https://file.aiquickdraw.com/mathbot/file/1729285899289vqmhj12w.jpg

Match the coefficient of determination to the scatter diagram. The scales on the​ x-axis and​ y-axis are the same for each scatter diagram. 

left parenthesis a right parenthesis Upper R squared equals 0.90(a) R2=0.90​, 

nbsp left parenthesis b right parenthesis Upper R squared equals 0.58 (b) R2=0.58​, 

left parenthesis c right parenthesis Upper R squared equals 1

To match the coefficient of determination (R²) to the scatter diagrams, let's analyze each scatter plot based on how well the data points fit the regression line. The coefficient of determination (R²) measures how well the explanatory variable (x-axis) predicts the response variable (y-axis), where:

- **R² = 1** means a perfect fit: all data points lie exactly on the regression line.
- **R² close to 0** means a very poor fit: the regression line doesn't explain much of the variation in the data.
- **R² between 0 and 1** means a partial fit, with the data points somewhat spread out around the line.

Looking at the scatter plots:

1. **Scatter plot I**: 
   - This shows all points lying exactly on the regression line, indicating a perfect fit.
   - **R² = 1**.
   
2. **Scatter plot II**: 
   - The points are quite close to the regression line, but there is some variation, indicating a strong fit but not perfect.
   - **R² = 0.90**.

3. **Scatter plot III**:
   - The points are more spread out, with significant deviation from the line, indicating a moderate fit.
   - **R² = 0.58**.

So, the matching would be:

- **(a) R² = 0.90** → Scatter plot II
- **(b) R² = 0.58** → Scatter plot III
- **(c) R² = 1** → Scatter plot I

Let me know if you'd like further details or if you have any questions! 

Here are five questions related to this explanation:
1. How is the coefficient of determination calculated?
2. What are other factors that affect the R² value in regression analysis?
3. Can R² ever be negative? Why or why not?
4. How does the R² value change if outliers are introduced to the data?
5. What is the relationship between R² and the correlation coefficient (r)?

**Tip**: When interpreting R² values, always consider the context of your data. A high R² does not always imply that the model is appropriate.

Match the coefficient of determination to the scatter diagram. The scales on the x-axis and y-axis are the same for each scatter diagram. 
(a) R² = 0.90 
(b) R² = 0.58 
(c) R² = 1

In this problem, match scatter plots to their respective R² values. Learn about regression analysis and the coefficient of determination, including how well data points fit a regression line. Suitable for students in Grades 10-12 and introductory undergraduate courses.

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