Question

Interpreting\({R^2}\)In Exercise 2, the quadratic model results in = 0.255. Identify the percentage of the variation in Super Bowl points that can be explained by the quadratic model relating the variable of year and the variable of points scored. (Hint: See Example 2.) What does the result suggest about the usefulness of the quadratic model?

Short answer

25.5% of the variation in Super Bowl points is explained by the quadratic model between the variable “year” and the variable “Super Bowl scores”.

Since the value of the \({R^2}\)is low, it can be said that the given quadratic model is not very useful.

Step-by-step solution

Given information

The relation between y (number of points scored in Super Bowl) and x (years starting from 1980 and coded as 1,2,3…..) is modelled using a quadratic model with \({R^2}\) value equal to 0.255.

Interpretation of \({R^2}\)

The value of\({R^2}\)represents the amount of variation explained by the relation between the response variable and the predictor variable.

Here, the response variable is the number of Super Bowl scores and the predictor variable is the year in which the game was played.

The above relation is modeledusing a quadratic model and the value of the\({R^2}\)is equal to 0.255.

Transforming the value of\({R^2}\),

\(\begin{array}{c}{R^2} = 0.255\\ = \frac{{255}}{{1000}} \times 100\% \\ = 25.5\% \end{array}\)

Thus, it can be said that 25.5% of the variation in the Super Bowl scores is explained by the quadratic relation between the year in which the game was played and the number of points scored.

Range of \({R^2}\)

The value of\({R^2}\)lies between 0 and 1. A value close to 1 signifies that the fitted model perfectly describes the actual relationship between the response variable and the predictor variable.

A value close to 0 signifies that the fitted model poorly describes the actual relationship between the response variable and the predictor variable.

Here, the value of\({R^2}\)equal to 0.255 is close to 0.

Thus, it can be said that the fitted quadratic model is not a good one and is not useful for explaining the relation between the number of points scored and the year in which the championship was played.