Question

An English professor assigns letter grades on a test according to the following scheme. A: Top 13% of scores B: Scores below the top 13% and above the bottom 61% C: Scores below the top 39% and above the bottom 20% D: Scores below the top 80% and above the bottom 10% F: Bottom 10% of scores Scores on the test are normally distributed with a mean of 72.8 and a standard deviation of 7.3. Find the numerical limits for a D grade. Round your answers to the nearest whole number, if necessary.

Answer 1

The test is distributed normally with mean of 72.8 and the standard deviation of 7.3

Finding numerical limits for the D grade.

D grade : Scores below the top 80% and above the bottom 10%.

Let the bottom limit for D grade be
`$D_1$` and the top limit for D grade be
`$D_2$`.

First find the bottom numerical limit for a D grade is :

`$P(X<D_1)= 0.10$`

`$P(X\leq D_1)= 0.10$`

`$P\left((X-\mu)/(\sigma) \leq (D_1-\mu)/(\sigma)\right) = 0.10$`

`$P\left(Z \leq (D_1-72.8)/(7.3)\right) = 0.10$` ..........(1)

From (1)

`$(D_1 - 72.8)/(7.3) = -1.28$`

`$D_1 = -1.28(7.3)+72.8$`

= 63.45

≈ 64

Now the top numerical limit for D grade :

`$P(X>D_2)= 0.80$`

`$1-P(X\leq D_2)= 0.80$`

`$P(X\leq D_2)= 1-0.80$`

`$P(X\leq D_2)= 0.20$`

`$P\left((X-\mu)/(\sigma) \leq (D_2-\mu)/(\sigma)\right) = 0.20$`

`$P\left(Z \leq (D_2-72.8)/(7.3)\right) = 0.20$` ..........(2)

From (2)

`$(D_2- 72.8)/(7.3) = -0.84$`

`$D_12= -0.84(7.3)+72.8$`

= 66.668

≈ 67

Therefore, the numerical limit for a D grade is 64 to 67.