Question

A high-risk group of 710 male volunteers was included in a major clinical trial for testing a new vaccine for type B hepatitis. The vaccine was given to 403 persons randomly selected from the group, and the others were injected with a neutral substance (placebo). 19 of the vaccinated people and 31 of the nonvaccinated ones later got the disease. We wish to test the hypothesis that the probability of getting type B hepatitis is different (higher or lower) for people who were vaccinated compared with people who were not, using the 5% significance level.

(a)	If we use the contingency table method to test this hypothesis, find the 4 values in the expected table.

(b)	Find the value of the test statistic (using the contingency table method) to 3 decimal places

(c)	Find the critical value (using the contingency table method).

(d)	Using the z-test for proportions, find the value of the test statistic (and verify that z2  =  χ2, when there is one degree of freedom). (2 decimal places)

(e)	Find the p-value to 4 decimal places

Answer 1

Applying chi square test of independence:

Expected	Ei=row total*column total/grand total	disease	no	Total
	vaccine	28.3803	374.6197	403.00
	placebo	21.6197	285.3803	307.00
	total	50.00	660.00	710.00
chi square    χ2	=(Oi-Ei)2/Ei	disease	no	Total
	vaccine	3.100	0.235	3.3353
	placebo	4.070	0.3083	4.3782
	total	7.1703	0.5432	7.713
test statistic X2 =	7.7135

a)

expected cell values =28.3803 , 374.6197 , 21.6197 , 285.3803

b)

test statistic X2 =

7.713

c)

for 1 df and 0.05 level , critical value       χ2=

3.841

d)

z =sqrt(7.713)=2.78

e_)

from excel: p value =2*(1-normsdist(2.78))= 0.0054