An English professor assigns letter grades on a test according to the following scheme.

A:  Top 14%
 of scores

B:  Scores below the top 14%
 and above the bottom 58%
C:  Scores below the top 42%
 and above the bottom 19%
D:  Scores below the top 81%
 and above the bottom 5%
F:  Bottom 5%
 of scores

Scores on the test are normally distributed with a mean of 82
 and a standard deviation of 8.5
.  Find the numerical limits for a D grade.  Round your answers to the nearest whole number, if necessary.



To find the numerical limits for a D grade, we need to use the fact that the scores are normally distributed. The key here is to convert the percentage ranges given into corresponding z-scores (standard normal values), and then use those z-scores to calculate the limits of the D grade in terms of the actual test scores. 

### Given:
- The mean \( \mu = 82 \)
- The standard deviation \( \sigma = 8.5 \)
- D grades are between the top 81% and the bottom 5%.

We'll solve this in two steps:

1. **Find the z-scores** that correspond to the 81st percentile and the 5th percentile.
   - The z-score for the 81st percentile is denoted as \( z_{81} \), and for the 5th percentile, it is denoted as \( z_5 \).
   
2. **Convert the z-scores to actual test scores** using the formula:
   \[
   X = \mu + z \cdot \sigma
   \]
   where \( X \) is the actual test score, \( \mu = 82 \), and \( \sigma = 8.5 \).

### Step 1: Find the z-scores
- For the 81st percentile (\( z_{81} \)), using standard normal distribution tables or a calculator:
  - \( z_{81} \approx 0.88 \)
  
- For the 5th percentile (\( z_5 \)), using standard normal distribution tables or a calculator:
  - \( z_5 \approx -1.645 \)

### Step 2: Convert the z-scores to actual test scores
Using the formula \( X = \mu + z \cdot \sigma \):

- For the 81st percentile:
  \[
  X_{81} = 82 + (0.88 \times 8.5) = 82 + 7.48 = 89.48
  \]
  Rounded to the nearest whole number:
  \[
  X_{81} \approx 89
  \]

- For the 5th percentile:
  \[
  X_5 = 82 + (-1.645 \times 8.5) = 82 - 13.98 = 68.02
  \]
  Rounded to the nearest whole number:
  \[
  X_5 \approx 68
  \]

### Conclusion:
The numerical limits for a D grade are between **68 and 89**.

Let me know if you would like further details or explanations on any part!

Here are 5 related questions:
1. How would the limits for a B grade be calculated in this case?
2. What is the formula for converting z-scores to raw scores in a normal distribution?
3. How can you calculate the percentage of students scoring above a certain value in a normal distribution?
4. What are the z-scores for other common percentiles, like 90th or 95th?
5. How does changing the standard deviation affect the spread of grades in a normal distribution?

**Tip:** In any normally distributed dataset, 68% of the data falls within one standard deviation of the mean, and 95% falls within two standard deviations!

An English professor assigns letter grades on a test according to the following scheme. A: Top 14% of scores, B: Scores below the top 14% and above the bottom 58%, C: Scores below the top 42% and above the bottom 19%, D: Scores below the top 81% and above the bottom 5%, F: Bottom 5% of scores. Scores on the test are normally distributed with a mean of 82 and a standard deviation of 8.5. Find the numerical limits for a D grade. Round your answers to the nearest whole number, if necessary.

Learn how to calculate the numerical limits for a D grade in a normally distributed test with a mean of 82 and a standard deviation of 8.5. The problem involves using z-scores for the 81st and 5th percentiles, with detailed steps on converting these to actual test scores. Suitable for college-level students.

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