12. Find the line y = C + Dt which gives the best least squares fitting line to the points (t, y) = (0, 6), (1, 4), (2, 0). Show and explain your work.
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.
![**Problem 12: Least Squares Fitting Line**
12. Find the line \( y = C + Dt \) which gives the best least squares fitting line to the points \((t,y) = (0, 6), (1, 4), (2, 0)\). Show and explain your work.
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### Solution:
1. **Define Variables:**
- The points are given as \((0, 6)\), \((1, 4)\), and \((2, 0)\).
- We aim to find constants \(C\) and \(D\) so that the line \( y = C + Dt \) fits these points in the least squares sense.
2. **Set Up the System of Equations:**
- Use the general form \( y = C + Dt \) for each point.
- For the point \((0, 6)\): \( 6 = C + D \cdot 0 \Rightarrow 6 = C \)
- For the point \((1, 4)\): \( 4 = C + D \cdot 1 \Rightarrow 4 = C + D \)
- For the point \((2, 0)\): \( 0 = C + D \cdot 2 \Rightarrow 0 = C + 2D \)
3. **Solve the System:**
- From the first equation: \( C = 6 \)
- Substitute \( C = 6 \) into the other two equations:
- \( 4 = 6 + D \Rightarrow D = 4 - 6 \Rightarrow D = -2 \)
- Check the third equation to ensure consistency:
\( 0 = 6 + 2(-2) = 6 - 4 = 2 \) which does not hold true directly, implying a need for least squares fitting process mathematically rigorous.
4. **Apply Least Squares Method (find \( C \) and \( D \) minimizing the squared differences):**
- Form normal equations by minimizing the sum of squares of the residuals i.e.,
\[ \min_{C, D} \sum (y_i - (C + Dt_i))^2 \]
- For each parameter includes first derivatives and setting to zero:
- Residuals are \(r_i = y_i - (C](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0d2fdd51-a813-4b36-89e9-f9581acfc2ee%2F38b85196-c6d0-475c-8cc1-17454fbac04a%2Flkqdq0i_processed.png&w=3840&q=75)
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