Question

(e) when the mean and median of the salaries of the employees of a company are $34,000 and all employees are given a 5% raise, what happens to the mean and median of the salaries? the mean would select, and the median would select.

Explanation:

Step1: Analyze the effect on the mean

Let the original salaries be \( x_1, x_2, \dots, x_n \). The original mean \( \bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}=34000 \). After a 5% raise, each salary becomes \( x_i'=x_i\times(1 + 0.05)=1.05x_i \). The new mean \( \bar{x}'=\frac{\sum_{i = 1}^{n}x_i'}{n}=\frac{\sum_{i = 1}^{n}1.05x_i}{n}=1.05\times\frac{\sum_{i = 1}^{n}x_i}{n}=1.05\times\bar{x} \). So the mean increases by 5%.

Step2: Analyze the effect on the median

Let the original median (for a sorted list \( x_{(1)}\leq x_{(2)}\leq\dots\leq x_{(n)} \)) be \( M \). After a 5% raise, the sorted list becomes \( 1.05x_{(1)}\leq1.05x_{(2)}\leq\dots\leq1.05x_{(n)} \). The new median is \( 1.05M \), so the median also increases by 5%.

Answer:

The mean would increase by 5% (or be multiplied by 1.05), and the median would increase by 5% (or be multiplied by 1.05).