Question

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

A: Top 5%
of scores

B: Scores below the top 5%
and above the bottom 63%
C: Scores below the top 37%
and above the bottom 16%
D: Scores below the top 84%
and above the bottom 5%
F: Bottom 5%
of scores

Scores on the test are normally distributed with a mean of 81.1
and a standard deviation of 8.9
. Find the numerical limits for a D grade. Round your answers to the nearest whole number, if necessary.

Answer 1

x = 66.46

Explanation:

To find the numerical limits for a D grade, we need to find the scores that fall below the top 84% and above the bottom 5%.

Step 1: Find the z-scores corresponding to the top 84% and bottom 5% of scores.

The top 84% of scores corresponds to a z-score of approximately 1.04. (This can be found using a standard normal distribution table or calculator.)

The bottom 5% of scores corresponds to a z-score of approximately -1.64.

Step 2: Use the z-score formula to find the corresponding scores.

For the top 84% of scores:

z = (x - μ) / σ

1.04 = (x - 81.1) / 8.9

x - 81.1 = 9.25

x = 90.35

For the bottom 5% of scores:

z = (x - μ) / σ

-1.64 = (x - 81.1) / 8.9

x - 81.1 = -14.64

x = 66.46

Therefore, the numerical limits for a D grade are between 66 and 90 (rounded to the nearest whole number).