Problem 16 - Differential Equations and Linear Algebra

Question

Find the equation of the least squares line to the given data points. $$(-1,5),(1,1),(2,1),(3,-3).$$

Step-by-Step Solution

Answer

The equation of the least squares line for the given data points is \(y = -1x + \frac{9}{4}\).

Step 1: Calculate the needed sums

Let's calculate the required sums, \(\sum{x}\), \(\sum{y}\), \(\sum{(x^2)}\), and \(\sum{(xy)}\) for the given data points. For \(\sum{x}\), add all the x-values together: (-1) + 1 + 2 + 3 = 5 For \(\sum{y}\), add all the y-values together: 5 + 1 + 1 + (-3) = 4 For \(\sum{(x^2)}\), square each x-value and then sum: (-1)^2 + 1^2 + 2^2 + 3^2 = 1 + 1 + 4 + 9 = 15 For \(\sum{(xy)}\), multiply each x-value with its corresponding y-value and then sum: (-1)(5) +(1)(1)+ (2)(1)+ (3)(-3)= -1 Now we have all the sums required for the slope and y-intercept formulae.

Step 2: Calculate the slope (m)

Use the formula for the slope and the calculated sums: \(m = \frac{4 \times (-1) - 5 \times 4}{4 \times 15 - 5^2} = \frac{-20}{20} = -1\) So the slope, m, of the least squares line is -1.

Step 3: Calculate the y-intercept (b)

Use the formula for the y-intercept and the calculated sums along with the slope: \(b = \frac{4 - (-1) \times 5}{4} = \frac{9}{4}\) So the y-intercept, b, of the least squares line is \(\frac{9}{4}\).

Step 4: Write the equation of the least squares line

Now that we have the slope (m) and y-intercept (b), we can write the equation of the least squares line in the form: \(y = mx + b\) Substitute the values of m and b: \(y = -1x + \frac{9}{4}\) This is the equation of the least squares line for the given data points.

Key Concepts

Slope Calculations

The process of finding the slope is crucial in forming the equation of the least squares line. The slope \(m\) of a line measures its steepness and direction. In the context of linear regression, it tells us how the dependent variable (often referred to as \(y\)) changes for each unit change in the independent variable (\(x\)). We find the slope using the formula:

• \(m = \frac{n(\sum{(xy)}) - (\sum{x})(\sum{y})}{n(\sum{x^2}) - (\sum{x})^2}\)

where:

• \(n\) is the number of data points.
• \(\sum{x}\) and \(\sum{y}\) are the sums of the x and y values, respectively.
• \(\sum{(xy)}\) is the sum of the products of each pair of x and y values.
• \(\sum{x^2}\) is the sum of the squares of the x values.

In our case, we calculated all these sums and numbers based on the given data points. Plugging these values into the formula, we found that the slope \(m\) equals -1.
This negative slope indicates that as x increases, y tends to decrease, highlighting an inverse relationship with the given data.

Y-intercept Calculation

Once the slope has been determined, the next step in linear regression is finding the y-intercept, \(b\). The y-intercept represents the point where the line crosses the y-axis. It is the value of \(y\) when \(x\) is zero. The formula for calculating the y-intercept is:

• \(b = \frac{\sum{y} - m\sum{x}}{n}\)

Here:

• \(m\) is the slope, which we calculated as -1.
• \(\sum{y}\) and \(\sum{x}\) are the sums of the y and x values, respectively.
• \(n\) is the number of data points.

By substituting the known values into this formula, we derived the y-intercept as \(\frac{9}{4}\).
This means our line begins at this point on the y-axis before descending with a slope of -1.

Linear Regression

Linear regression is a simple and effective method for modeling the relationship between two variables by fitting a linear equation to observed data. The main objective of linear regression is to find the line that best represents the data points, which is termed the least squares line. This line minimizes the sum of the squared differences between the observed values and the values predicted by the line.
The general equation of the least squares line is:

• \(y = mx + b\)

Where:

• \(m\) is the slope.
• \(b\) is the y-intercept.

In our exercise, we've identified the slope \(m\) as -1 and the y-intercept \(b\) as \(\frac{9}{4}\).
This produces the equation \(y = -1x + \frac{9}{4}\),
which represents the line that best fits our given data points.
Linear regression is a foundational concept in statistics and is broadly applicable in numerous fields, including economics, biology, and social sciences, for making informed predictions and decisions.