Problem 75 - Algebra and Trigonometry

Question

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 2 (see page \(171 ) .\) By using calculus, it can be shown that the line that best fits the \(n\) data points \(\left(x_{1}, = y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{m} y_{n}\right)\) is the line \(y=a x+b,\) where the coefficients \(a\) and \(b\) satisfy the following pair of linear equations. (The notation \(\sum_{k=1}^{n} x_{k}\) stands for the sum of all the \(X^{\prime}\) see Section 13.1 for a complete description of sigma ( \(\Sigma\) ) notation. $$\left(\sum_{k=1}^{n} x_{k}\right) a+n b=\sum_{k=1}^{n} y_{k}$$ $$\left(\sum_{k=1}^{n} x_{k}^{2}\right) a+\left(\sum_{k=1}^{n} x_{k}\right) b=\sum_{k=1}^{n} x_{k} y_{k}$$ 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

Line: \(y = \frac{49}{58}x + \frac{625}{145}\); fits data.

Step 1: Collect the Data Points

We have five data points: \((1,3), (2,5), (3,6), (5,6),\) and \((7,9)\). These points will be used to set up our equations for the least squares line.

Step 2: Calculate Summations

We need to find the sums: \(\sum x_k\), \(\sum y_k\), \(\sum x_k^2\), and \(\sum x_k y_k\). Calculating each: \(\sum x_k = 1 + 2 + 3 + 5 + 7 = 18\), \(\sum y_k = 3 + 5 + 6 + 6 + 9 = 29\), \(\sum x_k^2 = 1^2 + 2^2 + 3^2 + 5^2 + 7^2 = 1 + 4 + 9 + 25 + 49 = 88\), \(\sum x_k y_k = 1 \cdot 3 + 2 \cdot 5 + 3 \cdot 6 + 5 \cdot 6 + 7 \cdot 9 = 3 + 10 + 18 + 30 + 63 = 124\).

Step 3: Set Up the System of Equations

Based on the formulas given, set up the two equations with the calculated sums:1. \[18a + 5b = 29\]2. \[88a + 18b = 124\]

Step 4: Solve for Coefficient 'a' and 'b'

Use substitution or elimination to solve the system of linear equations:1. From equation 1: \(b = \frac{29 - 18a}{5}\)2. Substitute in equation 2: \[88a + 18\left(\frac{29 - 18a}{5}\right) = 124\]3. Solve for \(a\): \[88a + \frac{522 - 324a}{5} = 124\] \[440a + 522 - 324a = 620\] \[116a = 98\] \[a = \frac{98}{116} = \frac{49}{58}\]4. Substitute back to solve for \(b\): \[b = \frac{29 - 18(\frac{49}{58})}{5} = \frac{29 \times 58 - 18 \times 49}{290} = \frac{2132 - 882}{290} = \frac{1250}{290} = \frac{625}{145}\]

Step 5: Write the Least Squares Line Equation

The equation for the least squares line is \(y = \frac{49}{58}x + \frac{625}{145}\).

Step 6: Sketch and Confirm the Line

Plot the given points and sketch the line \(y = \frac{49}{58}x + \frac{625}{145}\). Make sure the line closely fits the points. If available, confirm by using a calculator with regression functionality to check the equation obtained.

Key Concepts

Linear Equations

Understanding linear equations is crucial when determining the least squares regression line. In general, a linear equation represents a line in two-dimensional space and can be expressed in the form \(y = ax + b\), where:

• \(y\) is the dependent variable,
• \(x\) is the independent variable,
• \(a\) is the slope of the line,
• \(b\) is the y-intercept, where the line crosses the y-axis.

In the context of least squares, the goal is to find the values of \(a\) and \(b\) such that the line minimizes the distances to each given data point. The specific distance we minimize is the vertical distance from each point to the line, squared, and then summed for all points. Linear equations help to provide a systematic way to find this best-fit line by utilizing these calculated sums.

Data Points

Data points are the individual measurements or observations collected, typically plotted on a graph. In the context of least squares regression analysis, these points represent the known values from which we want to predict the pattern or trend.

• Each data point consists of an \(x\)-value (independent) and a \(y\)-value (dependent).
• The goal is to determine the relationship between these values using a line that approximates this relationship.
• For example, given the points \(1,3\), \(2,5\), \(3,6\), \(5,6\), and \(7,9\), we seek a line that essentially passes "as close as possible" to each point.

Data points provide the essential information needed to solve for the coefficients \(a\) and \(b\) through the least squares method. The calculated line summarizes this collection of data, often used for predictive analysis.

Sigma Notation

Sigma notation (\(\Sigma\)) is a way to represent the summation of a set of terms. It simplifies the expressions where you need to add multiple terms together, as often seen in mathematical formulas for least squares regression.

• The symbol \(\Sigma_{k=1}^{n} x_k \) indicates that you add up all values of \(x_k \) from \(k=1 \) to \(n \).
• Similarly, \(\Sigma_{k=1}^{n} y_k \) represents the sum of all \(y_k \) values.
• This notation is crucial in formulating and understanding the linear equations for the least squares method. It's how we express the total sums needed for our calculations.

For the least squares line, sigma notation helps efficiently formulate the sums needed for the equations handling multiple data points in a concise mathematical expression. It allows for a clear and consistent approach to solving least squares problems.

Calculus in Regression

Calculus plays a significant role in deriving the least squares regression line, though its application might be under-the-hood in many standard problems. The main calculus concept involved is optimization.

• Optimization in calculus is used to find the minima or maxima of functions, such as when we want to minimize the sum of squared errors in least squares.
• By setting up the problem as minimizing the error, calculus provides the tools to derive the formulas used to calculate the slope \(a\) and intercept \(b\) for the best-fit line.
• The process involves differentiating the error function with respect to the coefficients and setting the derivatives to zero to find the minimum point.

Therefore, calculus ensures that the least squares formulae are derived correctly, establishing a line that best represents the data by minimizing the error effectively.