Question
An analysis of the monthly incentives received by 5 salesmen : The mean and median of the incentives is $7000. The only mode among the observations is $12,000. Incentives paid to each salesman were in full thousands. What is the difference between the highest and the lowest incentive received by the 5 salesmen in the month?
A. $4000
B. $13000
C. $9000
D. $5000
E. $11000
Correct Answer Choice E. $11,000
Explanatory Answer
The arithmetic mean of 5 observations is $7000.
Therefore, the sum of these 5 observations is 5 * 7000 = $35,000.
The median is $7000.
Let us say their incentives in ascending order are a, b, c, d and e.
So, c = 7000 and a + b + c + d + e = $35,000
The only mode is $12,000. So, the maximum number of observations among the 5 will be in $12,000.
The only possibility is that both d and e got an incentive of $12,000 each.
So, c + d + e = 7000 + 12000 + 12000 = $31,000
Therefore, a + b = 35,000 – 31,000 = 4000.
a and b have to be two different values as the only mode is $12,000.
So, a has to be $1000 and b has to be $3000.
The difference between the highest and the lowest is therefore 12,000 – 1000 = $11,000
I think the question should be maximum possible range of these five employees since they could get 3000, 3000 or 1000, 5000.
3000, 3000 is not a possibility as the question states that there is only one mode. If two employees get 3000, then the mode will not be a single mode.
Although not an option to choose, but I think $12000 is also possible. This would mean that salesmen A receives $0 incentive and salesmen B $4000.