Problem 58 - College Algebra and Calculus: An Applied Approach

Question

Fitting a Line to Data In Exercises \(55-60\), find the least squares regression line \(y=a x+b\) for the points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) by solving the system for \(a\) and \(b\). (If you are unfamiliar with summation notation, look at the discussion in Section 7.1.) $$ \left\\{\begin{array}{c} n b+\left(\sum_{i=1}^{n} x_{i}\right) a=\sum_{i=1}^{n} y_{i} \\ \left(\sum_{i=1}^{n} x_{i}\right) b+\left(\sum_{i=1}^{n} x_{i}^{2}\right) a=\sum_{i=1}^{n} x_{i} y_{i} \end{array}\right. $$ $$ \left\\{\begin{array}{r} 6 b+15 a=23.6 \\ 15 b+55 a=48.8 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution for the system of equations is \(a = 1.24\) and \(b = 0.36\)

Step 1: Write down the equations

Start by noting down the equations given. The two equations are \(\6b+15a=23.6\) and \(15b+55a=48.8\)

Step 2: Solving the system of equations

Find either the value of \(a\) or \(b\) from the first equation, and then substitute it into the second equation. From the first equation, you can express \(a\) as \(\a = (23.6 - 6b) / 15\) . Substituting \(a\) into the second equation, we have \(15b + 55((23.6 - 6b) / 15) = 48.8 \)

Step 3: Find value of \(b\)

On simplifying this equation, we get \(b = 0.36\)

Step 4: Substitute \(b\) into the first equation to solve for \(a\)

Substituting \(b = 0.36\) into the first equation, we get \(a = (23.6 - 6*0.36) / 15 \)

Step 5: Compute the value of \(a\)

This gives us \(a = 1.24\)

Key Concepts

Linear Regression

Linear regression is a fundamental concept in statistics and data analysis. It helps in predicting the relationship between independent and dependent variables through a linear equation. Imagine you have a set of data points, and you want to find a line that best fits these points. This course of action is what we refer to as linear regression.

The line is represented by an equation:

• \( y = ax + b \)

where \( y \) is the dependent variable, \( x \) is the independent variable, \( a \) is the slope, and \( b \) is the intercept.
In linear regression, the goal is to determine the values of the slope \( a \) and intercept \( b \) so that the line minimizes the distance from each data point. This minimization process is called the "least squares" method, where we sum the squares of the differences between observed values and the values predicted by the line.

This method is very useful in predictive modeling, helping us understand and forecast patterns in data.

System of Equations

A system of equations is a collection of two or more equations with the same set of variables. In the context of least squares regression, we end up with a system of equations when finding values for \( a \) and \( b \).

In the given exercise, the system of equations is:

• \( 6b + 15a = 23.6 \)
• \( 15b + 55a = 48.8 \)

Solving this system involves finding values of \( b \) and \( a \) that satisfy both equations simultaneously. You can use various methods like substitution or elimination to solve it.

For example, we can express one variable, say \( a \), in terms of the other variable \( b \) using one equation and substitute it into the other. This process allows us to solve for both variables and find the exact line that best fits the data. Knowing how to solve such equations is a key skill in mathematics and data analysis, enabling us to uncover relationships between variables effectively.

Summation Notation

Summation notation is a compact way of representing the sum of a sequence of numbers. In mathematical terms, it is expressed with the Greek letter sigma (\( \Sigma \)). It is a convenient tool for expressing calculations involving series or sequences without having to write out all the numbers.

For instance, the expression

• \( \sum_{i=1}^{n} x_i \)

represents the sum of all \( x \) values from \( x_1 \) to \( x_n \). In the context of least squares regression, it becomes particularly helpful when defining systems of equations. When we talk about the sums of products of variables or their squares, summation notation keeps expressions concise and easier to manage.

Understanding summation notation is crucial because it helps simplify otherwise complex mathematical expressions. This allows one to focus on the overall structure of equations and their solutions rather than getting bogged down in lengthy arithmetic calculations. Mastery of this notation is a vital part of mathematical literacy, especially in fields like statistics, calculus, and any data-driven discipline.