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The problem asks for the regression with the weakest linear relationship between \( x \) and \( y \), and the key to answering this question is the correlation coefficient, denoted by \( r \).

- The correlation coefficient \( r \) measures the strength and direction of a linear relationship between two variables. Its values range from \( -1 \) (perfect negative correlation) to \( +1 \) (perfect positive correlation), with values closer to 0 indicating a weaker linear relationship.

From the provided dataThe problem asks for the regression with the weakest linear relationship between \( x \) and \( y \), and the key to answering this question is the correlation coefficient, denoted by \( r \).

- The correlation coefficient \( r \) measures the strength and direction of a linear relationship between two variables. Its values range from \( -1 \) (perfect negative correlation) to \( +1 \) (perfect positive correlation), with values closer to 0 indicating a weaker linear relationship.

From the provided 

- Regression 1: \( r = -0.4712 \)
- Regression 2: \( r = 0.9868 \)
- Regression 3: \( r = -0.8876 \)
- Regression 4: \( r = 0.2151 \)

Since \( r = 0.2151 \) (from Regression 4) is the closest to zero, it represents the weakest linear relationship between \( x \) and \( y \).

**Answer: Regression 4**

Do you want more details about correlation coefficients or have any other questions?

Here are 5 related questions for further exploration:
1. How is the correlation coefficient \( r \) calculated from data?
2. What is the difference between positive and negative values of \( r \)?
3. What are the assumptions behind linear regression models?
4. How does the slope \( a \) influence the regression line's steepness?
5. Can a weak \( r \)-value indicate a nonlinear relationship?

**Tip:** A correlation coefficient near zero does not imply no relationship, just that it's not linear. Nonlinear relationships might still exist.

Which of the following regressions represents the weakest linear relationship between x and y? The correlation coefficients (r) are given for four regressions.

Learn how to identify the regression equation with the weakest linear relationship between variables x and y by analyzing the correlation coefficient (r). This problem involves linear regression analysis and understanding the significance of the r-value.

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