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.