Question

(d) when the mean of the salaries of a company is $34,000 and all eligible employees are given a $2500 raise, what happens to the mean and median of the salaries? the mean would select , and the median would select .

Explanation:

Step1: Recall the formula for the mean

The mean (average) of a set of values \( x_1, x_2, \dots, x_n \) is given by \( \bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n} \). Let the original salaries be \( x_1, x_2, \dots, x_n \) with mean \( \bar{x}=\$34000 \), so \( \sum_{i = 1}^{n}x_i=n\times34000 \). After a raise of \( \$2500 \) for each employee, the new salaries are \( y_i=x_i + 2500 \) for each \( i \). The new mean \( \bar{y}=\frac{\sum_{i = 1}^{n}y_i}{n}=\frac{\sum_{i = 1}^{n}(x_i + 2500)}{n}=\frac{\sum_{i = 1}^{n}x_i+\sum_{i = 1}^{n}2500}{n}=\frac{n\times34000 + 2500n}{n}=34000 + 2500 \). So the mean increases by \( \$2500 \).

Step2: Recall the definition of the median

The median is the middle value (or the average of the two middle values) when the data is ordered. If we order the original salaries \( x_1\leq x_2\leq\dots\leq x_n \), the median \( M \) is the middle value. After adding \( 2500 \) to each salary, the new ordered salaries are \( x_1 + 2500\leq x_2 + 2500\leq\dots\leq x_n + 2500 \). The middle value of the new set is \( M+ 2500 \), so the median also increases by \( \$2500 \).

Answer:

The mean would increase by $2500, and the median would increase by $2500.