Question

o A sports researcher is interested in determining if there is a relationship between the number of home team and visiting team wins and different sports. A random sample of 526 games is selected and the results are given below. Calculate the chi-square test statistie y used to test the claim that the number of home team and visiting team wins is independent of the sport. Use a 0.01 Home team wins Visiting team wins Football Basketball Soccer Baseball 39 156 25 83 31 98 19 75 OA. 4.192 OB 5.391 O c. 2.919 OD 3.290

Answer 1

Null hypothesis: The number of home team and visiting team wins are independent of the sport.

Alternate hypothesis:  The number of home team and visiting team wins are not independent of the sport.

Test statistic:

$\chi ^2=\sum_{l=1}^{2}\sum_{k=1}^{4}\frac{(O_i-E_i)^2}{E_i}$

where O_i is the observed value and E_i is the expected value.

Table for observed value

	Football	Basketball	Soccer	Baseball	Total
Home team wins	39	156	25	83	303
Visiting team wins	31	98	19	75	223
Total	70	254	44	158	526

Table for expected value

	Football	Basketball	Soccer	Baseball
Home team wins	70 * 303 - 40.32 526	$\frac{254*303}{526}=146.32$	$\frac{44*303}{526}=25.35$	$\frac{158*303}{526}=91.02$
Visiting team wins	$\frac{70*223}{526}=29.68$	$\frac{254*223}{526}=107.68$	$\frac{44*223}{526}=18.65$	$\frac{158*223}{526}=66.98$

Now, $\chi ^2=\sum_{l=1}^{2}\sum_{k=1}^{4}\frac{(O_i-E_i)^2}{E_i}$

$\chi ^2=0.043+0.641+0.005+0.706+0.059+0.871+0.006+0.959=3.29$

Value of test statistic = 3.29

degrees of freedom: $v=(l-1)*(k-1)=1*3=3$

level of significance: $\alpha =0.01$

Critical value: $\chi ^2_{\alpha ,v}=\chi ^2_{0.01,3}=11.345$

$\chi ^2<\chi ^2_{0.01,3}$

We fail to reject the null hypothesis at $\alpha =0.01$ level of significance and conclude that the number of home team and visiting team wins are independent of the sport.

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