Question

A high-risk group of 814 male volunteers was included in a major clinical trial for testing a new vaccine for type B hepatitis. The vaccine was given to 367 persons randomly selected from the group, and the others were injected with a neutral substance (placebo). 16 of the vaccinated people and 37 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).

(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.

Problem #3(a):

Separate your answers with a comma. The order doesn't matter. Answers correct to 4 decimals.

Problem #3(b):

test statistic (correct to 3 decimals)

Problem #3(c):

critical value (correct to 3 decimals)

Problem #3(d):

answer correct to 2 decimals

Problem #3(e):

p-value (correct to 4 decimals)

Answer 1

applying chi square test:

Expected	Ei=row total*column total/grand total	disease	no disease	Total
	vaccinated	23.896	343.104	367
	non vaccinated	29.104	417.896	447
	total	53	761	814
chi square    χ2	=(Oi-Ei)2/Ei	disease	no disease	Total
	vaccinated	2.6089	0.1817	2.791
	non vaccinated	2.1419	0.1492	2.291
	total	4.751	0.331	5.082

Problem #3(a): 23.896 , 29.104 , 343.104 ,417.896

Problem #3(b): test statistic =5.082

Problem #3(c): crtiical value =3.841

Problem #3(d): z=2.25

Problem #3(e ): p value =0.0242