Problem 4 - Advanced Engineering Mathematics

Question

Find the least squares line for the given data. $$ (0,0),(2,1.5),(3,3),(4,4.5),(5,5) $$

Step-by-Step Solution

Answer

The least squares line is \( y = 0.9074x + 0.2673 \).

Step 1: Understand the Problem

To find the least squares line (best-fit linear line), we need to determine the line \( y = mx + c \) that minimizes the sum of the squares of the vertical distances between the points and the line.

Step 2: Calculate Means

Calculate the means of the \(x\) values and \(y\) values.\[ \bar{x} = \frac{0 + 2 + 3 + 4 + 5}{5} = 2.8 \] \[ \bar{y} = \frac{0 + 1.5 + 3 + 4.5 + 5}{5} = 2.8 \]

Step 3: Compute the Slope (m)

Use the formula for the slope \( m \): \[ m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \] \[ m = \frac{(0-2.8)(0-2.8) + (2-2.8)(1.5-2.8) + (3-2.8)(3-2.8) + (4-2.8)(4.5-2.8) + (5-2.8)(5-2.8)}{(0-2.8)^2 + (2-2.8)^2 + (3-2.8)^2 + (4-2.8)^2 + (5-2.8)^2} \] \[ m = \frac{31.6}{10.8} = 0.9074 \]

Step 4: Compute the Intercept (c)

Use the means and the slope to find the y-intercept \( c \) with: \[ c = \bar{y} - m\bar{x} \] \[ c = 2.8 - 0.9074 \times 2.8 = 0.2673 \]

Step 5: State the Least Squares Line Equation

The least squares line equation is \( y = mx + c \) where \( m = 0.9074 \) and \( c = 0.2673 \). Therefore, \[ y = 0.9074x + 0.2673 \]

Key Concepts

Linear Regression

Linear regression is a fundamental concept in statistics that helps us understand the relationship between two variables by fitting a linear line to observed data. This process involves finding the line of best fit through data points plotted on a graph, where the line can be represented by the equation \( y = mx + c \). Here:

• \( m \) is the slope of the line, showing how much \( y \) changes for a unit change in \( x \).
• \( c \) is the y-intercept, the point where the line meets the y-axis when \( x = 0 \).

In the context of least squares regression, which is a type of linear regression, the goal is to minimize the sum of the squared differences between observed and predicted values. The smaller this sum, the closer the line fits the data.
To practically apply linear regression, you need data points and a method to calculate the best-fit line that represents the trend in your data. Identifying this trend is crucial for making predictions or understanding underlying patterns in any data set.

Data Analysis

Data analysis involves inspecting, cleaning, transforming, and modeling data to discover useful information, support decision-making, and draw conclusions. In the context of linear regression, data analysis helps you find relationships between variables. For example:

• The first step is understanding the data and determining its key characteristics.
• Next, compute basic statistics such as the mean of both the independent (\( x \)) and dependent (\( y \)) variables.

In our example, we calculated the mean of the \( x \) values and the \( y \) values as 2.8. The means are essential in determining the overall trend and patterns present in the data. Using these calculations, we further analyze the data to find consistent patterns or outliers that can affect the results of our regression analysis.
Successful data analysis requires clear understanding and careful computation to ensure the results are accurate and reliable. It transforms raw data into something meaningful, enabling you to draw conclusions and support statistical hypotheses.

Statistical Methods

Statistical methods are techniques used to collect, analyze, interpret, and present data. Among these, the least squares regression is a widely used statistical technique. It is effective in modeling and finding the line of best fit for data.

• The method begins with calculating the slope \( m \), which illustrates how change in one variable affects the other.
• The next step is determining the y-intercept \( c \), using the mean values and the computed slope.

The least squares method is analytically solving for \( m \) and \( c \) by minimizing the sum of the squares of the vertical distances of the points from the line. These calculations involve derivative calculus, but the formula provides a straightforward means to find the best-fit line.
Understanding and utilizing statistical methods like least squares regression are invaluable in making data-driven decisions. Whether for predicting trends, understanding relationships, or optimizing processes, these methods provide the necessary tools to extract insights effectively from complex data sets.