A technology company evaluated the annual performance scores of its employees across three… — ACT

A technology company evaluated the annual performance scores of its employees across three departments: Marketing, Engineering, and Customer Support.

- The Marketing department has $20$ employees, with a mean performance score of $92$ .
- The Engineering department has twice as many employees as the Marketing department, with a mean performance score of $86$ .
- The Customer Support department has $10$ more employees than the Marketing department.
- The overall mean performance score across all employees in all three departments combined is $84$ .

What is the mean performance score for the Customer Support department?

A
$60$
B
$74$
$76$ Answer
D
$79$
E
$84$

Answer

The mean performance score for the Customer Support department is 76.

To find the mean score for Customer Support, calculate the total points across all departments (

90 \times 84 = 7,560

) and subtract the known total points from Marketing (

20 \times 92 = 1,840

) and Engineering (

40 \times 86 = 3,440

). The remaining

2,280

points are divided by the

30

Customer Support employees, yielding a mean of

76

Step-by-Step Solution

Determine the number of employees in each department and the total employee count.

Marketing has

20

employees. Engineering has

2 \times 20 = 40

employees. Customer Support has

20 + 10 = 30

employees. Total employees =

20 + 40 + 30 = 90

Accurate employee counts per group are required to calculate weighted sums.

Calculate the total combined performance score for all 90 employees.

\text{Total Score} = 90 \times 84 = 7,560

The sum of all individual scores is equal to the overall mean multiplied by the total number of employees.

Calculate the total performance scores for Marketing and Engineering.

Marketing total =

20 \times 92 = 1,840

. Engineering total =

40 \times 86 = 3,440

. Combined Marketing + Engineering total =

1,840 + 3,440 = 5,280

Each department's score contribution is its employee count multiplied by its mean score.

Subtract the Marketing and Engineering scores from the total combined score to find the Customer Support total score.

Customer Support total score =

7,560 - 5,280 = 2,280

The remaining score points belong to the Customer Support department.

Divide the Customer Support total score by the number of Customer Support employees.

Customer Support mean score =

\frac{2,280}{30} = 76

The mean is the total department score divided by its employee count.

Key Concept

Weighted Mean and Subgroup Summaries

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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