Let’s start by solving this step-by-step, breaking down what we know and what is being asked.

Step 1: Understand the Given Information

• Mean salary of 5 employees: $41,400. This means:

  $\frac{\text{Sum of all salaries}}{5} = 41,400$

  So, the sum of all salaries is:

  $\text{Sum of salaries} = 41,400 \times 5 = 207,000$

• Median salary: $39,400. Since there are 5 salaries, the median is the third salary when the salaries are arranged in ascending order.

• Lowest salary: $31,500. After the raise of $3,700, the new salary for the lowest-paid employee will be:

  $31,500 + 3,700 = 35,200$

Step 2: Determine the Salaries

Let’s assume the 5 salaries in ascending order are $a \leq b \leq c \leq d \leq e$. Based on the given information:

• a (lowest salary) = $31,500 before the raise.
• After the raise, $a = 35,200$.
• c (the third salary) = $39,400 since it is the median.

Step 3: Calculate the New Mean

The new sum of the salaries will be the old sum plus the raise:

$\text{New Sum of Salaries} = 207,000 + 3,700 = 210,700$

Now, calculate the new mean:

$\text{New Mean} = \frac{\text{New Sum of Salaries}}{5} = \frac{210,700}{5} = 42,140$

So, the new mean is $42,140.

Step 4: Determine the New Median

The order of the salaries changes after the raise. After raising the lowest salary from $31,500 to $35,200, the new order of salaries is:

$35,200 \leq b \leq 39,400 \leq d \leq e$

Since the number of salaries is still 5, the median remains the third salary:

$\text{New Median} = 39,400$

So, the new median remains $39,400.

Final Answers

a) New Mean = $42,140
b) New Median = $39,400

Would you like more details or have any questions about this solution? Here are some related questions you might find useful:

1. How does a raise affect the mean of a dataset?
2. What is the difference between mean and median?
3. How would the median change if there was an even number of employees?
4. Why does the raise affect the mean but not necessarily the median?
5. How can the mean and median be used to describe the distribution of salaries?

Tip: When working with median values, it's essential to consider the ordered list of values and how additions or changes might shift the middle value.