Question

An English professor assigns letter grades on a test according to the following scheme. A: Top 5% of scores B: Scores below the top 5% and above the bottom 64% C: Scores below the top 36% and above the bottom 23% D: Scores below the top 77% and above the bottom 9% F: Bottom 9% of scores Scores on the test are normally distributed with a mean of 78.8 and a standard deviation of 7.1. Find the numerical limits for a D grade. Round your answers to the nearest whole number, if necessary.

Answer 1

Grade D score:

`69 \leq x \leq 74`

Explanation:

We are given the following information in the question:

Mean, μ = 78.8

Standard Deviation, σ = 7.1

We are given that the distribution of scores on test is a bell shaped distribution that is a normal distribution.

Formula:

`z_(score) = \displaystyle(x-\mu)/(\sigma)`

D: Scores below the top 77% and above the bottom 9%

We have to find the value of x such that the probability is 0.77

`P( X > x) = P( z > \displaystyle(x - 78.8)/(7.1))=0.77`

`= 1 -P( z \leq \displaystyle(x - 78.8)/(7.1))=0.77`

`=P( z \leq \displaystyle(x - 78.8)/(7.1))=0.23`

Calculation the value from standard normal z table, we have,

`\displaystyle(x - 78.8)/(7.1) =-0.739\\\\x = 73.55`

We have to find the value of x such that the probability is 0.09

`P(X < 0.09) = \\\\P( X < x) = P( z < \displaystyle(x - 78.8)/(7.1))=0.09`

Calculation the value from standard normal z table, we have,

`\displaystyle(x - 78.8)/(7.1) = -1.341\\\\x = 69.27`

Thus, the numerical value of score to achieve grade D is

`69 \leq x \leq 74`