The table below shows results since 2006 of challenged referee calls in the U.S. Open. Use a 0.05 significance level to test the claim that the gender of the tennis player is independent of whether the call is overturned. Do players of either gender appear to be better at challenging calls?

	Was the Challenge to the Call Successful?
	Yes	No
Men	161	376
Women	68	152

The data for the successfulresults who challenged referee calls in the U.S. Open and the gender is provided.

The level of significance is 0.05.

Formula forexpected frequencyis,

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

The observed frequencies along with row and column totals is,

Theexpected frequency tableis represented as,

The expected values are larger than 5. Assuming the players are randomly selected, the requirements of the chi-square test are satisfied.

To test if the successful call is independent of the gender of the player, the hypotheses are formulated as follows,

\({H_0}:\)The gender of the tennis player is independent of whether the call is overturned.

\({H_1}:\)The gender of the tennis player is dependent on whether the call is overturned.

The value of the test statisticis computed as,

\[\begin{aligned}{c}{\chi ^2} = \sum {\frac{{{{\left( {O - E} \right)}^2}}}{E}} \\ = \frac{{{{\left( {161 - 162.4478} \right)}^2}}}{{162.4478}} + \frac{{{{\left( {376 - 374.5522} \right)}^2}}}{{374.5522}} + ... + \frac{{{{\left( {152 - 153.4478} \right)}^2}}}{{153.4478}}\\ = 0.0637\\ \approx 0.064\end{aligned}\]

Therefore, the value of the test statistic is 0.064.

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 degree of freedom and at 0.05 level of significance is 3.841.

The p-value is computed as 0.801.

Since the critical (3.841) is greater than the value of the test statistic (0.064). In this case, the null hypothesis fails to be rejected.

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

There issufficient evidencethatthe gender of the tennis player is independentof the overturning of the call.

Thus, it can be concluded that neither of the genders appears to be better as the overturning of the call is not dependent on the player’s gender.