Final averages are typically approximately normally distributed with a mean of 72 and a standard deviation of 12.5. Your professor says that the top 8% of the class will receive an A, the next 20% a B, the next 42% a C, the next 18% a D, and the bottom 12% an F.

a. What average must you exceed to obtain an A?
b. What average must you exceed to receive a grade better than a C?
(By the way, we won’t follow this rule in this class, we’ll follow the syllabus instead).

What rule?

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.08A, .08+.20>C) and related Z scores. Insert in equation below to find raw score.

Z = (score-mean)/SD

To determine the average you must exceed to obtain a particular grade, we can use z-scores and the standard normal distribution.

a. To obtain an A, you need to be in the top 8% of the class. We can find the z-score corresponding to the top 8% by using the z-table or a calculator. The z-score is the number of standard deviations away from the mean a particular value is. We need to find the value of z such that P(Z > z) = 0.08.

Using the z-table or a calculator, we find that the z-score corresponding to a cumulative probability of 0.92 (1 - 0.08) is approximately 1.405. We can use the formula z = (x - μ) / σ to convert this z-score to the corresponding average:
1.405 = (x - 72) / 12.5

Solving for x (the average), we get:
x - 72 = 1.405 * 12.5
x = 72 + 1.405 * 12.5
x ≈ 90.56

Therefore, to obtain an A, you must exceed an average of approximately 90.56.

b. To receive a grade better than a C, you need to be in the top 8% + 20% = 28% of the class. We can find the z-score corresponding to the top 28% by using the z-table or a calculator. The z-score is the number of standard deviations away from the mean a particular value is. We need to find the value of z such that P(Z > z) = 0.28.

Using the z-table or a calculator, we find that the z-score corresponding to a cumulative probability of 0.72 (1 - 0.28) is approximately 0.608. We can use the formula z = (x - μ) / σ to convert this z-score to the corresponding average:
0.608 = (x - 72) / 12.5

Solving for x (the average), we get:
x - 72 = 0.608 * 12.5
x = 72 + 0.608 * 12.5
x ≈ 79.6

Therefore, to receive a grade better than a C, you must exceed an average of approximately 79.6.

To answer these questions, we need to find the corresponding cutoff scores for each grade category based on the given distribution.

a. To obtain an A, you need to be in the top 8% of the class. Since we are given that the final averages are approximately normally distributed with a mean of 72 and a standard deviation of 12.5, we can use the Z-score formula to determine the cutoff score.

The formula for calculating the Z-score is given by:
Z = (X - μ) / σ

Where:
Z = Z-score
X = Raw score (average in this case)
μ = Mean
σ = Standard deviation

First, we need to find the Z-score that corresponds to the top 8% of the distribution. We can look up this Z-score in a standard normal distribution table or use statistical software.

Using a standard normal distribution table or a Z-score calculator, we find that the Z-score corresponding to the top 8% is approximately 1.405. Now we can use the Z-score formula to solve for the average (X):

1.405 = (X - 72) / 12.5

Solving for X:
X - 72 = 1.405 * 12.5
X - 72 = 17.56
X = 72 + 17.56
X ≈ 89.56

Therefore, to obtain an A, you must exceed an average of approximately 89.56.

b. To receive a grade better than a C, we need to be in the top 50% (i.e., not in the bottom 50%). We can find the cutoff score for this category by calculating the Z-score corresponding to the 50th percentile.

The Z-score for the 50th percentile is 0. Using the Z-score formula:

0 = (X - 72) / 12.5

Solving for X:
X - 72 = 0
X = 72

Therefore, to receive a grade better than a C, you must exceed an average of 72.