more defects under these conditions. Since the probability of finding 8 or more defects is almost certain (99.75%), the quality standard is not being violated.

**Answer: True**  
The chance of finding 8 or more defects in the sample is indeed 0.9975.

Would you like a deeper explanation or more questions on this topic?  
Here are five related questions:

1. What is the expected number of defects for a 5-yard sample under the same standard?
2. How would the probability change if the standard allowed for 3 defects per square yard?
3. What is the probability of finding exactly 8 defects in this 3 square-yard sample?
4. How could we calculate the probability of finding fewer than 8 defects?
5. How would the analysis change if the sample size increased to 6 square yards?

**Tip**: When calculating probabilities of events, always compare them to expected outcomes to determine the likelihood of unusual occurrences.

The Brockingham Carpet Company prides itself on high quality carpets. At the end of each day, the company quality managers select 3 square yards for inspection. The quality standard requires an average of no more than 2.3 defects per square yard. Last night, the inspector found 8 defects in the sample of 3 square yards. The chance of finding 8 or more defects in the sample is 0.9975.

Explore how to calculate the probability of finding 8 or more defects in a 3-yard sample, based on a quality standard of no more than 2.3 defects per yard. Learn the concept of expected value and use probability distribution methods. Suitable for high school students in Grades 10-12.

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