Question

An art history 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 62% C: Scores below the top 38% and above the bottom 22% D: Scores below the top 78% and above the bottom 5% F: Bottom 5% of scores Scores on the test are normally distributed with a mean of 73.3 and a standard deviation of 9.7. Find the numerical limits for a B grade. Round your answers to the nearest whole number, if necessary.

Answer 1

Answer:

76 ≤ B ≤ 89

Step-by-step explanation:

Mean score (μ) = 73.3

Standard deviation (σ) = 9.7

If B Scores are below the top 5% and above the bottom 62%, then:

62% ≤ B ≤ 95%

In a normal distribution, the 62nd percentile has a corresponding z-score of z = 0.305, while the 95th percentile has a corresponding z-score of z = 1.650.

The grades X1 and X2 which are the limits for a B grade are given by:

$z= \frac{X-\mu}{\sigma}\\0.305= \frac{X_1-73.3}{9,7}\\X_1=76.2\\1.650= \frac{X_2-73.3}{9,7}\\X_2=89.3$

Rounding to the nearest whole number, a B grade is given to grades between 76 and 89.