Question

Critical Thinking: Did the NFL Rule Change Have the Desired Effect? Among 460 overtime National Football League (NFL) games between 1974 and 2011, 252 of the teams that won the overtime coin toss went on to win the game. During those years, a team could win the coin toss and march down the field to win the game with a field goal, and the other team would never get possession of the ball. That just didn’t seem fair. Starting in 2012, the overtime rules were changed. In the first three years with the new overtime rules, 47 games were decided in overtime and the team that won the coin toss won 24 of those games. Analyzing the Results

First explore the two proportions of overtime wins. Does there appear to be a difference? If so, how?

Short answer

The sample proportion of overtime wins before the rules were changed is equal to 0.548.

The sample proportion of overtime wins after the rules were changed is equal to 0.511.

There appears to be a difference in the two sample proportions of overtime wins. This implies that the change in the rules reduced the number of overtime wins that were previously won unfairly.

Step-by-step solution

Given information

In the years between 1974 and 2011, out of 460 overtime games, 252 games were won by the team that won the coin toss.

In the first 3 years beginning from 2012, there were 47 overtime games, and 24 of those games were won by the team that won the coin toss.

Sample Proportions

The sample proportion of overtime wins before the rules were changed (between 1974 and 2011) is computed below:

\(\begin{aligned}{c}{{\hat p}_1} &= \frac{{{\rm{Number}}\;{\rm{of}}\;{\rm{overtime}}\;{\rm{wins}}}}{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{overtime}}\;{\rm{games}}}}\\ &= \frac{{252}}{{460}}\\ &= 0.548\end{aligned}\)

Thus, the sample proportion of overtime wins between 1974 and 2011 is equal to 0.548.

The sample proportion of overtime wins after the rules were changed (in the first three years starting from 2012) is computed below:

\(\begin{aligned}{c}{{\hat p}_2} &= \frac{{{\rm{Number}}\;{\rm{of}}\;{\rm{overtime}}\;{\rm{wins}}}}{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{overtime}}\;{\rm{games}}}}\\ &= \frac{{24}}{{47}}\\ &= 0.511\end{aligned}\)

Thus, the sample proportion of overtime wins after the rules were changed is equal to 0.511.

Comparison

There appears to be a difference in the two sample proportions of overtime wins.

This suggests that the change in the rules lowered the number of overtime wins that were previously won unfairly.