To determine which model is best suited for making predictions about the given data set, we need to consider the nature of the data and the relationships among the variables. In this case, we have pairs of numerical variables (Variable 1 and Variable 2) and we are tasked with predicting one based on the other.
-
Linear Model: \( y = 2.0259 x + 0.167763 \)
- Simple and interpretable, typically used when the relationship appears linear.
-
Quadratic Model: \( y = 0.00319504 x^2 + 1.98351 x + 0.273178 \)
- Good for modeling relationships that have a curve to them (e.g., parabolic).
-
Cubic Model: \( y = -0.00329151 x^3 + 0.0689209 x^2 + 1.60841 x + 0.831659 \)
- Captures more complex relationships with a potential asymmetrical curve.
-
Exponential Model: \( y = 4.74127 * 1.15404^x \)
- Used if the data exhibits exponential growth or decay.
To select the best model, consider the following steps:
- Visual Inspection: Plot the data points and see how they relate to each type of model. Are they linear, quadratic, cubic, or exponential?
- Goodness of Fit: Use statistical measures such as R² (coefficient of determination) to assess how well the model explains the variability of the data.
- Error Analysis: Look at residuals to determine if the model is appropriate. They should be randomly distributed for a good model.
Since we do not have the ability to graph this data or calculate errors directly here, I suggest you start with the linear model for its simplicity and interpretability. If the data visibly fits a line closely, this would be the model to select. If you notice a clear curve or complexity in the data trend, consider trying the quadratic or cubic models.
Ultimately, the choice of model depends on empirical testing and validation to see which fit provides the best accuracy for prediction. Based on the information provided, a visual inspection or statistical analysis would be required to accurately determine the best model.