Question

A psychology professor assigns letter grades on a test according to the following scheme.

A: Top 10%of scores

B: Scores below the top 10%and above the bottom 55%

C: Scores below the top 45%and above the bottom 20%

D: Scores below the top 80%and above the bottom 7%

F: Bottom 7%of scores

Scores on the test are normally distributed with a mean of 67.6and a standard deviation of 9. Find the numerical limits for a D grade. Round your answers to the nearest whole number, if necessary.

Answer 1

Solution:

Given, the Normal distribution with,

   = 67.6

= 9

For D grade : Scores below the top 80%and above the bottom 7%

i.e. P(X > x) = 80% and P(X < x) = 7%

P(X > x) = 0.80 and P(X < x) = 0.07

First we find z values from z table.

P(Z > z) = 0.80

P(Z < z) = 1 - 0.80 = 0.20

Use z table and search 0.20 probability and see corresponding z value.

P(Z < -0.842 ) = 0.20

so , z = -0.842

Using z score formula,

x = + (z * ) = 67.6 + ( -0.842 * 9) = 60.022 = 60

Upper limit is 60.022

Now , using P(Z < z ) = 0.07

Using z table , z = -1.476

x = + (z * ) = 67.6 + ( -1.476 * 9) = 54.316 = 54

Answer: The numerical limits for D grade are 54 and 60