Question

Nominal Data. In Exercises 9–12, use the sign test for the claim involving nominal data.

Overtime Rule in Football Before the overtime rule in the National Football League was changed in 2011, among 460 overtime games, 252 were won by the team that won the coin toss at the beginning of overtime. Using a 0.05 significance level, test the claim that the coin toss is fair in the sense that neither team has an advantage by winning it. Does the coin toss appear to be fair?

Short answer

There is enough evidence to conclude that the team that wins the coin toss has an advantage of winning the game.

Thus, the coin toss is not fair.

Step-by-step solution

Given information

Out of 460 overtime games, the number of games won by the team that won the coin toss is equal to 252.

The significance level is 0.05.

Frame the statistical hypothesis

The sign test is the non-parametric test used to test the claim of difference between the proportions of winning teams that won the coin toss and those that did not win the coin toss.

Let p be the proportion of the teams that won the game but did not win the coin toss.

Considering that the proportions of both the types of teams should be the same, the null hypothesis is as follows:

\({H_0}:p = 0.5\)

The team that wins the coin toss has no advantage.

The alternative hypothesis is as follows:

\({H_0}:p \ne 0.5\)

The team that wins the coin toss has an advantage.

The test is two-tailed.

Define the sign of the two categories

The games won by the team that won the coin toss are denoted by a positive sign.

The games won by the team that did not win the coin toss are denoted by a negative sign.

The number of negative signs =208.

The number of positive signs =252.

The sample size (n) is equal to 460.

Define test statistic

Let x be the number of times the less frequent sign occurs.

The less frequent sign is the negative sign corresponding to the number of games won by the team that did not win the coin toss.

The value of x is equal to 208.

As the sample size n is greater than 25, the value of z is calculated.

The test statistic z is calculated as shown:

\(\begin{array}{c}z = \frac{{\left( {x + 0.5} \right) - \frac{n}{2}}}{{\frac{{\sqrt n }}{2}}}\\ = \frac{{\left( {208 + 0.5} \right) - \frac{{460}}{2}}}{{\frac{{\sqrt {460} }}{2}}}\\ = - 2.01\end{array}\)

Determine the result and the conclusion of the test

Critical value:

The critical value ofz from the table for a two-tailed test with a value of\(\alpha \)= 0.05 is equal to\( \pm 1.96\).

Since the absolute zequal to 2.01 is greater than the critical value, the null hypothesis is rejected.

P-value:

The corresponding p-value for a z-score of -2.01 and\(\alpha \)equal to 0.05 is equal to 0.0222.

As the test is two-tailed, the p-value becomes as follows:

\(\begin{array}{c}p{\rm{ - value}} = 2 \times 0.0222\\ = 0.0444\end{array}\)

Here, the p-value is less than 0.05; the null hypothesis is rejected.

There is enough evidence to conclude that the team that wins the coin toss has an advantage of winning the game.

Thus, the coin toss is not fair.