Problem 59 - Algebra and Trigonometry

Question

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 Focus on Modeling (see page 197 ). 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 \(\sum_{k=1}^{n} x_{k}\) stands for the sum of all the \(x^{\prime}\) . See Section 12.1 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

The least squares line is approximately \( y = 0.8448x + 2.5828 \).

Step 1: Calculate Summations

First, we need to calculate the following sums using the given data points:\[ \sum x_k = 1 + 2 + 3 + 5 + 7 = 18 \]\[ \sum y_k = 3 + 5 + 6 + 6 + 9 = 29 \]\[ \sum x_k^2 = 1^2 + 2^2 + 3^2 + 5^2 + 7^2 = 1 + 4 + 9 + 25 + 49 = 88 \]\[ \sum x_k y_k = 1\cdot 3 + 2\cdot 5 + 3\cdot 6 + 5\cdot 6 + 7\cdot 9 = 3 + 10 + 18 + 30 + 63 = 124 \]

Step 2: Set Up the System of Equations

Using the results from Step 1, substitute them into the linear equations for the least squares line:1. \( 18a + 5b = 29 \)2. \( 88a + 18b = 124 \)

Step 3: Solve the System of Equations

We solve the two equations simultaneously. First, solve the first equation for \( b \):\[ b = \frac{29 - 18a}{5} \]Substitute \( b \) into the second equation:\[ 88a + 18\left(\frac{29 - 18a}{5}\right) = 124 \]Simplify and solve for \( a \):\[ 88a + \frac{522 - 324a}{5} = 124 \]\[ 440a + 522 - 324a = 620 \]\[ 116a = 98 \]\[ a = \frac{98}{116} = \frac{49}{58} \approx 0.8448 \]

Step 4: Find the Value of b

Substitute \( a = \frac{49}{58} \) back into the expression for \( b \):\[ b = \frac{29 - 18\cdot\frac{49}{58}}{5} \]Calculate:\[ b = \frac{29 - \frac{882}{58}}{5} \]\[ b = \frac{1682}{290} \approx 2.5828 \]

Step 5: State the Equation of the Line

The least squares regression line is:\[ y = \frac{49}{58}x + \frac{1682}{290} \]or approximately:\[ y \approx 0.8448x + 2.5828 \]

Step 6: Plot and Verify

Plot the given data points and the calculated regression line to visually verify how well the line fits the data points. If using a calculator, check if its regression function produces similar values for \( a \) and \( b \).

Key Concepts

Calculus

The term calculus often brings to mind intricate functions and complex limits, but its role in least squares regression is quite intuitive. When we seek the best-fitting line, we're really trying to minimize the "error" between our data points and the regression line. This "error" is calculated as the sum of the squares of the differences between the actual data points and the estimated ones.

• Calculus allows us to determine points where functions like these reach their minimum or maximum values.
• In the context of least squares, we leverage calculus to derive formulas that ensure the sum of squared errors is minimized, leading to the optimal regression line.

The formulas for coefficients 'a' and 'b' of the regression line stem from this minimization process. Derivatives are taken with respect to these coefficients, leading to the system of linear equations used to compute 'a' and 'b'.

Linear Equations

Linear regression is all about finding the best-fitting line described by a linear equation. This equation has the form:
\[ y = ax + b \]
This equation depicts a straight line where:

• 'a' represents the slope of the line, indicating how much 'y' changes for a unit change in 'x'.
• 'b' represents the y-intercept, or the point where the line crosses the y-axis.

In the least squares method, we use the calculated sums from the data points to derive two linear equations:

• \( (\sum x_k) a + n b = \sum y_k \)
• \( (\sum x_k^2) a + (\sum x_k) b = \sum x_k y_k \)

Solving these equations simultaneously gives us the values of 'a' and 'b' needed for our regression line.

Sigma Notation

Sigma notation simplifies the representation of series and summations in mathematical equations. It is particularly useful in least squares regression for compactly expressing the sums involved. The capital sigma, \( \Sigma \), indicates that you are summing a series of values:

• \( \sum_{k=1}^{n} x_k \) means add up all \( x_k \) values from 1 to \( n \).
• \( \sum_{k=1}^{n} x_k y_k \) represents the sum of the products of paired values \( x_k \) and \( y_k \).

Using sigma notation helps keep mathematical expressions tidy and simplifies understanding the underlying math in regression problems. It's a powerful tool for showing patterns and repetitive calculations clearly and concisely.

Summation

The process of summation involves adding a sequence of numbers together. It's a fundamental operation in the calculation of the least squares regression line, necessary for computing the sums required by the line equations.
In the context of least squares, summations include:

• \( \sum x_k \): the total of the x-values from your data.
• \( \sum y_k \): the total of the y-values.
• \( \sum x_k^2 \): the total of each x-value squared.
• \( \sum x_k y_k \): the total of each x-value multiplied by its corresponding y-value.

By calculating these summations, we can insert them into our linear equations to find the specific coefficients 'a' and 'b' for the regression line. Summation simplifies data analysis, making it easier to track patterns and handle large datasets efficiently.