Question

linear association, positive correlation
linear association, negative correlation
nonlinear association, strong correlation
nonlinear association, weak correlation

Explanation:

• Linear vs Non - linear: In a linear association, the data points would approximately follow a straight - line pattern. Here, the data points do not follow a straight - line pattern. For example, if we consider a simple linear model \(y = mx + b\), the curvature in the data (the non - straight - line trend) indicates a non - linear association.
• Positive vs Negative Correlation: In a positive correlation (for linear cases), as \(x\) increases, \(y\) generally increases. In a negative correlation (for linear cases), as \(x\) increases, \(y\) generally decreases. Since the association is non - linear, we don't use the positive/negative terms in the traditional linear - correlation sense. But if we consider the overall trend (ignoring the linearity for a moment), as \(x\) increases from \(0\) to \(4\), \(y\) has an increasing trend. However, the key is the non - linearity.
• Strong vs Weak Correlation: A strong correlation (whether linear or non - linear) means that the data points are closely clustered around a "pattern" (a line for linear, or a curve for non - linear). Here, the data points are relatively close to each other and follow a discernible non - linear pattern. A weak correlation would have data points more spread out with no clear pattern.

Answer:

linear association, positive correlation