Question

A high-risk group of 803 male volunteers was included in a major clinical trial for testing a new vaccine for type B hepatitis. The vaccine was given to 320 persons randomly selected from the group, and the others were injected with a neutral substance (placebo). 13 of the vaccinated people and 30 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 10% 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).

(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).

(e)	Find the p-value.

Answer 1

Applying chi square test of independence:

Expected	Ei=row total*column total/grand total	disease	no disease	Total
	vaccine	17.1357	302.8643	320.00
	placebo	25.8643	457.1357	483.00
	total	43.00	760.00	803.00
chi square    χ2	=(Oi-Ei)2/Ei	disease	no disease	Total
	vaccine	0.998	0.056	1.0546
	placebo	0.661	0.037	0.6987
	total	1.6595	0.0939	1.753
test statistic X2 =	1.7534

a)

4 values in the expected table =17.1357 , 302.8643 , 25.8643 , 457.1357

b)

test statistic X2 =

1.7534

c)

for 1 df and 0.1 level , critical value       χ2=

2.706

d)

z =sqrt(1.7534)=1.32

e)

p value =2*P(Z>1.32)=0.1868