Problem 6 - Advanced Engineering Mathematics

Question

In Problems \(1-6\), find the least squares line for the given data. $$ (1,2),(2,2.5),(3,1),(4,1.5),(5,2),(6,3.2),(7,5) $$

Step-by-Step Solution

Answer

The least squares line is \( y = 0.4x + 0.6286 \).

Step 1: Understanding the Problem

We need to find the least squares line, or the line of best fit, for the given data points. This involves determining the slope, \(m\), and y-intercept, \(b\), of the linear equation \(y = mx + b\) that minimizes the sum of squared differences between the observed values (data points) and the values predicted by the line.

Step 2: Use the Least Squares Formulas

The formulas for the slope \(m\) and y-intercept \(b\) of the least squares line are:\[ m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2} \]\[ b = \frac{(\sum y) - m(\sum x)}{N} \]where \(N\) is the number of points, \(x\) and \(y\) are the data points, and the summations are over all data points.

Step 3: Calculate Summations Needed

Calculate the necessary summations:\(\sum x = 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28\)\(\sum y = 2 + 2.5 + 1 + 1.5 + 2 + 3.2 + 5 = 17.2\)\(\sum xy = 1 \cdot 2 + 2 \cdot 2.5 + 3 \cdot 1 + 4 \cdot 1.5 + 5 \cdot 2 + 6 \cdot 3.2 + 7 \cdot 5 = 80.4\)\(\sum x^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 = 140\)

Key Concepts

Linear Regression

Linear regression is a statistical method used to model the relationship between two variables by fitting a linear equation to observed data. The line that results from this process is known as the least squares line. This method aims to predict the outcome variable based on the input variable.

The basic idea is to find the equation of a line, which best captures the trend in the data. The line is usually represented as:

• \(y = mx + b\)
• Where \(y\) is the dependent variable, \(x\) is the independent variable, \(m\) is the slope of the line, and \(b\) is the y-intercept.

By using linear regression, we can predict future values by extrapolating the line. It is widely used in numerous fields such as economics, biology, and social sciences. For effective regression, it is paramount that the relationship between variables is linear.

Line of Best Fit

The line of best fit in linear regression refers to the straight line that best represents the data points on a scatter plot. It might not pass through all the points, but it seeks to minimize the distance from each point to the line.

This minimization is based on the principle of least squares, which entails calculating a line that minimizes the sum of squared vertical distances (errors) from the data points to the line itself. The line of best fit helps in understanding the trend in data and is used for making predictions.

• While determining this line, calculations are made to find the optimal slope \(m\) and y-intercept \(b\).
• These values are polished through the methods including calculations of the sums \( \sum x, \sum y, \sum xy, \sum x^2 \).

Understanding how to effectively draw and calculate this line is crucial for accurate data analysis.

Sum of Squared Differences

The sum of squared differences, also referred to as the sum of squared errors, is a method used in the least squares approach to determine how well the line fits the data. Here is how it works:

• For each data point, the vertical distance (residual) from the actual data point to the predicted value on the line is calculated.
• This distance is then squared, ensuring that positive and negative distances do not cancel each other out.
• Finally, these squared differences are all summed together to form the total sum of squared differences.

This sum showcases the fit of the line: smaller values indicate a more accurate fit. By minimizing this sum, using the least squares method, the line of best fit effectively describes the trend in the data and allows for future predictions. It’s a foundational element of statistical modeling.