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30 Mar 2024, 22:36
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
(N/A)
Question Stats:
33% (01:48) correct
66% (01:51) wrong based on 3 sessions
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Each employee of a certain company is in either Department X or Department Y, and there are more than twice as many employees in Department X as in Department Y. The average (arithmetic mean) salary is $25,000 for the employees in Department X and $35,000 for the employees in Department Y. Which of the following amounts could be the average salary for all of the employees of the company?
Indicate all such amounts.
A. $26,000
B. $28,000
C. $29,000
D. $30,000
E. $31,000
F. $32,000
G. $34,000
Indicate all such amounts.
A. $26,000
B. $28,000
C. $29,000
D. $30,000
E. $31,000
F. $32,000
G. $34,000
12 Apr 2024, 16:20
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Let's use weighted averages
Weighted average of groups combined = (group A proportion)(group A average) + (group B proportion)(group B average) + (group C proportion)(group C average) + ...
There are more than twice as many employees in Department X as in Department Y
Let's see what happens if there are EXACTLY twice as many employees in Department X as in Department Y
So, for every 3 employees, 2 work for Department X and 1 works for Department Y
In other words, 2/3 of the employees work for Department X
And 1/3 of the employees work for Department Y
The average (arithmetic mean) salary is $25,000 for the employees in Department X and $35,000 for the employees in Department Y
So, the average salary of ALL employees = (2/3)($25,000) + (1/3)($35,000)
≈ $28,333
So, the average salary of all employees = $28,333 if there are EXACTLY twice as many employees in Department X as in Department Y
Now let's see what happens if we MAXIMIZE the number of employees working for Department X
Let's say that ALL of the employees work for Department X
So, 1/1 employees work for Department X, and 0/1 employees work for Department Y
Now, the average salary of ALL employees = (1/1)($25,000) + (0/1)($35,000)
≈ $25,000
So, the average salary of all employees = $25,000 if there ALL of the employees work in Department X
Now that we've examined the EXTREME cases, we know that the average salary of all employees is BETWEEN $25,000 and $28,333
Answer: A, B
Weighted average of groups combined = (group A proportion)(group A average) + (group B proportion)(group B average) + (group C proportion)(group C average) + ...
There are more than twice as many employees in Department X as in Department Y
Let's see what happens if there are EXACTLY twice as many employees in Department X as in Department Y
So, for every 3 employees, 2 work for Department X and 1 works for Department Y
In other words, 2/3 of the employees work for Department X
And 1/3 of the employees work for Department Y
The average (arithmetic mean) salary is $25,000 for the employees in Department X and $35,000 for the employees in Department Y
So, the average salary of ALL employees = (2/3)($25,000) + (1/3)($35,000)
≈ $28,333
So, the average salary of all employees = $28,333 if there are EXACTLY twice as many employees in Department X as in Department Y
Now let's see what happens if we MAXIMIZE the number of employees working for Department X
Let's say that ALL of the employees work for Department X
So, 1/1 employees work for Department X, and 0/1 employees work for Department Y
Now, the average salary of ALL employees = (1/1)($25,000) + (0/1)($35,000)
≈ $25,000
So, the average salary of all employees = $25,000 if there ALL of the employees work in Department X
Now that we've examined the EXTREME cases, we know that the average salary of all employees is BETWEEN $25,000 and $28,333
Answer: A, B
23 Jun 2024, 11:30
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Official Explanation
One strategy for answering this kind of question is to find the least and/or greatest possible value. Clearly the average salary is between $25,000 and $35,000, and all of the answer choices are in this interval. Since you are told that there are more employees with the lower average salary, the average salary of all employees must be less than the average of $25,000 and $35,000, which is $30,000. If there were exactly twice as many employees in Department X as in Department Y, then the average salary for all employees would be, to the nearest dollar, the following weighted mean,
\(\frac{(2)(25000)+(1)(35000)}{2+1} ≈ 28333\) dollars
where the weight for $25,000 is 2 and the weight for $35,000 is 1. Since there are more than twice as many employees in Department X as in Department Y, the actual average salary must be even closer to $25,000 because the weight for $25,000 is greater than 2. This means that $28,333 is the greatest possible average. Among the choices given, the possible values of the average are therefore $26,000 and $28,000.
Thus, the correct answer consists of Choices A ($26,000) and B ($28,000).
Intuitively, you might expect that any amount between $25,000 and $28,333 is a possible value of the average salary. To see that $26,000 is possible, in the weighted mean above, use the respective weights 9 and 1 instead of 2 and 1. To see that $28,000 is possible, use the respective weights 7 and 3.
One strategy for answering this kind of question is to find the least and/or greatest possible value. Clearly the average salary is between $25,000 and $35,000, and all of the answer choices are in this interval. Since you are told that there are more employees with the lower average salary, the average salary of all employees must be less than the average of $25,000 and $35,000, which is $30,000. If there were exactly twice as many employees in Department X as in Department Y, then the average salary for all employees would be, to the nearest dollar, the following weighted mean,
\(\frac{(2)(25000)+(1)(35000)}{2+1} ≈ 28333\) dollars
where the weight for $25,000 is 2 and the weight for $35,000 is 1. Since there are more than twice as many employees in Department X as in Department Y, the actual average salary must be even closer to $25,000 because the weight for $25,000 is greater than 2. This means that $28,333 is the greatest possible average. Among the choices given, the possible values of the average are therefore $26,000 and $28,000.
Thus, the correct answer consists of Choices A ($26,000) and B ($28,000).
Intuitively, you might expect that any amount between $25,000 and $28,333 is a possible value of the average salary. To see that $26,000 is possible, in the weighted mean above, use the respective weights 9 and 1 instead of 2 and 1. To see that $28,000 is possible, use the respective weights 7 and 3.
