To determine the best mathematical model to use for the data, we should look at the \( R^2 \) values of the different regressions. The \( R^2 \) value indicates the proportion of the variance for a dependent variable that's explained by an independent variable or variables in a regression model. A higher \( R^2 \) value suggests a better fit of the model to the data.
Here are the \( R^2 \) values for the models provided:
A) Power model: \( R^2 = 0.522 \)
B) Linear model: \( R^2 = 0.237 \)
C) Exponential model: \( R^2 = 0.712 \)
D) Quadratic model: \( R^2 = 0.698 \)
Among these options, the exponential model (C), with an \( R^2 \) value of 0.712, has the highest \( R^2 \) value. This indicates that it explains the most variance in the data compared to the other models listed.
Therefore, the best mathematical model to use for the data is:
C) Exponential model for an exponential regression \( R^2 = 0.712 \).