Question

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

A: Top 8% of scores

B: Scores below the top 8% and above the bottom 58%

C: Scores below the top 42% and above the bottom 22%

D: Scores below the top 78% and above the bottom 7%

F: Bottom 7% of scores

Scores on the test are normally distributed with a mean of 77.1 and a standard deviation of 7.4.

Find the minimum score required for an A grade. Round your answer to the nearest whole number, if necessary.

Answer 1

Here we have = 77.1, = 7.4

A: Top 8% of scores

p ( Z > z ) = 0.08

p ( Z > z ) = 1- p ( Z z ) = 0.08

p ( Z z ) = 0.92

( We use excel formula "=norm.s.inv(0.92)" )

z = 1.41

x = 87.53

Minimum score required for an A grade is 87.53.

B: Below the top 8% and above the bottom 58%:

Here we need to find values a and b.

P ( Z < z ) = 0.58

( We use excel formula "=norm.s.inv(0.58)" )

We get z = 0.20

x_a =78.58

And

p ( Z > z ) = 0.08

p ( Z > z ) = 1 - p ( Z z ) = 0.08

p ( Z z ) = 1 - 0.08 = 0.92

( We use excel formula "=norm.s.inv(0.92)" )

z = 1.41

x_b = 87.53

C: Scores below the top 42% and above the bottom 22%

Here we need to find values a and b.

P ( Z < z ) = 0.22

( We use excel formula "=norm.s.inv(0.22)" )

We get z = -0.77

x_a =71.40

And

p ( Z > z ) = 0.42

p ( Z > z ) = 1 - p ( Z z ) = 0.42

p ( Z z ) = 1 - 0.42 = 0.58

( We use excel formula "=norm.s.inv(0.58)" )

z = 0.20

x_b = 78.58

D: Scores below the top 78% and above the bottom 7%

Here we need to find values a and b.

P ( Z < z ) = 0.07

( We use excel formula "=norm.s.inv(0.07)" )

We get z = -1.48

x_a =66.15

And

p ( Z > z ) = 0.78

p ( Z > z ) = 1 - p ( Z z ) = 0.78

p ( Z z ) = 1 - 0.78 = 0.22

( We use excel formula "=norm.s.inv(0.22)" )

z = -0.77

x_b = 71.40

F: Bottom 7% of scores

Here we need to find values a.

P ( Z < z ) = 0.07

( We use excel formula "=norm.s.inv(0.07)" )

We get z = -1.48

x_a =66.15