https://file.aiquickdraw.com/mathbot/file/1729396427592xkvc48kj.jpg

Let's go through each part of the problem shown in the image systematically. We have a dataset and calculations related to the least-squares regression line. Here’s how to solve each question:

### Part (a)
The total variation in the sample \( y \)-values is given by the **total sum of squares (SST)**, which is represented by \( \sum (y_i - \bar{y})^2 \). For this data, we find the value from the column sum under \((y_i - \bar{y})^2\), which is **77.8920**.

### Part (b)
The value \( r^2 \) is the proportion of the total variation in the sample \( y \)-values that is explained by the linear relationship between \( x \) and \( y \). It is calculated as:
\[
r^2 = \frac{\text{SSReg}}{\text{SST}}
\]
where \( \text{SSReg} = \sum (\hat{y}_i - \bar{y})^2 = 63.9627 \) and \( \text{SST} = 77.8920 \).

\[
r^2 = \frac{63.9627}{77.8920} \approx 0.82
\]
So, \( r^2 \) rounded to two decimal places is **0.82**.

### Part (c)
The least-squares regression line minimizes the **sum of the squared residuals**, which is the column sum under \((y_i - \hat{y})^2\), equal to **15.9561**. This is why it’s called the “best fit” line.

### Summary Answers
1. **Part (a)**: Total variation = \((y_i - \bar{y})^2\), value: 77.8920
2. **Part (b)**: \( r^2 = 0.82 \)
3. **Part (c)**: The minimized quantity is \((y_i - \hat{y})^2\), value: 15.9561

Would you like further details or explanations on any of these points?

### Relative Questions:
1. How is the least-squares regression line derived mathematically?
2. What is the significance of \( r^2 \) in interpreting the fit of a regression model?
3. Why is minimizing the sum of squared residuals important in regression analysis?
4. How can you determine if a regression model is appropriate for the given data?
5. How does changing the slope or intercept of the line affect the \( r^2 \) value?

### Tip:
Always verify the sums and computations when working with least-squares regression, as small errors in the sums can lead to incorrect conclusions about the model's accuracy.

The problem involves calculating the total variation in y-values, finding the coefficient of determination (r^2), and explaining why the least-squares regression line is considered the 'best fit' line.

Explore how to calculate the total variation in sample y-values, determine the coefficient of determination (r^2), and understand why the least-squares regression line is considered the 'best fit' line. This problem covers essential concepts in regression analysis, suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation