Problem 63 - Basic Statistics for Business and Economics with Student

Question

The Split-A-Rail Fence Company sells three types of fence to homeowners in suburban Seattle, Washington. Grade A costs \(\$ 5.00\) per running foot to install, Grade \(B\) costs \(\$ 6.50\) per running foot, and Grade \(C,\) the premium quality, costs \(\$ 8.00\) per running foot. Yesterday, Split-ARail installed 270 feet of Grade \(\mathrm{A}, 300\) feet of Grade \(\mathrm{B}\), and 100 feet of Grade \(C\). What was the mean cost per foot of fence installed?

Step-by-Step Solution

Answer

The mean cost per foot of fence installed is approximately \$6.12.

Step 1: Calculate Total Cost for Each Grade

First, calculate the total cost for each grade of fence installed. For Grade A, the cost per foot is \(5.00, so the total cost is \( 270 \, \text{feet} \times 5.00 \\) = 1350\\( \). For Grade B, the cost per foot is \)6.50, so the total cost is \( 300 \, \text{feet} \times 6.50 \\( = 1950\\) \). For Grade C, the cost per foot is \(8.00, so the total cost is \( 100 \, \text{feet} \times 8.00 \\) = 800\$ \).

Step 2: Calculate Total Cost and Total Feet Installed

Add together the total costs of all grades to find the total cost for all fences: \( 1350\\( + 1950\\) + 800\\( = 4100\\) \). Also, sum up all the feet of fence installed: \( 270 \, \text{feet} + 300 \, \text{feet} + 100 \, \text{feet} = 670 \, \text{feet} \).

Step 3: Find the Mean Cost per Foot

Divide the total cost by the total number of feet installed to find the mean cost per foot: \( \frac{4100\\(}{670 \, \text{feet}} \approx 6.12\\) \).

Key Concepts

Understanding Statistics in Fence Installation

Statistics is a branch of mathematics dealing with data collection, analysis, interpretation, and presentation. In this problem, statistics is used to understand the overall expense of a fence installation by determining the mean cost per foot.
Understanding how to handle such data efficiently allows us to manage resources better and optimize cost.
In this exercise, we're not just interested in the quality of the fences installed but specifically in finding the average cost that customers are paying per foot for the different grades combined.
By applying statistical knowledge, we turn raw numbers into valuable insight, highlighting the balance between different grades and their respective costs.

Average Cost Calculation

Calculating an average is a foundational element in mathematics and statistics, often referred to as "mean."
The mean provides a summary number that identifies the central point or typical value of the dataset.
In our example, the mean cost per foot of fence provides a practical understanding of the overall pricing strategy. To calculate this mean, follow these straightforward steps:

• First, calculate the overall cost for each grade by multiplying the price per foot by the total feet installed.
• Next, add up these totals to find the combined cost for all grades.
• Finally, add up all feet installed and divide the total cost by this total number of feet.

The mean cost serves as an indicator of the cost-effectiveness of the installation, showing us the typical cost per foot that a customer might encounter.

Brainstorming Through Problem Solving

Problem solving is a valuable skill that applies technique and logic to overcome challenges. For this fence installation cost problem, the goal is clear: find a standardized cost measure.
Tackling this problem effectively requires structured steps and clear thinking. When approaching a problem:

• Break the problem into smaller parts, such as calculating individual costs first.
• Use logical reasoning to combine these parts into a comprehensive solution, like summing up totals and dividing for the mean.
• Check your steps for accuracy to ensure your solution effectively addresses the question.

This structured approach to problem solving not only helps to achieve accuracy but also improves efficiency in calculating complex setups, like determining the mean cost per foot of fencing across different product categories.