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Difficulty: MediumDescriptive Statistics and Data Representations

At a regional debate tournament, Team Alpha has 66 members with a mean score of 8484 points, and Team Beta has 44 members with a mean score of 9494 points. If 22 new members join Team Alpha such that the overall mean score for all 1212 members on both teams increases to 8989 points, what is the mean score of the 22 new members?

  1. A
    89
  2. B
    90
  3. 94Answer
  4. D
    104
  5. E
    188

Answer

The mean score of the 2 new members is 94.
To find the mean score of the 2 new members, first determine the total points scored by the initial 10 members. Team Alpha scored 6×84=5046 \times 84 = 504 points, and Team Beta scored 4×94=3764 \times 94 = 376 points, giving an initial combined total of 880880 points. Next, determine the total points required for all 12 members to have a mean score of 89, which is 12×89=106812 \times 89 = 1068 points. The difference, 1068−880=1881068 - 880 = 188 points, represents the total points scored by the 2 new members. Dividing this sum by 2 yields a mean score of 94 points.

Step-by-Step Solution

1
Calculate the total initial points for both teams combined.
Team Alpha total = 6×84=5046 \times 84 = 504. Team Beta total = 4×94=3764 \times 94 = 376. Combined initial total = 504+376=880504 + 376 = 880 points.
The sum of data values equals the mean multiplied by the number of values.
2
Calculate the required total points for all 12 members to achieve the target overall mean.
Target total points = 12×89=106812 \times 89 = 1068 points.
The total points required for 1212 members with a mean of 8989 is 12×8912 \times 89.
3
Determine the total points contributed by the 2 new members.
Points contributed = 1068−880=1881068 - 880 = 188 points.
Subtracting the initial combined points from the required total points isolates the points added by the 22 new members.
4
Calculate the mean score of the 2 new members.
Mean score = 188÷2=94188 \div 2 = 94 points.
Dividing the total points of the new members by the count of new members (22) yields their mean score.

Key Concept

Weighted Mean and Target Mean Problem Solving
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Topics

Descriptive Statistics and Data Representations

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