Final averages are typically approximately normally distributed with a mean of 66 and a standard deviation of 10.5. Your professor says that the top 5% of the class will receive an A; the next 15%, a B; the next 47%, a C; the next 22%, a D; and the bottom 11%, an F.
(a) What average must you exceed to obtain an A?
(b) What average must you exceed to receive a grade of C or better?
(c) What average must you obtain to pass the course? (You'll need a D or better.)
To solve these questions, we need to refer to the concept of the standard normal distribution and the z-score.
The z-score is a measure of how many standard deviations a data point is from the mean. It can be calculated using the formula:
z = (x - μ) / σ
Where:
x is the value we want to find the z-score for,
μ is the mean of the distribution, and
σ is the standard deviation of the distribution.
To obtain the A grade, we need to find the z-score corresponding to the top 5%. In other words, we want to find the z-score that separates the top 5% of the distribution from the rest.
(a) To find the average required to obtain an A, we can use the z-table or a calculator to look up the z-score that corresponds to the cumulative area of 0.95 (since we need the top 5%). Let's call this z_a. We can then solve for x using the formula:
z_a = (x - μ) / σ
Substituting the given values:
z_a = (x - 66) / 10.5
From the z-table, we find the z-score that corresponds to the cumulative area of 0.95 is approximately 1.645.
1.645 = (x - 66) / 10.5
Solving for x:
1.645 * 10.5 = x - 66
17.2275 = x - 66
x = 66 + 17.2275
x ≈ 83.23
Therefore, you must exceed approximately 83.23 to obtain an A grade.
(b) To receive a grade of C or better, we need to find the average that corresponds to the cumulative area of 0.47 + 0.15 = 0.62. Let's call this z_c. Using a similar approach as in part (a):
z_c = (x - 66) / 10.5
From the z-table, we find the z-score that corresponds to a cumulative area of 0.62 is approximately 0.32.
0.32 = (x - 66) / 10.5
Solving for x:
0.32 * 10.5 = x - 66
3.36 = x - 66
x = 66 + 3.36
x ≈ 69.36
Therefore, you must exceed approximately 69.36 to receive a grade of C or better.
(c) To pass the course (D or better), we need to find the average that corresponds to the cumulative area of 1 - 0.11 = 0.89. Let's call this z_d. Using a similar approach:
z_d = (x - 66) / 10.5
From the z-table, we find the z-score that corresponds to a cumulative area of 0.89 is approximately 1.23.
1.23 = (x - 66) / 10.5
Solving for x:
1.23 * 10.5 = x - 66
12.915 = x - 66
x = 66 + 12.915
x ≈ 78.915
Therefore, you must obtain approximately 78.915 to pass the course with a D or better.