Problem 18 - Differential Equations and Linear Algebra

Question

Find the equation of the least squares line to the given data points. $$(-3,1),(-2,0),(-1,1),(0,-1),(2,-1).$$

Step-by-Step Solution

Answer

The equation of the least squares line that best fits the given data points is \(y \approx -0.4x + 0.32\).

Step 1: Find the average of the x and y coordinates

Calculate the average of the x and y values for the given data points. This will help us to determine the slope and y-intercept of the least squares line. $$ \begin{aligned} \bar{x} &= \frac{-3+-2+-1+0+2}{5} = -\frac{4}{5}\\ \bar{y} &= \frac{1+0+1+-1+-1}{5} = 0 \end{aligned} $$

Step 2: Calculate the slope (m) of the line

To find the slope of the least squares line, we can use the following formula: $$ m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} $$ Substitute the values of x and y into the formula and calculate the slope: $$ \begin{aligned} m &= \frac{(-3-(-\frac{4}{5}))(1-0) + (-2-(-\frac{4}{5}))(0-0) + (-1-(-\frac{4}{5}))(1-0)+\\ &(0-(-\frac{4}{5}))(-1-0)+(2-(-\frac{4}{5}))(-1-0)}{(-3-(-\frac{4}{5}))^2+(-2-(-\frac{4}{5}))^2+(-1-(-\frac{4}{5}))^2+(0-(-\frac{4}{5}))^2+(2-(-\frac{4}{5}))^2}\\ m & \approx -0.4 \end{aligned} $$

Step 3: Calculate the y-intercept (b) of the line

To find the y-intercept of the least squares line, we can use the following formula: $$ b = \bar{y} - m\bar{x} $$ Substitute the values of \(m\) and \(\bar{x}\) into the formula and calculate the y-intercept: $$ \begin{aligned} b &= 0 - (-0.4)\left(-\frac{4}{5}\right)\\ b & \approx 0.32 \end{aligned} $$

Step 4: Write the equation of the least squares line

Now that we have calculated the slope and y-intercept, we can write the equation of the least squares line in the form y = mx + b: $$ y \approx -0.4x + 0.32 $$ This is the equation of the least squares line that best fits the given data points.

Key Concepts

Data Points

In the context of finding a least squares line, data points are the specific coordinates on a graph that you want to analyze and understand. These are the

• measured values,
• experimental results,
• or observed outcomes,

that are plotted, usually in the form of \((-3,1), (-2,0), (-1,1), (0,-1), (2,-1)\).
Each data point consists of an

• x-value (horizontal axis),
• y-value (vertical axis).

For example, the point \((-3,1)\) represents an x-value of -3 and a y-value of 1.

These points are used to establish a relationship between variables by approximating a line. The line's purpose is to give a "best fit," predicting how the values might relate if they were to continue. It helps identify trends or make forecasts about

• future points,
• missing data,
• or general patterns.

Understanding the position and arrangement of data points is crucial to creating an accurate least squares line.

Slope Calculation

The slope of a line in the context of least squares represents the rate at which one variable changes with respect to another. It is a measure of how steep the line is.
To calculate the slope (\(m\)) of the least squares line, you use the formula:
\[ m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \]This formula takes into account all of the data points, as well as their deviations from their averages (\(\bar{x}\) and \(\bar{y}\)).

Here's how it works:

• You calculate the deviation for each x-value by subtracting the mean of x (\(\bar{x}\)) from each \(x_i\).
• Similarly, compute the deviation for each y-value by subtracting the mean of y (\(\bar{y}\)) from each \(y_i\).
• The sum of the products of these deviations gives the numerator.
• The sum of the squared deviations of x-values gives the denominator.

In our example, the slope was calculated to be approximately \(-0.4\) , indicating a downward trend as you move from left to right along the line.

Y-Intercept

The y-intercept is a key component of the equation of a line, famously expressed as \(y = mx + b\). This component, denoted by \(b\), represents the point where the line crosses the y-axis. It's the y-value when \(x = 0\).
Finding \(b\) for a least squares line involves using the values of the slope (\(m\)) and the mean x-value (\(\bar{x}\)), calculated as follows:\[ b = \bar{y} - m\bar{x} \]This formula essentially adjusts the average \(y\)-value downward or upward based on the slope of the line.

In our particular exercise, the y-intercept was found to be approximately \(0.32\). This means that when \(x = 0\), the prediction based on the least squares line would be \(y = 0.32\). Understanding this can help conceptualize how high or low the line starts before being influenced by the slope's direction.

This y-intercept indicates how the line positions itself vertically on the graph, providing crucial information about the general level of y-values independent of the x-variable.