Math Problem Statement
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.
Solution
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
- The standard deviation
- D grades are between the top 81% and the bottom 5%.
We'll solve this in two steps:
-
Find the z-scores that correspond to the 81st percentile and the 5th percentile.
- The z-score for the 81st percentile is denoted as , and for the 5th percentile, it is denoted as .
-
Convert the z-scores to actual test scores using the formula: where is the actual test score, , and .
Step 1: Find the z-scores
-
For the 81st percentile (), using standard normal distribution tables or a calculator:
-
For the 5th percentile (), using standard normal distribution tables or a calculator:
Step 2: Convert the z-scores to actual test scores
Using the formula :
-
For the 81st percentile: Rounded to the nearest whole number:
-
For the 5th percentile: Rounded to the nearest whole number:
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:
- How would the limits for a B grade be calculated in this case?
- What is the formula for converting z-scores to raw scores in a normal distribution?
- How can you calculate the percentage of students scoring above a certain value in a normal distribution?
- What are the z-scores for other common percentiles, like 90th or 95th?
- 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!
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Normal Distribution
Percentiles
Z-scores
Formulas
X = μ + z * σ
Percentile-to-z-score conversion
Theorems
Standard Normal Distribution
Suitable Grade Level
College/University level
Related Recommendation
Calculate Z-Score for Normally Distributed Test Scores with Mean 65.2 and Standard Deviation 8.3
Grading on a Curve: Calculating Letter Grade Limits Based on a Normal Distribution
Calculating Minimum Score for an A Grade Using Normal Distribution
Probability of Average Final Exam Grade Between 76 and 82 in Skewed Distribution
Normal Distribution: Bottom 5% and Middle 90% Score Analysis