Question

Sample surveys on sensitive issues can give different results depending on how the question is asked. A University of Wisconsin study divided 2400 respondents into three groups at random. All were asked if they had ever used cocaine. One group of 800 was interviewed by phone; 21% said they had used cocaine. Another 800 people were asked the question in a one-on-one personal interview; 25% said "Yes". The remaining 800 were allowed to make an anonymous written response; 28% said "Yes". Are there statistically significant differences among these proportions? State the hypothesis, convert that information given into a two-way table of counts, give the test statistic and its P-value, and state your conclusion.

Answer 1

null hypothesis:Ho the proportion are equal for all three groups

Alternative hypothesis:Ha :the proportion are not equal for all three groups

degree of freedom(df) =(rows-1)*(columns-1)=		2
for 2 df and 0.05 level of signifcance critical region       χ2=			5.991
Applying chi square test of independence:

Observed	Oi	1st group	2nd group	3rd group	Total
	Use	168	200	224	592
	not use	632	600	576	1808
	total	800	800	800	2400
Expected	Ei=row total*column total/grand total	1st group	2nd group	3rd group	Total
	Use	197.3333	197.3333	197.3333	592
	not use	602.6667	602.6667	602.6667	1808
	total	800	800	800	2400
chi square    χ2	=(Oi-Ei)2/Ei	1st group	2nd group	3rd group	Total
	Use	4.3604	0.0360	3.6036	8.0000
	not use	1.4277	0.0118	1.1799	2.6195
	total	5.7881	0.0478	4.7835	10.6195
test statistic X2 =		10.619
p value =		0.0049

since p value is significantly low, we reject null hypothesis

we have sufficient evidence to conclude that  the proportion are not equal for all three groups