Problem 52 - Calculus Early Transcendentals: Pearson New International Edition

Question

Brass is produced in long rolls of a thin sheet. To monitor the quality, inspectors select at random a piece of the sheet, measure its area, and count the number of surface imperfections on that piece. The area varies from piece to piece. The following table gives data on the area (in square feet) of the selected piece and the number of surface imperfections found on that piece. $$ \begin{array}{ccc} \hline \text { Piece } & \begin{array}{c} \text { Area in } \\ \text { Square Feet } \end{array} & \begin{array}{c} \text { Number of } \\ \text { Surface Imperfections } \end{array} \\ \hline 1 & 1.0 & 3 \\ 2 & 4.0 & 12 \\ 3 & 3.6 & 9 \\ 4 & 1.5 & 5 \\ 5 & 3.0 & 8 \\ \hline \end{array} $$ (a) Make a scatter plot with area on the horizontal axis and number of surface imperfections on the vertical axis. (b) Does it look like a line through the origin would be a good model for these data? Explain. (c) Find the equation of the least-squares line through the origin. (d) Use the result of part (c) to predict how many surface imperfections there would be on a sheet with area \(2.0\) square feet.

Step-by-Step Solution

Answer

Plot the scatter, fit a line through the origin. Equation: \( y = 3x \). Predicted imperfections for 2.0 sq ft = 6.

Step 1: Understanding the Problem

We need to make a scatter plot for the given data, determine if a line through the origin would fit the data well, find the least-squares line equation through the origin, and use this equation to predict the number of surface imperfections for a given area.

Step 2: Constructing the Scatter Plot

Plot the points with 'Area in Square Feet' on the x-axis and 'Number of Surface Imperfections' on the y-axis. The points are (1.0, 3), (4.0, 12), (3.6, 9), (1.5, 5), and (3.0, 8).

Step 3: Evaluating Line Through the Origin

Observe the scatter plot. A line through the origin suggests that as the area increases, the number of imperfections is proportional. Check if the points seem to roughly align or trend in such a way.

Step 4: Finding the Least-Squares Line Through the Origin

The least-squares line through the origin has the form \( y = bx \), where \( b = \frac{\sum (xy)}{\sum (x^2)} \). Calculate \( \sum (xy) = 3*1 + 12*4 + 9*3.6 + 5*1.5 + 8*3 \). Calculate \( \sum (x^2) = 1^2 + 4^2 + 3.6^2 + 1.5^2 + 3^2 \). Substitute in values to find \( b \) and thus the equation.

Step 5: Predicting Future Surface Imperfections

Use the equation from Step 4. Substitute \( x = 2.0 \) into \( y = bx \) to predict the number of surface imperfections for 2.0 square feet.

Key Concepts

Scatter Plot Analysis

When analyzing data, one of the most visual and insightful tools you can use is a scatter plot. A scatter plot displays pairs of data points on a graph, with one variable on the x-axis and the other on the y-axis. For this exercise, the scatter plot allows us to see the relationship between the area of the brass sheet (in square feet) and the number of surface imperfections. By plotting the given data points, we create an intuitive visual representation of this relationship.

Once the data is on the graph as points, you can observe patterns or trends. If you see the points clustering towards a line, you can infer a linear relationship. Specifically, for brass production, observing how these points align can suggest if a proportional relationship exists, meaning as the sheet size increases, the number of imperfections might increase in a predictable manner.

Surface Imperfections

Surface imperfections in materials like brass can affect the quality and utility of the product, leading to a need for careful monitoring and control. These imperfections might include scratches, dents, or any irregularities on the surface that could impair functionality or aesthetic appeal. Inspectors count these flaws to gauge the quality of production.

This exercise provides a practical scenario where imperfections are tracked in relation to the sheet's area. Keeping track of imperfections is essential for manufacturers looking to maintain high standards and implement corrective action when necessary. By understanding the frequency and distribution of imperfections, companies can better predict potential issues and improve processes.

Quality Control

Quality control is crucial in manufacturing, ensuring products meet specific standards and customers receive defect-free items. This task's data collection is a part of the broader quality control effort. By selecting random samples and analyzing surface imperfections, manufacturers can assess the overall quality of the production process.

Incorporating statistical methods like least-squares regression analysis aids in making informed decisions about quality. If a pattern in data suggests a need for adjustment, such as more imperfections with larger sheet sizes, the process can be altered to improve quality. Effective quality control not only enhances product standards but also helps reduce waste and increase customer satisfaction.

Data Modeling

Data modeling is an essential technique in statistics to describe the relationships between variables. This exercise features linear modeling using least-squares regression, specifically finding a best-fit line that predicts imperfections based on sheet area.

The least-squares method minimizes the sum of the squares of the differences between the observed values and those predicted by the line. The formula used here for a line through the origin is \( y = bx \), where \( b \) is calculated based on the product of the data sets. By using this model, manufacturers can anticipate how changes in sheet size might impact imperfections, allowing for better planning and process adjustments. Data modeling helps uncover insights that are not immediately apparent from raw data, providing a mathematical framework for understanding and predicting outcomes.