Problem 59 - Precalculus: Mathematical for Calculus

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 240). 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 \(x^{\prime}\) 's. See Section \(11.1 \text { for a complete description of sigma }(\Sigma) \text { notation. }]\) $$\left(\sum_{k=1}^{n} x_{k}\right) a+n b=\sum_{k=1}^{n} y_{k}$$ $$\left(\sum_{k=1}^{n} x_{k}^{2}\right) a+\left(\sum_{k=1}^{n} x_{k}\right) b=\sum_{k=1}^{n} x_{k} y_{k}$$ 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 equation of the least squares line is \(y = \frac{49}{58}x + 2.34\).

Step 1: Calculate Sums

First, let's calculate the necessary sums: \( \sum x_k \), \( \sum y_k \), \( \sum x_k^2 \), and \( \sum x_k y_k \). For the given data points \((1,3), (2,5), (3,6), (5,6), (7,9)\), compute:1. \( \sum x_k = 1 + 2 + 3 + 5 + 7 = 18 \)2. \( \sum y_k = 3 + 5 + 6 + 6 + 9 = 29 \)3. \( \sum x_k^2 = 1^2 + 2^2 + 3^2 + 5^2 + 7^2 = 1 + 4 + 9 + 25 + 49 = 88 \)4. \( \sum x_k y_k = 1\cdot3 + 2\cdot5 + 3\cdot6 + 5\cdot6 + 7\cdot9 = 3 + 10 + 18 + 30 + 63 = 124 \)

Step 2: Set Up the Equations

Using the calculations from Step 1, plug these values into the equations:1. \( 18a + 5b = 29 \) 2. \( 88a + 18b = 124 \)

Step 3: Solve the System of Equations

Solve the linear system to find values of \(a\) and \(b\):- Solve equation (1) for \(b\): \( b = \frac{29 - 18a}{5} \)- Substitute \(b\) in equation (2): \[ 88a + 18\left(\frac{29 - 18a}{5}\right) = 124 \]- Simplify and solve for \(a\): \[ 88a + \frac{522 - 324a}{5} = 124 \]- Multiply through by 5 to clear the fraction: \[ 440a + 522 - 324a = 620 \] \[ 116a = 98 \] \[ a = \frac{49}{58} \]- Substitute \(a\) back to find \(b\): \[ b = \frac{29 - 18\left(\frac{49}{58}\right)}{5} \approx 2.34 \]

Step 4: Write the Least Squares Line Equation

Now, write the equation of the line using the values of \(a\) and \(b\):The least squares line is: \[ y = \frac{49}{58}x + 2.34 \]

Step 5: Sketch the Points and Line

Plot the given points on a graph and draw the line \(y = \frac{49}{58}x + 2.34\). The line should pass near all points, visually confirming it fits well.

Key Concepts

Linear Equations

Linear equations form the backbone of understanding equations of lines. A linear equation in two variables, typically written as \( y = ax + b \), represents a straight line when plotted on a graph. Here, \(a\) is the slope, dictating how steep the line is, and \(b\) is the y-intercept, showing where the line crosses the y-axis.
To find a line that best fits a given set of data points, we need to determine the specific values of \(a\) and \(b\). In least squares regression, the line we fit minimizes the total squared vertical distances from each data point to the line.
For any given x-value, the corresponding y-value can be estimated using this line. It's a crucial method in data analysis, allowing us to predict trends and gain insights from data. Understanding this equation helps us apply it to real-world situations, like predicting sales or scientific measurements.

Calculus

Calculus, specifically concepts from differential calculus, is pivotal in least squares regression. We use calculus principles to derive the formulas needed to find the coefficients \(a\) and \(b\) of the least squares line.
The least squares method involves calculating derivatives to minimize the error between the observed data points and the line we are trying to fit. This process ensures we find the real line that reduces discrepancies in the data points' vertical distances from the line.
By understanding derivatives, one can comprehend how small changes in the slope and intercept affect the error, thus allowing precise adjustments for the best fit. Calculus not only aids in mathematical understanding but also enhances the application of these theoretical concepts in practical regression analysis.

Sigma Notation

Sigma notation, symbolized as \(\Sigma\), is a compact way to write long sums. In the context of least squares regression, we encounter sigma notation to express sums of variables or their products.
For example, \( \sum_{k=1}^n x_k \) denotes the sum of all \(x\)-values in the dataset. Similarly, \( \sum_{k=1}^n x_k^2 \) and \( \sum_{k=1}^n x_k y_k \) indicate the sum of squares of \(x\)-values and the sum of the product of respective \(x\) and \(y\) values, respectively.
Using sigma notation makes calculations in regression analysis manageable, especially when dealing with large datasets. It streamlines processes, allowing for efficient computation of needed sums, which are pivotal in forming and solving the equations to find the best-fit line.

Data Points Fitting

Fitting data points with a line involves finding a geometric representation that best summarizes the relationship between the variables. The least squares method is particularly significant in this regard.
The goal is to minimize the distances between the data points and the regression line, resulting in a line that represents the central trend of the data. By using methods such as least squares, the variance of the points from the line is minimized, providing a reliable model for predictions.
A well-fitted line is key in data analytics, aiding in forecasting and making informed decisions based on historical data. In practice, this involves using statistical software to calculate the regression line or utilizing calculators with built-in regression functions to verify manual calculations.