Question

a small accounting firm has 4 accountants who each earn a different salary between 52,000 dollars and 58,000 dollars. for extra help during tax season, they hire a 5th accountant who earns 10,000 dollars. how will hiring the 5th accountant affect the mean and median? choose 1 answer: a both the mean and median will decrease, but the median will decrease by more than the mean. b both the mean and median will decrease, but the mean will decrease by more than the median. c the mean will decrease, and the median will increase. d the mean will increase, and the median will decrease.

Explanation:

Step1: Recall mean and median concepts

Mean is sum of values divided by number of values. Median is middle - value when data is ordered.

Step2: Analyze the original data

Original 4 accountants have salaries between 52000 and 58000. Let the salaries be \(a,b,c,d\) where \(52000\leq a < b < c < d\leq58000\). The median of 4 values is \(\frac{b + c}{2}\), and the mean \(\bar{x}_1=\frac{a + b + c + d}{4}\).

Step3: Analyze the new data

After adding a 5th accountant with salary \(e = 10000\), the ordered data is \(e,a,b,c,d\). The median of 5 values is \(b\). The new mean \(\bar{x}_2=\frac{a + b + c + d+e}{5}\).
Since \(e = 10000\) is much smaller than the original salaries, the sum in the mean formula increases by a small amount (\(e\)) while the number of values increases by 1. So the mean decreases.
The original median was \(\frac{b + c}{2}\) and the new median is \(b\). Since \(b<\frac{b + c}{2}\) (because \(c>b\)), the median also decreases.
The mean is affected more by the out - lier value of 10000 as it is part of the sum used in the mean calculation. The median is just the middle value and is less affected by extreme values.

Answer:

B. Both the mean and median will decrease, but the mean will decrease by more than the median.