Question

Interpreting a Graph The accompanying graph plots the numbers of points scored in each Super Bowl to the last Super Bowl at the time of this writing. The graph of the quadratic equation that best fits the data is also shown in red. What feature of the graph justifies the value of\({R^2}\)= 0.255 for the quadratic model?

Short answer

Since a lot of points on the graph are away from the fitted line representing the quadratic equation, it justifies the value of \({R^2} = 0.255\), which is very low indicating that the quadratic model does not describe the relationship between the two variables well.

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. the graph of the values is plotted.

Interpretation of \({R^2}\)using scatter plot

A scatterplot of the two variables depicts the pattern of the relationship between the two variables.

Here, a scatterplot is plotted between the two variables with “year” on the horizontal scale and “number of points scored” on the vertical scale.

The quadratic equation of the two variables is fitted and plotted using a red line.

If the data points lie close to the line, then the relation between the two variables is well modelled by the quadratic equation.

If the data points lie far from the line, then the relation between the two variables is poorly modelled by the quadratic equation.

Here, the value of\({R^2}\)equal to 0.255 (close to 0) for the quadratic model indicates that the relation between the number of points scored and the year is poorly represented by the quadratic equation.

Moreover, upon observing the scatterplot,it can be seen that the points do not lie close to the red line.

Thus, it can be concluded that the fitted quadratic model does not appropriately represent the relationship between the number of points scored and the year.