Question

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

A: Top 11% of scores

B: Scores below the top 11% and above the bottom 65%

C: Scores below the top 35% and above the bottom 24%

D: Scores below the top 76% and above the bottom 7%

F: Bottom 7% of scores

Scores on the test are normally distributed with a mean of 70.4 and a standard deviation of 8.2. Find the numerical limits for a C grade. Round your answers to the nearest whole number, if necessary.

Answer 1

Mean = 70.4

Standard deviation = 8.2

Let the lower limit for C be C₁ and upper limit for C be C₂

P(X < A) = P(Z < (A - mean)/standard deviation)

P(X < C₁) = 0.24

P(Z < (C₁ - 70.4)/8.2) = 0.24

Take value of z corresponding to probability of 0.24 from standard normal distribution table

(C₁ - 70.4)/8.2 = -0.71

C₁ = 65

P(X > C₂) = 0.35

P(X < C₂) = 1 - 0.35 = 0.65

P(Z < (C₂ - 70.4)/8.2) = 0.65

(C₂ - 70.4)/8.2 = 0.385

C₂ = 74

The numerical limits for C grade is 65 to 74