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 65% C: Scores below the top 35% and above the bottom 25% D: Scores below the top 75% and above the bottom 9% F: Bottom 9% of scores Scores on the test are normally distributed with a mean of 66.9 and 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:

A: Top 10% of scores

B: Scores below the top 10% and above the bottom 65%

C: Scores below the top 35% and above the bottom 25%

D: Scores below the top 75% and above the bottom 9%

F: Bottom 9% of scores

Scores on the test are normally distributed with a mean of 66.9 and a standard deviation of 9.

That is: Mean = and standard deviation =

We have to find the numerical limits for a D grade.

We know D: Scores below the top 75% and above the bottom 9%

If D is below top 75% and above bottom 9% , then percentage of D = 100 - ( 75 + 9) = 100 - 84 = 16%.

Now find z scores for both limits of D grade.

That is: Find z value for d₁.

We have: P( X < d₁ ) = 9% = 0.0900

Thus find z value such that:

P( Z < z) =0.0900

Look in z table for area = 0.0900 or its closest area and find z value.

Area 0.0901 is closest to 0.0900 and it corresponds to -1.3 and 0.04

Thus z = -1.34

Now use following formula to find value of d₁:

Now to find d₂ , find corresponding z value.

We have:

P( X < d₂) = 9% + 16%

P( X < d₂) = 25%

P( X < d₂) = 0.2500

Thus find z value such that:

P( Z < z ) = 0.2500

Look in z table for area = 0.2500 or its closest area and find z value.

Area 0.2514 closest to 0.2500 and it corresponds to -0.6 and 0.07

thus z = -0.67

Now use following formula to find value of d₂:

Thus the numerical limits for a D grade are : 55 and 61.