Question

An English professor assigns letter grades on a test according to the following scheme. A: Top 15% of scores B: Scores below the top 15% and above the bottom 55% C: Scores below the top 45% and above the bottom 19% D: Scores below the top 81% and above the bottom 6% F: Bottom 6% of scores Scores on the test are normally distributed with a mean of 74.2 and a standard deviation of 9.6. Find the numerical limits for a D grade. Round your answers to the nearest whole number, if necessar

Answer 1

Answer:

59 to 66

Step-by-step explanation:

Mean test scores = u = 74.2

Standard Deviation = $\sigma$ = 9.6

According to the given data, following is the range of grades:

Grade A: 85% to 100%

Grade B: 55% to 85%

Grade C: 19% to 55%

Grade D: 6% to 19%

Grade F: 0% to 6%

So, the grade D will be given to the students from 6% to 19% scores. We can convert these percentages to numerical limits using the z scores. First we need to to identify the corresponding z scores of these limits.

6% to 19% in decimal form would be 0.06 to 0.19. Corresponding z score for  0.06 is -1.56 and that for 0.19 is -0.88 (From the z table)

The formula for z score is:

$z=\frac{x-u}{\sigma}$

For z = -1.56, we get:

$-1.56=\frac{x-74.2}{9.6}\\\\ x = 59$

For z = -0.88, we get:

$-0.88=\frac{x-74.2}{9.6}\\\\ x = 66$

Therefore, a numerical limits for a D grade would be from 59 to 66 (rounded to nearest whole numbers)