Problem 75 - Precalculus Mathematics for Calculus

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 1 (see page 130 ). 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 \(x\) '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 is \(y \approx 0.845x + 2.759\).

Step 1: Calculate necessary sums

To find the least squares line, we need several sums. First, calculate the sums of the individual components: \( \sum_{k=1}^{n} x_k = 1 + 2 + 3 + 5 + 7 = 18 \). Next, find \( \sum_{k=1}^{n} y_k = 3 + 5 + 6 + 6 + 9 = 29 \). Then calculate the sum of squares for \( x \): \( \sum_{k=1}^{n} x_k^2 = 1^2 + 2^2 + 3^2 + 5^2 + 7^2 = 1 + 4 + 9 + 25 + 49 = 88 \). Finally, compute the sum of \( x_k y_k \): \( \sum_{k=1}^{n} x_k y_k = 1 \cdot 3 + 2 \cdot 5 + 3 \cdot 6 + 5 \cdot 6 + 7 \cdot 9 = 3 + 10 + 18 + 30 + 63 = 124 \).

Step 2: Set up the linear equations

Using the equations provided, substitute the computed sums:\[\begin{align*}18a + 5b &= 29 \88a + 18b &= 124\end{align*}\]

Step 3: Solve the linear equations for \(a\) and \(b\)

Start with the first equation: \( 18a + 5b = 29 \).Solution for \(b\) in terms of \(a\):\( b = \frac{29 - 18a}{5} \).Substitute \(b\) into the second equation:\( 88a + 18\left(\frac{29 - 18a}{5}\right) = 124 \).Simplify:\( 88a + \frac{522 - 324a}{5} = 124 \).Clear the fraction by multiplying through by 5:\( 440a + 522 - 324a = 620 \).Combine like terms:\( 116a = 620 - 522 \).Calculate \(a\):\( 116a = 98 \)\( a = \frac{98}{116} = \frac{49}{58} \approx 0.845 \).

Step 4: Solve for \(b\) using \(a\)

Substitute \(a = \frac{49}{58}\) back into the expression for \(b\) from Step 3:\( b = \frac{29 - 18\left(\frac{49}{58}\right)}{5} \).Simplify:\( b = \frac{29 - \frac{882}{58}}{5} = \frac{1682 - 882}{290} = \frac{800}{290} = \frac{80}{29} \approx 2.759 \).

Step 5: Write the equation of the least squares line

Now that we have \(a\) and \(b\), substitute them into \(y = ax + b\):\( y = \frac{49}{58}x + \frac{80}{29} \). Convert them back to approximate decimals if needed:\( y \approx 0.845x + 2.759 \).

Step 6: Sketch the graph and verify

Plot the data points \((1,3), (2,5), (3,6), (5,6), (7,9)\) on a coordinate plane. Draw the line \(y \approx 0.845x + 2.759\). Confirm visually that the line fits the data points well. This line should pass close to most of the points, indicating a good fit.

Key Concepts

Linear Equations

To understand the least squares regression line, it is essential to grasp the basics of linear equations. A linear equation is an algebraic expression representing a straight line on a graph. It typically has the form \( y = ax + b \), where:

• \(y\) is the dependent variable, often influenced by the independent variable \(x\).
• \(a\) is the slope. It describes how \(y\) changes when \(x\) changes by one unit.
• \(b\) is the y-intercept. It is the point where the line crosses the y-axis when \(x = 0\).

Linear equations are crucial in modeling relationships between two quantities, giving us a way to predict one based on the other. In the context of the least squares, the goal is to find the line that best represents the set of data points provided. This involves calculating specific coefficients \(a\) and \(b\) to minimize the difference between the data points and the predicted values on the line. The least squares method ensures that the sum of these squared differences is as small as possible, making the line a very close estimate of the overall trend of the data.

Data Points

In the least squares regression exercise, data points are used to determine the best-fitting line. Data points are sets of values that represent real-world observations. Each data point comprises an \((x,y)\) pair:

• \(x\) represents an independent variable, which can be a measurement or quantity you control or influence.
• \(y\) is the dependent variable, which responds to the values of \(x\).

For example, in our problem, we have the data points: \((1,3)\), \((2,5)\), \((3,6)\), \((5,6)\), \((7,9)\). Each pair suggests a relationship between \(x\) and \(y\), showing a pattern or trend that we aim to model with a line. By representing these points on a graph, one can visually assess how closely they might fit to a straight line. The closer and more consistently the points deviate towards a straight line, the more accurate the linear model is likely to be.

Sigma Notation

Sigma notation, represented as \( \Sigma \), is a concise way to express the sum of a sequence of numbers. It is vital in the process of determining the least squares line, particularly when calculating sums required for setting up the linear equations.In our exercise, sigma notation was used to calculate:

• \( \sum_{k=1}^{n} x_k \): the sum of all \(x\)-values in our dataset.
• \( \sum_{k=1}^{n} y_k \): 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.

Each of these calculations is essential because they contribute to the coefficients \(a\) and \(b\) in the linear equation for the least squares line. Understanding how to use sigma notation efficiently simplifies handling large data sets, allowing for organized and understandable summation of values critical for analysis.