Question

The data below show sport preference and age of participant from a random sample of members of a sports club. Is there evidence to suggest that they are related?

Frequencies of Sport Preference and Age

	Tennis	Swimming	Basketball
18-25	88	94	74
26-30	107	84	91
31-40	79	62	55
Over 40	76	71	43

What can be concluded at the α = 0.10 significance level?

1. What is the correct statistical test to use?
   • Paired t-test
   • Homogeneity
   • Goodness-of-Fit
   • Independence
2. What are the null and alternative hypotheses?
   H0:
   • Age and sport preference are independent.
   • The age distribution is not the same for each sport.
   • Age and sport preference are dependent.
   • The age distribution is the same for each sport.
   
   
   
   H1:
   • Age and sport preference are independent.
   • The age distribution is the same for each sport.
   • Age and sport preference are dependent.
   • The age distribution is not the same for each sport.
3. The test-statistic for this data =  (Please show your answer to 2 decimal places.)
   
4. The p-value for this sample = (Please show your answer to 4 decimal places.)
   
5. The p-value is Select an answer greater than less than (or equal to)  αα
   
6. Based on this, we should
   • reject the null
   • fail to reject the null
   • accept the null
7. Thus, the final conclusion is...
   • There is sufficient evidence to conclude that the age distribution is not the same for each sport.
   • There is insufficient evidence to conclude that age and sport are dependent.
   • There is sufficient evidence to conclude that age and sport are independent.
   • There is sufficient evidence to conclude that age and sport are dependent.
   • There is insufficient evidence to conclude that the age distribution is not the same for each sport

Answer 1

The Observed and expected values are given below. Each Expected value = Row Total * Column Total / N

	Observed
	Tennis	Swimming	Basketball	Total
18-25	88	94	74	256
26-30	107	84	91	282
31-40	79	62	55	196
Over 40	76	71	43	190
Total	350	311	263	924
	Expected
	Tennis	Swimming	Basketball	Total
18-25	96.97	86.16	72.87	256
26-30	106.82	94.92	80.27	282
31-40	74.24	65.97	55.79	196
Over 40	71.97	63.95	54.08	190
Total	350	311	263	924

______________________________________

The correct test to be used is : Test of Independence

_______________________________________

The Hypothesis:

Option 1: H0: Age and sport preference are independent.

Option 3: H1: Age and sport preference are dependent.

_________________________________________

The Test Statistic

#	Observed	Expected	(O-E)	(O-E)2	(O-E)2/E
1	88	96.97	8.969697	80.45546	0.829697
2	107	106.82	-0.18182	0.033058	0.000309
3	79	74.24	-4.75758	22.63453	0.304873
4	76	71.97	-4.0303	16.24334	0.225697
5	94	86.16	-7.8355	61.39503	0.712533
6	84	94.92	10.91558	119.15	1.255326
7	62	65.97	3.969697	15.75849	0.238875
8	71	63.95	-7.04978	49.69945	0.777158
9	74	72.87	-1.1342	1.286408	0.017654
10	91	80.27	-10.7338	115.2137	1.435395
11	55	55.79	0.787879	0.620753	0.011127
12	43	54.08	11.08009	122.7683	2.270121
			Total		8.079

test = 8.08

____________________

The p value: For (8.08, 6) = 0.2323

___________________

The p value is greater than (0.10)

_____________________

Based on this we should: Option 2: Fail to reject the Null.

______________________

The Conclusion: Option 2: There is insufficient evidence to conclude that age and sports are dependent.

_______________________