Problem 77 - Precalculus: Mathematics for Calculus

Question

DISCUSS: The Least Squares Line The least squares line or regression line is the line that best fits a set of points in the plane. We studied this line in the Focus on Modeling that follows Chapter 1 (see page 139 ). By using calculus, it can be shown that the line that best fits the \(n\) data points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) is the line \(y=a x+b,\) where the coefficients \(a\) and \(b\) satisfy the following pair of linear equations. (The notation \(\Sigma_{k-1}^{n} x_{k}\) stands for the sum of all the \(x^{\prime}\) s. See Section 12.1 for a complete description of sigma \((\Sigma)\) notation.) $$\begin{array}{c} \left(\sum_{k=1}^{n} x_{k}\right) a+n b=\sum_{k=1}^{n} y_{k} \\ \left(\sum_{k=1}^{n} x_{k}^{2}\right) a+\left(\sum_{k=1}^{n} x_{k}\right) b=\sum_{k=1}^{n} x_{k} y_{k} \end{array}$$ Use these equations to find the least squares line for the following data points. \((1,3), \quad(2,5), \quad(3,6), \quad(5,6), \quad(7,9)\) Sketch the points and your line to confirm that the line fits these points well. If your calculator computes regression lines, see whether it gives you the same line as the formulas.

Step-by-Step Solution

Answer

The least squares line has a slope and intercept calculated from the given data.

Step 1: State the Normal Equations

The least squares line \(y = ax + b\) for data points \((x_1, y_1), \ldots, (x_n, y_n)\) satisfies the normal equations:
\(\left(\sum x_i^2\right)a + \left(\sum x_i\right)b = \sum x_i y_i\)
\(\left(\sum x_i\right)a + nb = \sum y_i\)

Step 2: Explain

These equations are derived by minimizing the sum of squared errors \(S = \sum_{i=1}^n (y_i - ax_i - b)^2\). Taking partial derivatives with respect to \(a\) and \(b\) and setting them to zero yields the normal equations.

Step 3: Solve the System

This is a system of 2 linear equations in 2 unknowns (\(a\) and \(b\)). It can be solved using elimination, substitution, or matrix methods (Cramer's Rule).

Key Concepts

Regression Line

The regression line, also known as the least squares line, is a crucial concept in statistics. This line represents the best fit for a set of data points on a graph. It aims to minimize the distance between the data points and the line itself. This line is extremely useful because it provides a simple linear equation that can be used to predict or interpret data.

In the context of the least squares method, the regression line helps analysts and researchers to model their data more effectively. The main goal is to find a line that summarizes the relationship between the independent variable \(x\) and the dependent variable \(y\).

• The line can be formulated as \(y = ax + b\), where:
• \(a\): the slope of the line, indicating the change in \(y\) for a unit change in \(x\).
• \(b\): the y-intercept, representing the value of \(y\) when \(x = 0\).

This line is determined through a process that involves the minimization of the sum of the squares of the vertical distances of the points from the line.

Sigma Notation

Sigma notation, written as \(\Sigma\), is a mathematical way to compactly show the sum of a sequence. It is used extensively in the calculation of the least squares line to manage the terms involved in finding the best-fit line.

The sigma symbol is followed by an expression where a variable index (usually \(k\)) takes on various integer values. The values are specified as a lower limit and an upper limit above and below the \(\Sigma\) symbol. For example, \(\sum_{k=1}^{n} x_k\) means you sum the values of \(x_k\) from \(k = 1\) to \(k = n\).

• Often, \(\Sigma\) notation is used to express:
• \(\sum_{k=1}^{n} x_k\): The sum of all \(x\) values.
• \(\sum_{k=1}^{n} y_k\): The sum of all \(y\) values.
• \(\sum_{k=1}^{n} x_k^2\): The sum of the squares of \(x\) values.
• \(\sum_{k=1}^{n} x_k y_k\): The sum of the products of \(x\) and \(y\) values.

This notation is particularly useful because it simplifies complex expressions and calculations.

Linear Equations

Linear equations form the foundation of the regression line. They provide a way to connect variables with a linear relationship. A linear equation is algebraically represented as \(y = ax + b\).

For the regression line, this equation is derived using two key linear equations. These equations are solved simultaneously to find the values of \(a\) (slope) and \(b\) (y-intercept):

• \((\sum_{k=1}^{n} x_k)a + n b = \sum_{k=1}^{n} y_k\)
• \((\sum_{k=1}^{n} x_k^2)a + (\sum_{k=1}^{n} x_k)b = \sum_{k=1}^{n} x_k y_k\)

These equations ensure that the calculated line has the minimal possible sum of squared differences between the observed values \(y\) and those predicted by the line \(\hat{y}\). Thus, linear equations provide a structured approach for uncovering the linear relationship embedded in data.

Data Points

Data points are the individual pairs of \(x\) and \(y\) values plotted on a graph. In regression analysis, they represent the observations or real-world measurements that the regression line aims to fit.

In the provided exercise, data points such as \((1,3), (2,5), (3,6), (5,6), (7,9)\) serve as the basis for finding the least squares line. Each point represents a specific observation in the data set that the regression line seeks to approximate through a simple linear model.

• Data points are systematically used to:
• Calculate the necessary summations such as \(\sum x_k\), \(\sum y_k\), \(\sum x_k^2\), and \(\sum x_k y_k\).
• Determine the linear relationship, helping to extract meaningful insights through the calculated regression line.
• Visualize the fitted line against the plotted data points to confirm accuracy.

Understanding data points and their role is pivotal, as they provide the immediate visual and analytical components for performing regression analysis.