Find the least-squares cubic fit to the given data.
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.
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## Fitting a Least-Squares Cubic Polynomial
### Problem Statement:
Given the table of data:
\[
\begin{array}{c|ccccc}
x & -2 & -1 & 0 & 1 & 2 \\
\hline
y & 1 & 7 & 3 & -2 & 6 \\
\end{array}
\]
Find the least-squares cubic fit to the given data.
### Explanation:
The table provides a set of \(x\) and \(y\) values, which represent points in a coordinate system. The goal is to find a cubic polynomial (a polynomial of degree 3) that best fits the given points in the least-squares sense. This means that the sum of the squares of the residuals (the differences between the actual \(y\) values and those predicted by the polynomial) should be minimized.
A cubic polynomial can be represented as:
\[
y = ax^3 + bx^2 + cx + d
\]
Where \(a\), \(b\), \(c\), and \(d\) are coefficients that need to be determined.
### Steps to Find the Least-Squares Cubic Fit:
1. **Setup Equations:**
Create a system of linear equations based on the given \(x\) and \(y\) values.
2. **Matrix Representation:**
Represent the system in matrix form and use matrix operations to solve for the coefficients \(a\), \(b\), \(c\), and \(d\).
3. **Minimize Residuals:**
Solve the normal equations derived from the least-squares condition to minimize the sum of the squared residuals.
### Conclusion:
By solving for these coefficients, you will obtain the cubic polynomial that best fits the given data points in the least-squares sense. This polynomial can then be used to predict \(y\) values for other \(x\) values within the range of the given data.
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This explanation and problem statement can serve as a basis for lessons on polynomial fitting, least-squares methods, and numerical analysis techniques.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fda2e0952-5073-44c4-aa37-c19a81cec12c%2F77cd0bd3-2a4d-4224-b41f-cbe2fc6a1762%2Fxyt4huj_processed.jpeg&w=3840&q=75)
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