Question

Determine the test statistic?

what is the P-Value?

Men Women Was the Challenge to the Call Successful? Yes No i 241 499 521 769

Answer 1

Solution:

Hypothesis are:

H₀: The gender of the tennis player is independent of whether a call is overturned
H₁:The gender of the tennis player is not independent of whether a call is overturned.

Test statistic:

Formula for Chi square independence

$\chi ^{2}=\sum_{i=1}^{2}\sum_{j=1}^{2}\frac{O_{ij}^{2}}{E_{ij}}\: \: -\: \: N$

Where

O_ij = Observed frequencies for i^th row and j^th column.

E_ij = Expected frequencies for i^th row and j^th column.

Where

$E_{ij}=\frac{R_{i}\times C_{j}}{N}$

	Observed Frequencies
	Yes	No	Total
Men	241	769	R1=1010
Women	499	521	R2=1020
Total	C1=740	C2=1290	N =2030

$E_{ij}=\frac{R_{i}\times C_{j}}{N}$

E11 R1 x C1 N 1010 x 740 2030 368.1773

E12 Ri X C2 N 1010 x 1290 2030 641.8227

$E_{21}=\frac{R_{2}\times C_{1}}{N}=\frac{1020 \times 740}{2030}=371.8227$

$E_{22}=\frac{R_{2}\times C_{2}}{N}=\frac{1020\times 1290}{2030}=648.1773$

Thus

	Expected Frequencies
	Yes	No
Men	368.1773	641.8227
Women	371.8227	648.1773

Thus

Oij	Eij	Oij2/Eij
241	368.1773	157.753
769	641.8227	921.378
499	371.8227	669.677
521	648.1773	418.776
		$\sum_{i=1}^{2}\sum_{j=1}^{2}\frac{O_{ij}^{2}}{E_{ij}}\: \: = 2167.583$

$\chi ^{2}=\sum_{i=1}^{2}\sum_{j=1}^{2}\frac{O_{ij}^{2}}{E_{ij}}\: \: -\: \: N$

$\chi ^{2}= 2167.583 -2030$

$\mathbf{{\color{DarkOrange} \chi ^{2}= 137.583}}$

P-value:

Use following Excel command:

=CHISQ.DIST.RT( X² , df )

where

df = ( R - 1) X (C - 1)

R = Number of Rows = 2

C = Number of Columns = 2

thus

df = ( 2 - 1) X (2 - 1)

df = 1 X 1

df = 1

thus

=CHISQ.DIST.RT(137.583,1)

=0.0000

Thus

P-value = 0.0000

Conclusion:

Decision Rule:
Reject null hypothesis H₀, if P-value < 0.05 level of significance, otherwise we fail to reject H₀

Since P-value = 0.0000 < 0.05, we reject null hypothesis H₀.

There is sufficient evidence to warrant the rejection of the claim that the gender of the tennis player is independent of whether a call is overturned.