Question

TDaP is a booster shot that prevents Diphtheria, Tetanus, and Pertussis in adults and adolescents. It should be administered every 8 years for it to remain effective. A random sample of 550 people living in a town that experienced a pertussis outbreak this year were divided into two groups.

Group 1 was made up of 145 individuals who had not had the TDaP booster in the past 8 years, and Group 2 consisted of 355 individuals who had. In Group 1, 18 individuals caught pertussis during the outbreak, and in Group 2, 13 individuals caught pertussis. Is there evidence to suggest that the proportion of individuals who caught pertussis and were not up to date on their booster shot is higher than those that were? Test at the 0.05 level of significance.

Enter the p-value - round to 5 decimal places.

Answer 1

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

p1cap = X1/N1 = 18/145 = 0.1241
p1cap = X2/N2 = 13/355 = 0.0366
pcap = (X1 + X2)/(N1 + N2) = (18+13)/(145+355) = 0.062

Test statistic
z = (p1cap - p2cap)/sqrt(pcap * (1-pcap) * (1/N1 + 1/N2))
z = (0.1241-0.0366)/sqrt(0.062*(1-0.062)*(1/145 + 1/355))
z = 3.68

P-value Approach
P-value = 0.00012
As P-value < 0.05, reject the null hypothesis.