Question

A test is given and the average (μ_X) score out of 100 was only a 53.1, with a SD (σ_X) of 8.9. Assuming the grades followed a normal distribution, use the Z table or Excel and formulas to find the dividing line (test scores) between the A's, B's, C's, D's, and E's. Starting from the top, the teacher will give the highest 15% A's, the next 10% B's, the next 30% C's, the next 25% D's, and the bottom 20% will receive E's.

Answer 1

Given Mean x = 53.1

Standard Deviation x = 8.9

We need to find the dividing line (test scores) between the A's, B's, C's, D's, and E's

Test score that divides the 15% A's from the rest that are lower than A's

We will find the test score that dvides the highest 15% A's from the rest that are lower than A's

Given that the top 15% will be given A's

In our z-tables we have the area to left of the z-score. So here 85% will be to the left or below the test-score that divides A's from the rest

So we need to find a z-score that has an area of 0.85 to its left from the below attached positive z-table

The area to the left of z-score 1.03 is 0.84849 and to the left of z-score 1.04 is 0.85083

The exact z-score for which the area to the left of it is 0.85 can be found using online calculator = 1.03643

We know that

Z-score = (X - x) / x

1.03643 = (X - 53.1) / 8.9

X = 1.03643 * 8.9 + 53.1

X = 62.32423

So a test score of 62.32423 divides the 15% A's from the rest that are lower than A's

Test score that divides the next 10% B's from the rest that are lower than B's and higher than B's

We will find the test score that dvides the next 10% B's from the rest that are lower than B's and higher than B's

Given that the top 15% will be given A's and next 10% will be B's

In our z-tables we have the area to left of the z-score. So here 75% will be to the left or below the test-score that divides B''s from the rest that are lower than B's and higher than B's

So we need to find a z-score that has an area of 0.75 to its left from the below attached positive z-table

The area to the left of z-score 0.67 is 0.74857 and to the left of z-score 0.68 is 0.75175

The exact z-score for which the area to the left of it is 0.75 can be found using online calculator = 0.67449

We know that

Z-score = (X - x) / x

0.67449 = (X - 53.1) / 8.9

X = 0.67449 * 8.9 + 53.1

X = 59.10296

So a test score of 59.10296 divides the 10% B's from the rest that are lower than B's and higher than B's

Test score that divides the next 30% C's from the rest that are lower than C's and higher than C's

We will find the test score that dvides the next 30% C's from the rest that are lower than C's and higher than C's

Given that the top 15% will be given A's and next 10% will be B's and next 30% will be C's

In our z-tables we have the area to left of the z-score. So here 45% will be to the left or below the test-score that divides C's from the rest that are lower than C's and higher than C's

So we need to find a z-score that has an area of 0.45 to its left from the below attached negative z-table

The area to the left of z-score -0.12 is 0.45224 and to the left of z-score -0.13 is 0.44828

The exact z-score for which the area to the left of it is 0.45 can be found using online calculator = -0.12566

We know that

Z-score = (X - x) / x

-0.12566 = (X - 53.1) / 8.9

X = -0.12566 * 8.9 + 53.1

X = 51.98163

So a test score of 51.98163 divides the 30% C's from the rest that are lower than C's and higher than C's

Test score that divides the next 25% D's from the rest that are lower than D's and higher than D's

We will find the test score that dvides the next 25% D's from the rest that are lower than D's and higher than D's

Given that the top 15% will be given A's and next 10% will be B's ,next 30% will be C's and next 25% will be D's

In our z-tables we have the area to left of the z-score. So here 20% will be to the left or below the test-score that divides D's from the rest that are lower than D's and higher than D's

So we need to find a z-score that has an area of 0.2 to its left from the below attached negative z-table

The area to the left of z-score -0.84 is 0.20045 and to the left of z-score -0.85 is 0.19766

The exact z-score for which the area to the left of it is 0.45 can be found using online calculator = -0.84162

We know that

Z-score = (X - x) / x

-0.84162 = (X - 53.1) / 8.9

X = -0.84162 * 8.9 + 53.1

X = 45.60958

So a test score of 45.60958 divides the 25% D's from the rest that are lower than D's and higher than D's

In short rounding to 2 decimals,

The test score that Seperates A's and B's is 62.32

The test score that Seperates B's and C's is 59.10

The test score that Seperates C's and D's is 51.98

The test score that Seperates D's and E's is 45.61