Problem 6 - Basic Statistics for Business and Economics with Student

Question

Compute the mean hourly wage paid to carpenters who earned the following hourly wages: \(\$ 15.40, \$ 20.10\), \(\$ 18.75, \$ 22.76, \$ 30.67, \$ 18.00 .\)

Step-by-Step Solution

Answer

The mean hourly wage is approximately \$20.95.

Step 1: Identify the Data Points

Start by listing out all the hourly wages given in the exercise: \\(15.40, \\)20.10, \\(18.75, \\)22.76, \\(30.67, \\)18.00.

Step 2: Understand the Formula for Mean

The mean (average) is calculated by adding all the numbers in a set of data together and then dividing by the number of data points.

Step 3: Calculate the Sum of Wages

Add together all the hourly wages: \(15.40 + 20.10 + 18.75 + 22.76 + 30.67 + 18.00\).

Step 4: Sum Calculation

Evaluate the sum: \(15.40 + 20.10 = 35.50\); \(35.50 + 18.75 = 54.25\); \(54.25 + 22.76 = 77.01\); \(77.01 + 30.67 = 107.68\); \(107.68 + 18.00 = 125.68\). The total sum is \$125.68.

Step 5: Determine the Number of Data Points

Count the number of hourly wages: There are 6 carpenter wages given.

Step 6: Calculate the Mean Wage

Divide the sum of the wages by the number of data points: \(\frac{125.68}{6}\).

Step 7: Mean Calculation

Perform the division: \(\frac{125.68}{6} = 20.9466\). Rounding to two decimal places, this is approximately \$20.95.

Key Concepts

Step-by-Step Solutions

Breaking down complex problems into manageable steps can significantly enhance understanding and eases the problem-solving process. This is especially true for calculations involving averages or means, like finding the mean hourly wage of carpenters. By using a step-by-step approach, students can systematically address each smaller task within the larger problem.
Here's why it's beneficial:

• Each step allows for focus on one part of the solution at a time.
• It creates a clear path from the problem statement to the final answer.
• If mistakes are made, it's easier to pinpoint which step went wrong.

This method not only helps in solving the current problem but also strengthens problem-solving skills for future challenges.

Data Points Identification

The first essential step in calculating a statistical measure like the mean is to identify and list all the data points clearly. In our exercise, each hourly wage is a data point. Recognizing and organizing these correctly is crucial because they form the basis for further calculation steps.
Consider why organizing data points is important:

• Provides a clear view of all values involved, ensuring none are overlooked.
• Facilitates accurate calculation of sum and average by centralizing the values.
• Helps in clearly identifying the number of data points, which is crucial for accurate mean calculation.

Keeping all data clearly listed prevents errors and allows for a structured approach to data analysis.

Sum Calculation

Once data points have been listed, the next step is calculating their sum, which is foundational for computing averages. Adding each hourly wage accurately provides the total amount, which you will divide by the number of data points to get the mean.
The process involves:

• Addition of all data points, ensuring no numbers are omitted or miscalculated.
• Breaking down the addition into smaller groups, if necessary, to prevent errors.

In our problem, adding all hourly wages gave a correct total of \(\$125.68\). This sum is what you will use to find the average wage. Precise sum calculation is vital as a small error here could lead to an incorrect average.

Statistical Formula

To find the mean, a well-known statistical formula is used: Add all the data points, then divide by the quantity of points. This formula is the backbone for calculating averages in statistics and is formulated as:
\[ \text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}} \]In our exercise for calculating the mean hourly wage:

• The sum of carpenter wages, \(\\(125.68\), was divided by the number of data points, which is 6.
• The division \(\frac{125.68}{6}\) gave us approximately \(\\)20.95\), after rounding.

Understanding this formula is crucial, as it applies not only to wages but to any set of quantitative data when calculating an average.