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 μ=82\mu = 82
  • The standard deviation σ=8.5\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 z81z_{81}, and for the 5th percentile, it is denoted as z5z_5.
  2. Convert the z-scores to actual test scores using the formula: X=μ+zσX = \mu + z \cdot \sigma where XX is the actual test score, μ=82\mu = 82, and σ=8.5\sigma = 8.5.

Step 1: Find the z-scores

  • For the 81st percentile (z81z_{81}), using standard normal distribution tables or a calculator:

    • z810.88z_{81} \approx 0.88
  • For the 5th percentile (z5z_5), using standard normal distribution tables or a calculator:

    • z51.645z_5 \approx -1.645

Step 2: Convert the z-scores to actual test scores

Using the formula X=μ+zσX = \mu + z \cdot \sigma:

  • For the 81st percentile: X81=82+(0.88×8.5)=82+7.48=89.48X_{81} = 82 + (0.88 \times 8.5) = 82 + 7.48 = 89.48 Rounded to the nearest whole number: X8189X_{81} \approx 89

  • For the 5th percentile: X5=82+(1.645×8.5)=8213.98=68.02X_5 = 82 + (-1.645 \times 8.5) = 82 - 13.98 = 68.02 Rounded to the nearest whole number: X568X_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!

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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