Question

Let x represent years coded as​ 1, 2,​ 3, ... for years starting in​ 1980, and let y represent the numbers of points scored in each annual Super Bowl beginning in 1980. Using the data from 1980 to the last Super Bowl at the time of this​ writing, one can obtain the following values of for the different​ models: linear:​ 0.008; quadratic:​ 0.023; logarithmic:​ 0.0004; exponential:​ 0.027; power: 0.007. 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?

Answer 1

The logarithmic model has the best fit for predicting Super Bowl scores with a value of 0.0004, indicating the least error among the models. Extrapolation outside the model's range can lead to inaccurate results, as predictive models are usually only reliable within the range of data they were built upon.

Step-by-step explanation:

To determine which model is best for predicting the number of points scored in the annual Super Bowl starting in 1980, we examine the values for the different models: linear: 0.008; quadratic: 0.023; logarithmic: 0.0004; exponential: 0.027; power: 0.007. The best model is typically the one with the smallest value, as it indicates the least error in prediction. Thus, the logarithmic model has the best fit with a value of 0.0004. While the logarithmic model is the best among these, we must still assess its usefulness. Considering the context and the nature of sporting events, predicting future scores can be complex due to many variables, therefore further validation of this model's predictive accuracy is needed before using it for future game predictions.

When applying these models for extrapolation, like predicting for a year not within the range of 1981 to 2002, the result may not be accurate or sensible, as shown by the negative value of -725 for the year 1970. This indicates the limitations of the model and the importance of using data within the appropriate range.