Of the 13 employees in a certain department, 1 has an annual salary of $38{,000}, 2 have an annual salary...

1. Translate the problem requirements: We need to find the median salary, which is the middle value when all 13 salaries are arranged in ascending order. Since we have 13 employees (odd number), the median will be the 7th value in the ordered list.
2. Organize salary data by frequency: Group the given salary information to understand how many employees earn each amount, making it easier to create an ordered list.
3. Arrange salaries in ascending order: Create a sequential list of all 13 salaries from lowest to highest to identify positions.
4. Locate the middle position: Find the 7th salary in the ordered list, which represents the median for our 13-employee dataset.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're looking for in everyday terms. The median is simply the "middle value" when we line up all the numbers from smallest to largest. Think of it like arranging 13 people by height - the median would be the height of the person standing exactly in the middle of the line.

Since we have 13 employees (which is an odd number), finding the middle is straightforward. We need to count to the 7th position because:

• There will be 6 employees with salaries below the median
• 1 employee at the median position (7th)
• 6 employees with salaries above the median

So our goal is to find the salary of the 7th employee when all 13 salaries are arranged from lowest to highest.

Process Skill: TRANSLATE - Converting the median concept into a clear positional requirement

2. Organize salary data by frequency

Let's organize the salary information we're given to make it easier to work with:

• 1 employee earning $38,000
• 2 employees earning $42,500 each
• 3 employees earning $40,000 each
• 2 employees earning $45,000 each
• 5 employees earning $48,500 each

Let's verify we have all 13 employees: \(1 + 2 + 3 + 2 + 5 = 13\) ✓

Now I'll rearrange this information from lowest salary to highest salary:

• $38,000: 1 employee
• $40,000: 3 employees
• $42,500: 2 employees
• $45,000: 2 employees
• $48,500: 5 employees

3. Arrange salaries in ascending order

Now let's create our ordered list by writing out each individual salary from position 1 to 13:

Position 1: $38,000 (the single employee at this salary)
Positions 2-4: $40,000, $40,000, $40,000 (the three employees at this salary)
Positions 5-6: $42,500, $42,500 (the two employees at this salary)
Positions 7-8: $45,000, $45,000 (the two employees at this salary)
Positions 9-13: $48,500, $48,500, $48,500, $48,500, $48,500 (the five employees at this salary)

To make this crystal clear, here's our complete ordered list:
1st: $38,000
2nd: $40,000
3rd: $40,000
4th: $40,000
5th: $42,500
6th: $42,500
7th: $45,000 ← This is our median!
8th: $45,000
9th: $48,500
10th: $48,500
11th: $48,500
12th: $48,500
13th: $48,500

4. Locate the middle position

Looking at our ordered list, the 7th position contains a salary of $45,000.

Let's double-check this makes sense:

• Positions 1-6 contain salaries of $38,000, $40,000, $40,000, $40,000, $42,500, $42,500
• Position 7 contains $45,000
• Positions 8-13 contain salaries of $45,000, $48,500, $48,500, $48,500, $48,500, $48,500

Perfect! We have exactly 6 salaries below $45,000 and 6 salaries above or equal to $45,000, with $45,000 in the middle position.

Final Answer

The median annual salary for the 13 employees is $45,000.

Looking at our answer choices, this corresponds to choice D.

Answer: D. 45,000

Common Faltering Points

Errors while devising the approach

1. Confusing median with mean (average)

Students often mix up median and mean, especially when dealing with salary data. They might try to calculate the average salary by adding all salaries and dividing by 13, instead of finding the middle value when salaries are arranged in order.

2. Incorrectly determining the median position

With 13 employees, students might think the median is at position 6 or 8, forgetting that for an odd number of data points, the median position is \((n+1)/2 = (13+1)/2 = 7\)th position.

3. Misunderstanding how to handle repeated values

Seeing that multiple employees have the same salary, students might incorrectly think they need to average these repeated values or treat them differently, rather than understanding that each employee contributes one position in the ordered list.

Errors while executing the approach

1. Miscounting employee totals

When organizing the salary data, students might incorrectly add up the number of employees (1+2+2+3+5) and not verify they have exactly 13 employees, leading to errors in positioning.

2. Incorrectly ordering the salary values

Students might arrange salaries in the wrong order (descending instead of ascending) or misplace salary values when creating their ordered list, such as putting $42,500 before $40,000.

3. Losing track of positions when creating the ordered list

When expanding from frequency data to individual positions, students might miscount and place the 7th position incorrectly, especially when transitioning between different salary groups.

Errors while selecting the answer

No likely faltering points - once the 7th position is correctly identified as $45,000, the answer selection is straightforward as it directly matches choice D.