Question

Cheever College offers several online courses via remote computer connection, in addition to traditional classroom-based courses. A study of student performance at Cheever found that, overall, the average student grade for online courses matched that for classroom-based courses. In this calculation of the average grade, course withdrawals were weighted as equivalent to a course failure, and the rate of withdrawal was much lower for students enrolled in classroom-based courses than for students enrolled in online courses.
If the statements above are true, which of the following must also be true of Cheever College?

• A. Among students who did not withdraw, students enrolled in online courses got higher grades, on average, than students enrolled in classroom-based courses.
• B. The number of students enrolled per course at the start of the school term is much higher, on average, for the online courses than for the classroom-based courses.
• C. There are no students who take both an online and a classroom-based course in the same school term.
• D. Among Cheever College students with the best grades, a significant majority take online, rather than classroom-based, courses.
• E. Courses offered online tend to deal with subject matter that is less challenging than that of classroom-based courses.

Hint:

For questions involving averages and percentages, it's often helpful to think of a simple numerical example. Suppose there are 100 students in each group. Classroom: 5 withdraw (5%). Online: 20 withdraw (20%). Let the overall average grade for both be 70. For Classroom: Avg Grade = (Avg of 95 finishers 95 + Avg of 5 withdrawers 5)/100. 70 = (A_C 95 + 0 5) / 100 -> A_C = 7000/95 \approx 73.7. For Online: Avg Grade = (Avg of 80 finishers 80 + Avg of 20 withdrawers 20)/100. 70 = (A_O 80 + 0 20) / 100 -> A_O = 7000/80 = 87.5. The example shows that A_O (87.5) > A_C (73.7).

Answer 1

The Correct Option is A

Step 1: Understanding the Premises
This is a logical deduction question based on statistics. Let's break down the information given:
Premise 1: Average Grade (Online) = Average Grade (Classroom).
Premise 2: The calculation of these averages includes withdrawals, which are counted as failures (let's say a grade of 'F' or 0).
Premise 3: Withdrawal Rate (Online) > Withdrawal Rate (Classroom). This means a higher percentage of the "grades" in the online average are failures due to withdrawal.

Step 2: Key Formula or Approach
Let \(A_{O}\) be the average grade for online students who completed the course, and \(W_{O}\) be the withdrawal rate (as a percentage).
Let \(A_{C}\) be the average grade for classroom students who completed the course, and \(W_{C}\) be the withdrawal rate.
The overall average grade is a weighted average of the grades of those who completed and those who withdrew (who get a grade of 0).
Overall Avg (Online) = \(A_{O} \times (1 - W_{O}) + 0 \times W_{O} = A_{O} \times (1 - W_{O})\).
Overall Avg (Classroom) = \(A_{C} \times (1 - W_{C}) + 0 \times W_{C} = A_{C} \times (1 - W_{C})\).
From Premise 1, we know:
\[ A_{O} \times (1 - W_{O}) = A_{C} \times (1 - W_{C}) \] From Premise 3, we know:
\[ W_{O} > W_{C} \] This implies that:
\[ (1 - W_{O}) < (1 - W_{C}) \]

Step 3: Detailed Explanation
We have the equation \( A_{O} \times (1 - W_{O}) = A_{C} \times (1 - W_{C}) \).
We are trying to compare \(A_{O}\) and \(A_{C}\), which are the average grades for the students who did not withdraw.
Let's rearrange the equation to solve for the ratio of \(A_{O}\) to \(A_{C}\):
\[ \frac{A_{O}}{A_{C}} = \frac{1 - W_{C}}{1 - W_{O}} \] Since we know that \( W_{O} > W_{C} \), the denominator \( (1 - W_{O}) \) is smaller than the numerator \( (1 - W_{C}) \).
When the numerator of a fraction is larger than the denominator (and both are positive), the value of the fraction is greater than 1.
Therefore, \( \frac{A_{O}}{A_{C}} > 1 \), which means \( A_{O} > A_{C} \).
This translates to: The average grade for online students who completed the course was higher than the average grade for classroom students who completed the course.

Step 4: Evaluating the Options
(A) This statement matches our logical deduction perfectly.
(B) We have no information about the absolute number of students, only about grades and withdrawal rates (percentages).
(C) The passage does not give any information to support this absolute claim.
(D) We only know about averages, not about the distribution of grades or the performance of the very top students.
(E) The passage gives no information about the difficulty of the subject matter. The difference in grades could be due to student motivation, teaching methods, etc. This is a possible explanation, but it is not something that must be true.