Question

Suppose you are interested in investigating the factors that affect the prevalence of tuberculosis among intravenous drug users. In a group of 97 individuals who admit to sharing needles, 24 had a positive tuberculin skin test result; among 161 drug users who deny sharing needles, 28 had a positive test result. At a 1% level of significance, test the hypothesis that the proportion of intravenous drug users who have a positive tuberculin skin test result are identical for those who share needles and those who do not. Write the null and alternative hypotheses for the test, as well as the decision and conclusion for the significance test.

Answer 1

p1cap = X1/N1 = 24/97 = 0.2474
p1cap = X2/N2 = 28/161 = 0.1739
pcap = (X1 + X2)/(N1 + N2) = (24+28)/(97+161) = 0.2016

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: p1 = p2
Alternate Hypothesis, Ha: p1 ≠ p2

Test statistic
z = (p1cap - p2cap)/sqrt(pcap * (1-pcap) * (1/N1 + 1/N2))
z = (0.2474-0.1739)/sqrt(0.2016*(1-0.2016)*(1/97 + 1/161))
z = 1.43

P-value Approach
P-value = 0.1527
As P-value >= 0.01, fail to reject null hypothesis.

There is not sufficient evidence to conclude that the proportion of intravenous drug users who have a positive tuberculin skin test result are identical for those who share needles and those who do not.