The lengths of nails produced in a factory are normally distributed with a mean of 4.814.81 centimeters and a standard deviation of 0.050.05 centimeters.  Find the two lengths that separate the top 10%10% and the bottom 10%10%.  These lengths could serve as limits used to identify which nails should be rejected.  Round your answer to the nearest hundredth, if necessary.

The lengths of nails are normally distributed, with a mean (\(\mu\)) of 4.81 cm and a standard deviation (\(\sigma\)) of 0.05 cm. To find the two lengths that separate the top 10% and the bottom 10%, we are essentially looking for the 10th and 90th percentiles of this normal distribution.

1. **Determine the z-scores** for the 10th and 90th percentiles. These values correspond to the points in the normal distribution where 10% of the data falls below (bottom 10%) and 10% of the data falls above (top 10%).

From standard normal distribution tables or using statistical software, we know:
- The z-score for the 10th percentile is approximately \(-1.28\).
- The z-score for the 90th percentile is approximately \(+1.28\).

2. **Convert the z-scores into actual lengths** using the formula for a normally distributed variable:

\[
x = \mu + z \cdot \sigma
\]

- For the bottom 10%:
  \[
  x_{10} = 4.81 + (-1.28) \cdot 0.05 = 4.81 - 0.064 = 4.75 \, \text{cm}
  \]

- For the top 10%:
  \[
  x_{90} = 4.81 + (1.28) \cdot 0.05 = 4.81 + 0.064 = 4.87 \, \text{cm}
  \]

Thus, the two lengths that separate the top 10% and the bottom 10% are approximately:
- Bottom 10%: **4.75 cm**
- Top 10%: **4.87 cm**

These lengths could be used as limits to identify nails that should be rejected.

Would you like more details on this or have any questions?

Here are 5 related questions:
1. How do you calculate z-scores for other percentiles?
2. What happens to the rejection limits if the standard deviation increases?
3. How would you use a normal distribution table to find specific z-scores?
4. Can you explain the impact of changing the mean on the rejection limits?
5. How can you use this concept in quality control for other manufacturing processes?

**Tip**: Understanding z-scores helps in quickly determining probabilities and percentiles for normally distributed data.

The lengths of nails produced in a factory are normally distributed with a mean of 4.81 centimeters and a standard deviation of 0.05 centimeters. Find the two lengths that separate the top 10% and the bottom 10%. These lengths could serve as limits used to identify which nails should be rejected. Round your answer to the nearest hundredth, if necessary.

Learn how to calculate the lengths that separate the top and bottom 10% of nails produced in a factory, based on a normal distribution with a mean of 4.81 cm and a standard deviation of 0.05 cm. This problem uses z-scores and is suitable for high school students.

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