Question

The accompanying table shows results of challenged referee calls in a major tennis tournament. Use a 0.05 significance level to test the claim that the gender of the tennis player is independent of whether a call is overturned. Was the Challenge to the Call​ Successful?

	yes	no
Men	297	848
Women	457	649

Determine the null and alternative hypotheses

Determine the​ P-value of the test statistic.

Use a 0.05 significance level to test the claim that the gender of the tennis player is independent of whether a call is overturned. Choose the correct answer below.

Answer 1

Null hypothesis : H_o : gender of the tennis player is independent of whether a call is overturned

Alternate Hypothesis : H₁ : gender of the tennis player is dependent of whether a call is overturned

$\small Test\; \; Statistic \; \; \; \; \; \chi ^{2}_{stat} =\sum \frac{(O-E)^{2}}{E}$

O : Observed Frequency

E: Expected Frequency

$\small E= \frac{Row\; \; Total\; \; \times Column\; \; Total}{Grand\; \; Total}$

Given,

O: Observed Frequency:

	Was the Challenge to the Call​ Successful
	yes	no
Men	297	848
Women	457	649

Expected frequency: E

	Was the Challenge to the Call​ Successful
	yes	no	Total
Men	(1145*754)/2251	(1145*1497)/2251	1145
Women	(1106*754)/2251	(1106*1497)/2251	1106
Total	754	1497	2251

	Was the Challenge to the Call​ Successful
	yes	no	Total
Men	383.5318	761.4682	1145
Women	370.4682	735.5318	1106
Total	754	1497	2251

O	E	O-E	(O-E)2	(O-E)2/E
297	383.5318	-86.5318	7487.7461	19.5231
457	370.4682	86.53176	7487.7461	20.2116
848	761.4682	86.53176	7487.7461	9.8333
649	735.5318	-86.5318	7487.7461	10.1800
			Total	59.7481

$\small Test\; \; Statistic \; \; \; \; \; \chi ^{2}_{stat} =\sum \frac{(O-E)^{2}}{E} = 59.7481$

$\small p-value = P(\chi ^{2}>\chi ^{2}_{stat})=P(\chi ^{2}>59.7481)$

Degrees of freedom = (Number of rows - 1) * (Number of columns - 1) = (2-1)*(2-1)=1

For '1' degrees of freedom ; $\small P(\chi ^{2}>59.7481)=0.000$

P-value of the test statistic = 0.000

significance level : $\small \alpha$ = 0.05 ;

As p-value: 0.000 < $\small \alpha$ : 0.05; Reject the null hypothesis;

There is sufficient evidence to conclude that the gender of the tennis player is dependent of whether a call is overturned