91影视

The data for overtime football games before and after the overtime rule was changed in the National Football League in 2011 is provided.

Theexpected frequencyis computed as,

\(E = \frac{{\left( {row\;total} \right)\left( {column\;total} \right)}}{{\left( {grand\;total} \right)}}\)

The observed frequencies(O), along with row and column totals is tabulated below,

Theexpected ( E) frequency table is represented as,

Assume the subjects are randomly selected for the study.

Since all the expected values are larger than 5, the requirement for chi-square test are fulfilled.

The claim to test the independence of the two variables; the hypotheses are formulated as follows,

\({H_0}:\)Winning an overtime game and whether playing under the old rule or the new rule are independent.

\({H_a}:\)Winning an overtime game and whether playing under the old rule or the new rule are dependent.

The value of the test statisticis computed as,

\[\begin{aligned}{c}{\chi ^2} = \sum {\frac{{{{\left( {O - E} \right)}^2}}}{E}} \\ = \frac{{{{\left( {252 - 250.4142} \right)}^2}}}{{250.4142}} + \frac{{{{\left( {24 - 25.5858} \right)}^2}}}{{25.5858}} + ... + \frac{{{{\left( {23 - 21.4142} \right)}^2}}}{{21.4142}}\\ = 0.2378\end{aligned}\]

Therefore, the value of the test statistic is 0.2378.

The degrees of freedomare computed as,

\(\begin{aligned}{c}\left( {r - 1} \right)\left( {c - 1} \right) = \left( {2 - 1} \right)\left( {2 - 1} \right)\\ = 1\end{aligned}\)

Therefore, the degrees of freedom are 1.

From chi-square table, the critical value for the row corresponding to 1 degrees of freedom and at 0.05 level of significance is 3.841.

Therefore, the critical value is 3.841.

Also, the p-value is obtained from the table as 0.626.

Since the critical (3.841) is greater than the value of the test statistic (0.2378), in case the null hypothesis fails to be rejected.

Therefore, the decision is that we fail to reject the null hypothesis.

There issufficient evidence to support the claim that winning an overtime game and whether playing under the old rule or the new rule are independent.

The results suggest that the change in rule is not effective on the winning in over time game.