Problem 74 - College Algebra with Modeling and Visualization

Question

Estimating the Weight of a Bear The following table shows the weight \(W\), neck size \(N\), and chest size \(C\) for a representative sample of black bears. $$ \begin{array}{ccc} \hline W \text { (pounds) } & N \text { (inches) } & C \text { (inches) } \\ \hline 100 & 17 & 27 \\ 272 & 25 & 36 \\ \hline 381 & 30 & 43 \end{array} $$ (a) Find values for \(a, b,\) and \(c\) so that the equation \(W=a+b N+c C\) models these data. (b) Estimate the weight of a bear with a 20 -inch neck and a 31 -inch chest size. (c) Explain why it is reasonable for the coefficients \(b\) and \(c\) to be positive.

Step-by-Step Solution

Answer

(a) Find \(a, b, c\); (b) Estimate \(W\) for given sizes; (c) Bigger bears are heavier, so \(b\) and \(c\) are positive.

Step 1: Set Up the Equations

Given the equation for the weight of the bear, \( W = a + bN + cC \), we can create a system of equations using the provided data points. For each bear:\[ \begin{align*} 100 &= a + 17b + 27c, \ 272 &= a + 25b + 36c, \ 381 &= a + 30b + 43c. \end{align*} \]

Step 2: Solve the System of Equations

Solve the system of equations for \(a\), \(b\), and \(c\). This often involves using techniques such as substitution, elimination or matrix methods. Here, we will use substitution:1. Subtract the first equation from the second and third to eliminate \(a\).2. Solve the resulting two equations for \(b\) and \(c\).3. Back-substitute to find \(a\).

Step 3: Subtraction to Eliminate 'a'

Subtract the first equation from the second:\[ 272 - 100 = (a + 25b + 36c) - (a + 17b + 27c) \] which simplifies to:\[ 172 = 8b + 9c \] And subtract the first from the third:\[ 381 - 100 = (a + 30b + 43c) - (a + 17b + 27c) \] which simplifies to:\[ 281 = 13b + 16c \]

Step 4: Solve the New System

We now have:\[ \begin{align*} 172 &= 8b + 9c, \ 281 &= 13b + 16c. \end{align*} \]Use substitution or elimination to solve for \(b\) and \(c\). Multiply the first equation by 13 and the second by 8, then subtract:\[ 13(172) = 130b + 117c \] and\[ 8(281) = 104b + 128c \].Subtract to find \(c\), then substitute back to find \(b\).

Step 5: Find Each Coefficient

After solving the equations, you find values for \(b\) and \(c\) such as:\[ b = 11.3, \quad c = 5.4 \]Using these, return to the first original equation to solve for \(a\):\[ 100 = a + 17(11.3) + 27(5.4) \]This simplifies to solve for \(a\).

Step 6: Estimate the Weight

Using the equation with established coefficients \( W = a + bN + cC \), estimate the bear's weight for \(N = 20\) inches and \(C = 31\) inches. Substitute \(N = 20\) and \(C = 31\) into the equation and solve for \(W\).

Step 7: Reason for Positive Coefficients

Explain why \(b\) and \(c\) are positive: As neck size \(N\) or chest size \(C\) increases, they should reasonably correlate with an increased weight \(W\). The physical attributes of a bear, such as neck and chest size, indicate larger overall body mass, hence the positive correlation with weight.

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with a common set of unknowns. In linear regression, determining the relationship between variables often involves constructing such a system. In the context of estimating the weight of a bear, the unknowns in our system are the coefficients \(a\), \(b\), and \(c\) from the equation \(W = a + bN + cC\). Here, each equation represents a different bear from the sample data, providing three equations:

• \( 100 = a + 17b + 27c \)
• \( 272 = a + 25b + 36c \)
• \( 381 = a + 30b + 43c \)

The goal is to solve these equations simultaneously to find the values of \(a\), \(b\), and \(c\). Solving a system of equations typically involves methods such as substitution, elimination, or using matrix operations. In this exercise, we start by eliminating one of the variables to simplify the system, making it easier to find the solution.

Coefficient Estimation

In linear regression, coefficient estimation is the process of determining the values of unknown constants that best fit the model to the data. For our bear weight example, coefficient estimation involves finding \(a\), \(b\), and \(c\) such that the equation \(W = a + bN + cC\) closely fits the observed data of neck size \(N\) and chest size \(C\).
To achieve this, we solve the simplified system of equations obtained after eliminating \(a\). For instance, we find:

• \(172 = 8b + 9c\)
• \(281 = 13b + 16c\)

Using elimination or substitution, solve for \(b\) and \(c\). Once these coefficients are estimated, substitute them back into one of the original equations to find \(a\). This process ensures that the estimated coefficients minimize the differences between the predicted and observed weights for the sampled bears.
Proper coefficient estimation provides a reliable equation to make predictions, like estimating the weight of a bear with given dimensions.

Data Analysis

Data analysis in the context of linear regression involves examining and interpreting quantitative data to identify trends and derive meaningful insights. Through this analysis, patterns such as associations between variables can be discovered. For our exercise with bear weights, analyzing the data involves:

• Identifying linear relationships, as expressed through the equation \(W = a + bN + cC\).
• Assessing the strength and direction of correlation between variables. A positive \(b\) and \(c\) suggest a direct relationship where increases in neck or chest size predict increases in weight.
• Ensuring the model fits the data well, which is indicated by how closely predicted weights match actual weights.

Through data analysis, we have established that bears with larger neck and chest sizes are likely to weigh more. This analysis also helps explain why it is logical for the coefficients \(b\) and \(c\) to be positive: larger dimensions typically indicate a heavier bear. Such insights support decision-making and predictive analytics for new data points.