The question is a Problem Solving question on Mean, Median and Mode - Descriptive Statistics.
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?
The correct choice is (E) and the correct answer is $11,000
The arithmetic mean of the incentives of the 5 salesmen is $7000.
Therefore, the sum of the incentives received by all 5 taken together is 5 * 7000 = $35,000.
The median for these 5 observations 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 among the 5 observations is $12,000.
So, the maximum number of observations among the 5 will be in $12,000.
We have deduced that c has got $7000.
Hence, the only possibility is that both d and e got an incentive of $12,000 each.
So, incentives received by 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 incentives is, therefore 12,000 - 1000 = $11,000