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

Yes, a line through the origin is suitable; the equation is \( y = 2.37x \). Predicted imperfections for 2.0 square feet are about 5.

Step 1: Make a scatter plot

Plot each pair of values from the table on a graph with 'Area in Square Feet' on the x-axis and 'Number of Surface Imperfections' on the y-axis. Create points for each piece: (1.0, 3), (4.0, 12), (3.6, 9), (1.5, 5), and (3.0, 8).

Step 2: Evaluate linearity through the origin

Visually inspect the scatter plot to determine if the data points roughly align with a line that passes through the origin (0,0). If the data follows a straight line trend that appears consistent with a line through the origin, the linear model is reasonable.

Step 3: Compute the slope for the line through the origin

For a line through the origin, use the formula for slope \( b = \frac{\sum (x_i y_i)}{\sum (x_i^2)} \). Calculate \( \sum (x_i y_i) = 1.0 \times 3 + 4.0 \times 12 + 3.6 \times 9 + 1.5 \times 5 + 3.0 \times 8 = 95.4 \) and \( \sum (x_i^2) = 1.0^2 + 4.0^2 + 3.6^2 + 1.5^2 + 3.0^2 = 40.21 \). Then, find the slope \( b = \frac{95.4}{40.21} \approx 2.37 \).

Step 4: Write the equation of the least-squares line

The equation of the least-squares line through the origin is given by \( y = bx \). With the slope calculated as \( b \approx 2.37 \), the equation becomes \( y = 2.37x \).

Step 5: Predict imperfections for a sheet with area 2.0 square feet

Using the equation \( y = 2.37x \), substitute \( x = 2.0 \) to find \( y \), yielding \( y = 2.37 \times 2.0 = 4.74 \). Therefore, approximately 5 surface imperfections are expected, rounding to the nearest whole number.

Key Concepts

Scatter Plot

A scatter plot is a graphical representation of data, where individual values from a dataset are shown as dots on a two-dimensional graph. In this exercise, the x-axis represents the area of brass sheets in square feet, and the y-axis represents the number of surface imperfections.

Here's how to create a scatter plot:

• List down the pair of values for each data point. For example, from the provided data table (1.0, 3), (4.0, 12), etc.
• On graph paper or any plotting software, mark each value pair.

Scatter plots are crucial for visually analyzing the relationship between two variables. They display trends, relationships, and spread of data points. If a scatter plot shows data points roughly forming a line, this suggests a potential linear relationship that can be modeled with a linear equation.

Linear Relationship

A linear relationship is present when two variables move in a consistent direction, forming a straight line on a scatter plot. In this scenario, we are asked to evaluate if the number of imperfections is linearly dependent on the area of the sheet, especially if this line passes through the origin (0, 0).

Checking for a linear relationship involves:

• Visually inspecting the scatter plot to see if the data points align with a straight line.
• Determining if this alignment seems consistent enough to proceed with linear modeling.

In the example provided, if the scatter plot shows that data points tend to cluster around a direct line passing through the origin, this suggests a proportional increase in surface imperfections with the increase in area, indicating a linear relationship.

Slope Calculation

The slope of a line in the context of linear regression through the origin represents how much the dependent variable (surface imperfections) changes with a one-unit change in the independent variable (area). To calculate the slope for the least-squares line through the origin, the formula is used: \( b = \frac{\sum (x_i y_i)}{\sum (x_i^2)} \).

Steps for calculating the slope:

• Calculate the product of each pair (area and imperfections) and sum them: \( \sum (x_i y_i) = 95.4 \).
• Compute the sum of squares of each area value: \( \sum (x_i^2) = 40.21 \).
• Divide these sums to find the slope \( b = \frac{95.4}{40.21} \approx 2.37 \).

A slope of 2.37 suggests that for every additional square foot increase in the area of the sheet, the number of surface imperfections increases by approximately 2.37.

Data Analysis

Data analysis involves interpreting the results obtained from statistical methods like the least-squares line, and making predictions based on the trend derived from the data. For this exercise, the goal is to predict how many imperfections exist for a sheet of a given area.

To predict using the least-squares equation \( y = 2.37x \):

• Substitute your area of interest into the equation. For example, when \( x = 2.0 \) square feet, solve \( y = 2.37 \times 2.0 = 4.74 \).
• Round the result to the nearest whole number as imperfections can only be whole numbers.

Therefore, for a sheet of 2.0 square feet, you would predict approximately 5 imperfections. Analyzing data this way helps in quality control and expecting outcomes from variations in production parameters.