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To solve this, we need to compute the numerical limits for each letter grade based on the provided normal distribution, with a mean of 31 and a standard deviation of 8.2.

### Steps:
1. **Determine the Z-scores** for the given percentiles (15%, 30%, 70%, 85%, etc.).
   - The Z-score tells us how many standard deviations a particular value is from the mean.

2. **Convert Z-scores to actual test scores** using the formula:
   \[
   X = \mu + Z \cdot \sigma
   \]
   Where:
   - \(X\) = test score
   - \(\mu = 31\) (mean score)
   - \(\sigma = 8.2\) (standard deviation)

### Percentiles:
- For **A** (Top 15%): We need the Z-score for the 85th percentile.
- For **B** (Between bottom 70% and top 15%): Find Z-scores for the 30th and 85th percentiles.
- For **C** (Between bottom 30% and top 30%): Z-scores for 30th and 70th percentiles.
- For **D** (Between bottom 15% and top 70%): Z-scores for 15th and 30th percentiles.
- For **F** (Bottom 15%): Z-score for the 15th percentile.

I'll now calculate the Z-scores and the corresponding intervals.Here are the computed numerical limits for each grade based on the given normal distribution:

- **A** (Top 15%): Scores above **39.50**
- **B** (Above bottom 70% and below top 15%): Scores between **35.30** and **39.50**
- **C** (Above bottom 30% and below top 30%): Scores between **26.70** and **35.30**
- **D** (Above bottom 15% and below top 70%): Scores between **22.50** and **26.70**
- **F** (Bottom 15%): Scores below **22.50**

Would you like more details or further clarification on any part?

### Related Questions:
1. How do we interpret Z-scores in a normal distribution?
2. How can we compute percentile ranks for other distributions?
3. What is the impact of increasing the standard deviation on grade cutoffs?
4. How do we calculate Z-scores for other percentiles?
5. How is a curved grading system different from a fixed grading system?

**Tip:** Z-scores are a powerful tool to standardize values and compare them across different normal distributions.

A teacher informs his computational physics class (of 500+ students) that a test was very difficult, but the grades would be curved. Scores on the test were normally distributed with a mean of 31 and a standard deviation of 8.2. The maximum possible score on the test was 100 points. The grades are curved according to the following scheme: Find the numerical limits for each letter grade (A, B, C, D, F).

Learn how to calculate the numerical limits for letter grades (A, B, C, D, F) in a computational physics class based on a normal distribution with a mean of 31 and a standard deviation of 8.2. This example demonstrates how to use Z-scores and percentiles for curved grading.

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