Question 1182170: The final grades of students in Foreign Language are normally distributed. records show that the average grade of students is 85 with the standard deviation of 2.3; suppose the professor wants to convert the grades of the students into an alpha-grade; A as the highest and D as the lowest grades, and DECIDES to assign: 10% A's, 20% B's, 30% C's, and 40% D's. What shall be the cut-off grades of these grade ranges?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to determine the cut-off grades for the Foreign Language class:
**Understanding the Problem**
The professor wants to assign letter grades based on a normal distribution. This means we need to find the scores that correspond to the given percentages of each grade.
**Steps**
1. **Percentiles:** We'll use the given percentages to find the corresponding percentiles in the normal distribution.
* 10% A's: This means the top 10% of scores are A's. So, the cutoff for a B is at the 90th percentile.
* 20% B's: The cutoff for a C is at the 70th percentile (90th percentile - 20%).
* 30% C's: The cutoff for a D is at the 40th percentile (70th percentile - 30%).
2. **Z-scores:** We need to find the z-scores that correspond to these percentiles. A z-score tells us how many standard deviations a score is away from the mean. You can use a z-table or a calculator with statistical functions to find these z-scores.
* 90th percentile: z ≈ 1.28
* 70th percentile: z ≈ 0.52
* 40th percentile: z ≈ -0.25
3. **Cut-off Scores:** Now, use the z-scores, the mean (85), and the standard deviation (2.3) to calculate the cut-off scores:
* B cutoff (90th percentile): Score = Mean + (z-score * Standard Deviation) = 85 + (1.28 * 2.3) ≈ 87.94
* C cutoff (70th percentile): Score = 85 + (0.52 * 2.3) ≈ 86.20
* D cutoff (40th percentile): Score = 85 + (-0.25 * 2.3) ≈ 84.43
**Results**
* **A:** 87.94 and above
* **B:** Between 86.20 and 87.94
* **C:** Between 84.43 and 86.20
* **D:** Below 84.43
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