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Difficulty: MediumWeighted Average and Combined Sets

A university department offers a specialized statistics course to two groups of students: undergraduate students and graduate students. A total of 120120 undergraduate students took the final examination, scoring an average of 8080 points. The remaining students in the course were graduate students, who achieved an average score of 9595 points on the same examination. If the combined average score for all students in the course was 8686 points, how many graduate students were enrolled in the course?

  1. A
    72
  2. 80Answer
  3. C
    90
  4. D
    120
  5. E
    180

Answer

80 graduate students were enrolled in the course.
The correct answer is obtained by applying the weighted average relationship: Nundergrad×(Meancombined−Meanundergrad)=Ngrad×(Meangrad−Meancombined)N_{\text{undergrad}} \times (\text{Mean}_{\text{combined}} - \text{Mean}_{\text{undergrad}}) = N_{\text{grad}} \times (\text{Mean}_{\text{grad}} - \text{Mean}_{\text{combined}}). Substituting the given values gives 120×(86−80)=Ngrad×(95−86)120 \times (86 - 80) = N_{\text{grad}} \times (95 - 86), which simplifies to 720=9Ngrad720 = 9 N_{\text{grad}}, yielding Ngrad=80N_{\text{grad}} = 80.

Step-by-Step Solution

1
Determine the distances of each group's mean score from the combined mean score.
Undergraduate distance: 86−80=686 - 80 = 6; Graduate distance: 95−86=995 - 86 = 9.
In a weighted average of two groups, the total deviation below the combined mean must equal the total deviation above the combined mean.
2
Set up the balance equation using the group counts and their respective distances from the combined mean.
120×6=N×9120 \times 6 = N \times 9, where NN represents the number of graduate students.
The weight of each group is inversely proportional to the distance of its mean from the overall combined mean.
3
Solve the linear equation for NN.
720=9N  ⟹  N=80720 = 9N \implies N = 80.
Dividing the total undergraduate score deviation (720720) by the graduate score deviation per student (99) gives the number of graduate students.

Key Concept

Weighted Average of Two Combined Sets
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Topics

Weighted Average and Combined Sets

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