Question

Using chi-square, test the null hypothesis that the prevalence of serious mental illness does not differ by type of substance abuse.

A sample of 118 college students is asked whether they are involved in campus activities. Using the following cross tabulation depicting student responses by the region in which their colleges are located, conduct a chi-square test of significance for regional differences.

Campus Activity Participation

Region     Involved      Uninvolved

East            19                     10

South         25                      6

Midwest        15                   15

West              8                       20

Answer 1

null hypothesis: Ho: prevalence of serious mental illness and type of substance abuse are independent.

alternate hypothesis:Ha: prevalence of serious mental illness and type of substance abuse are dependent.

degree of freedom =(row-1)*(column-1)=(4-1)*(2-1)=3

for 3 degree of freedom and 0.05 level rejection region: >7.815

applying chi square test of independence:

Observed	Oi	Involved	Uninvolved	Total
	east	19	10	29
	south	25	6	31
	midwest	15	15	30
	west	8	20	28
	Total	67	51	118
Expected	Ei=Σrow*Σcolumn/Σtotal	Involved	Uninvolved	Total
	east	16.466	12.534	29
	south	17.602	13.398	31
	midwest	17.034	12.966	30
	west	15.898	12.102	28
	Total	67	51	118
chi square    χ2	=(Oi-Ei)2/Ei	Involved	Uninvolved	Total
	east	0.390	0.510	0.900
	south	3.110	4.090	7.200
	midwest	0.240	0.320	0.560
	west	3.920	5.150	9.070
	Total	7.660	10.070	17.730

as test statisic 17.730 is in rejection region we reject null hypothesis

we have sufficient evidence at 0.05 level to conclude that  prevalence of serious mental illness and type of substance abuse are dependent.