Problem 4 - Differential Equations and Linear Algebra

Question

Find the equation of the least squares line associated with the given set of data points. (2,-2),(1,-3),(0,0).

Step-by-Step Solution

Answer

The equation of the least squares line associated with the given set of data points is: \[y = -x - \frac{2}{3}\]

Step 1: Find the required sums

First, find the sums of the x-values, y-values, product of x and y, and square of x-values over all given data points. \(\sum{x} = 2 + 1 + 0 = 3\) \(\sum{y} = -2 - 3 + 0 = -5\) \(\sum(xy) = (2)(-2) + (1)(-3) + (0)(0) = -4 - 3 = -7\) \(\sum{x^2} = 2^2 + 1^2 + 0^2 = 4 + 1 = 5\)

Step 2: Calculate the slope (m)

Now, use the formula for the slope (m) with the sums found in Step 1 and the number of data points (n = 3): \[m = \frac{n \sum{(xy)} - \sum{x} \sum{y}}{n \sum{x^2} - (\sum{x})^2}\] \[m = \frac{3(-7) - (3)(-5)}{3(5) - (3)^2} = \frac{-21 + 15}{15 - 9} = \frac{-6}{6} = -1\]

Step 3: Calculate the y-intercept (b)

Now, use the formula for the y-intercept (b) with the slope (m = -1), the sums found in Step 1, and the number of data points (n=3): \[b = \frac{\sum{y} - m\sum{x}}{n}\] \[b = \frac{-5 - (-1)(3)}{3} = \frac{-5 + 3}{3} = \frac{-2}{3}\]

Step 4: Write the equation of the least squares line

Using the values for the slope (m = -1) and the y-intercept (b = -2/3), we can now write the equation of the least squares line in the form \(y = mx + b\): \[y = (-1)x - \frac{2}{3}\] The equation of the least squares line associated with the given set of data points is: \[y = -x - \frac{2}{3}\]

Key Concepts

Data Points

When dealing with linear regression, data points are crucial. They are essentially pairs of numbers that tell you what values you are plotting on a graph.
Each data point consists of an x-value (independent variable) and a y-value (dependent variable).
In this exercise, the data points are (2,-2), (1,-3), and (0,0).

• The x-values are 2, 1, and 0.
• The y-values are -2, -3, and 0.

These data points lay the foundation for calculating the least squares line, as they help to determine how the line should ideally fit through the data.

Slope Calculation

The slope of a line tells you how steep the line is. It indicates how much the y-value increases or decreases as the x-value increases by one unit. Calculating the slope in least squares involves a specific formula.
Based on our data points, the formula is: \[m = \frac{n \sum{(xy)} - \sum{x} \sum{y}}{n \sum{x^2} - (\sum{x})^2}\] Here:

• \(n\) is the number of data points, which is 3.
• \(\sum{xy} = -7\).
• \(\sum{x} = 3\).
• \(\sum{y} = -5\).
• \(\sum{x^2} = 5\).

Plugging those into the formula gives us a slope \(m = -1\).
This tells us that the line decreases by one unit in y for every unit increase in x.

Y-intercept

The y-intercept is the point where the line crosses the y-axis.
For calculating it in least squares, use this formula: \[b = \frac{\sum{y} - m\sum{x}}{n}\] In our case:

• \(m = -1\) (from our slope calculation).
• \(\sum{x} = 3\).
• \(\sum{y} = -5\).
• \(n = 3\).

Inserting these into the formula, we find the y-intercept \(b = -\frac{2}{3}\).
This means that the line crosses the y-axis at \(-\frac{2}{3}\).

Equation of a Line

Once we have both the slope and the y-intercept, we can write the equation of the line. The standard form of the equation is: \[y = mx + b\]Here, \(m\) is the slope and \(b\) is the y-intercept.
For our least squares line, with \(m = -1\) and \(b = -\frac{2}{3}\), the equation becomes: \[y = -x - \frac{2}{3}\] This equation represents a line on a graph that best fits the given data points. This is the least squares line, minimizing the distance from each point to the line.