Problem 59 - 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.) $$ \left\\{\begin{array}{c} n b+\left(\sum_{i=1}^{n} x_{i}\right) a=\sum_{i=1}^{n} y_{i} \\ \left(\sum_{i=1}^{n} x_{i}\right) b+\left(\sum_{i=1}^{n} x_{i}^{2}\right) a=\sum_{i=1}^{n} x_{i} y_{i} \end{array}\right. $$ $$ (0,4),(1,3),(1,1),(2,0) $$

Step-by-Step Solution

Answer

The least squares regression line is \(y=-2x+4\).

Step 1: Data Analysis

Inspect the given data; there are four points (0,4),(1,3),(1,1), and(2,0). These points will be used to fill in the given equations.

Step 2: Substitute Into The First Equation

Substitute the x and y values into the first equation. In this case, \(n=4\), \(\sum_{i=1}^{n} x_{i}=0+1+1+2=4\), and \(\sum_{i=1}^{n} y_{i}=4+3+1+0=8\). Thus, the first equation becomes: \(4b + 4a = 8\). Divide through by 4: \(b + a = 2\).

Step 3: Substitute Into The Second Equation

Now, substitute the x and y values into the second equation. The sum of \(x_{i}^{2}\) is \(0^{2}+1^{2}+1^{2}+2^{2}=6\), and the sum of \(x_{i}y_{i}\) is \(0*4+1*3+1*1+2*0=4\). Thus, the second equation becomes: \(4b + 6a = 4\). Divide through by 2: \(2b + 3a = 2\).

Step 4: Solve The Two Equations

The final step is to solve the two equations \(b+a = 2\) and \(2b + 3a = 2\). Solving this system, we find \(a=-2\) and \(b=4\).

Key Concepts

System of Equations

A system of equations is a set of two or more equations that involve the same set of variables. In the context of finding the least squares regression line, we deal with a system composed of two equations. These equations emerge from the method of minimizing the squared differences between the actual data points and the points on the proposed linear model, usually denoted as the line \(y = ax + b\).

To solve such a system, substitution or elimination methods are commonly used. These techniques help us find the values of \(a\) (the slope) and \(b\) (the y-intercept) that make both equations true simultaneously. The solution to this system provides us with the best-fitting line that minimizes the sum of the squares of the vertical distances of the points from the line.

Summation Notation

Summation notation, represented by the Greek letter Sigma (\(\Sigma\)), is a convenient way to express the addition of a series of numbers following a pattern. In the given exercise, we see symbols like \(\sum_{i=1}^{n} x_{i}\), which signifies that we are to sum up all the \(x\) values from the first data point (when \(i = 1\)) to the nth data point.

For example, if we're adding the squared \(x\) values from our data set, we'd write \(\sum_{i=1}^{n} x_{i}^2\), and this would translate to adding \(x_1^2\), \(x_2^2\), and so on, up to \(x_n^2\). It's an efficient way to represent long sums and especially useful in statistical formulas, including those used for regression analysis.

Data Fitting

Data fitting refers to the process of finding a model, such as a line or curve, that best represents a set of data. In the context of this exercise, we're utilizing a linear model, also known as a straight line, to describe the relationship between the given data points. The goal of data fitting is to produce a line that closely follows the trend of the data.

In finding the best fit, we aim to minimize the discrepancy between the actual data points and the estimated values provided by our model. This minimization typically implies the computation of the least squares, which involves calculating the smallest possible sum of the squares of the errors (differences) between the actual and the estimated values.

Regression Analysis

Regression analysis is a powerful statistical tool used to understand the relationship between variables. In the case of the least squares regression line with which we're working, it is a method used to estimate the straight-line equation that best fits a set of bivariate data -– meaning data with one independent variable (\(x\)) and one dependent variable (\(y\)).

The regression line is typically written in the form \(y = ax + b\) where \(a\) represents the slope of the line and \(b\) represents the y-intercept. The slope indicates how much the dependent variable \(y\) changes for a unit change in the independent variable \(x\). The y-intercept is the value of \(y\) when \(x\) is zero. The regression analysis involves finding these parameters such that the line is the best possible fit for the data, in the sense that it reduces the sum of the squared residuals—essentially quantifying the line's predictive accuracy.