4. Find the parameters and equation of the least squares regression line using the points (1,0), (3,3), and (5,6). 1 1] where X = |1 3 and Y = 3 [0] A = (XTX)-'x"Y = L1 5] L61 answer 3 3 y = -,+5X 2
Correlation
Correlation defines a relationship between two independent variables. It tells the degree to which variables move in relation to each other. When two sets of data are related to each other, there is a correlation between them.
Linear Correlation
A correlation is used to determine the relationships between numerical and categorical variables. In other words, it is an indicator of how things are connected to one another. The correlation analysis is the study of how variables are related.
Regression Analysis
Regression analysis is a statistical method in which it estimates the relationship between a dependent variable and one or more independent variable. In simple terms dependent variable is called as outcome variable and independent variable is called as predictors. Regression analysis is one of the methods to find the trends in data. The independent variable used in Regression analysis is named Predictor variable. It offers data of an associated dependent variable regarding a particular outcome.
![**Title: Understanding Least Squares Regression**
**Introduction to Least Squares Regression**
In mathematical modeling, least squares regression is a statistical method used to determine the best-fit line through a set of points. Here, we discuss how to find the parameters and equation of the least squares regression line using given data points.
**Problem Statement**
Using the points (1,0), (3,3), and (5,6), find the parameters and equation of the least squares regression line.
**Solution**
1. **Matrices Definition**:
To find the regression line using the least squares method, you must define matrices \(X\) and \(Y\):
\[
X = \begin{bmatrix} 1 & 1 \\ 1 & 3 \\ 1 & 5 \end{bmatrix}, \quad Y = \begin{bmatrix} 0 \\ 3 \\ 6 \end{bmatrix}
\]
2. **Regression Equation**:
The formula for the regression equation in matrix form is:
\[
A = (X^TX)^{-1}X^TY
\]
Where \(A\) represents the parameters of the model.
3. **Calculation of Parameters**:
By solving the matrix equation, the result is:
\[
A = \begin{bmatrix} 3 \\ 2 \end{bmatrix}
\]
4. **Equation of the Regression Line**:
Using the parameters found:
- The equation of the regression line is \(y = 3 + 2x\).
**Conclusion**
The least squares regression line provides the most accurate fit for a set of data points. In this example, the derived regression equation, \(y = 3 + 2x\), demonstrates how changes in \(x\) will impact \(y\) using the identified slope and y-intercept.
Understanding the least squares method aids in various fields, allowing for predictions and insights when analyzing data trends.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffe09358e-0876-4919-81c1-67cd5ad8f5f7%2F8ce33eb4-821e-4fae-a61c-636de4d84f39%2Fwse7v9p_processed.jpeg&w=3840&q=75)
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