Let x represent years coded as 1,2,3, …. for years starting in 1980 and let y represent the numb

step 1 Based on these results, the model that is best would be the quadratic. And that's because it's t

Let x represent years coded as 1,2,3, …. for years starting...

Let $x$ represent years coded as $1,2,3, \ldots$. for years starting in 1980 and let $y$ represent the numbers of points scored in each Super Bowl from $1980 .$ Using the data from 1980 to the last Super Bowl at the time of this writing, we obtain the following values of $R^{2}$ for the different models: linear: 0.147; quadratic: 0.255; logarithmic: 0.176; exponential: 0.175; power: $0.203 .$ Based on these results, which model is best? Is the best model a good model? What do the results suggest about predicting the number of points scored in a future Super Bowl game?

Key Concepts

Coefficient of Determination (R²)

R² is a statistical measure in regression that indicates the proportion of the variance in the dependent variable that is predictable from the independent variable(s). It is used to assess the goodness-of-fit of a model, with higher values generally indicating that the model explains a greater portion of the variability of the response data around its mean. However, a higher R² does not always imply that the model is the best choice, especially if the value is relatively low overall.

Model Selection Based on R²

When comparing multiple regression models (such as linear, quadratic, logarithmic, exponential, and power models), using R² values is one method of selecting the best model. The model with the highest R² value is typically seen as providing the best fit to the data. In this context, the quadratic model, having the highest R², is considered the best among those tested.

Goodness-of-Fit and Model Quality

Even though the quadratic model has the highest R² value, its goodness-of-fit may still be debatable if the R² value is low overall. A low R² suggests that a significant amount of variation is unexplained by the model, meaning that while one model may be better than the others, it may still not be a particularly robust or reliable model for capturing the underlying relationship.

Predictive Limitations and Extrapolation

The fact that the best fitting model explains only a small proportion of the variance indicates that predictions, especially for future outcomes, should be approached with caution. Extrapolating from a model that captures limited variability may lead to unreliable or uncertain forecasts, highlighting the inherent difficulty in accurately predicting future events based on this analysis.

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