Question

In a study of treatments for very painful​ "cluster" headaches, 154 patients were treated with oxygen and 155 other patients were given a placebo consisting of ordinary air. Among the 154 patients in the oxygen treatment​ group, 114 were free from headaches 15 minutes after treatment. Among the 155 patients given the​ placebo, 30 were free from headaches 15 minutes after treatment. Use a 0.05 significance level to test the claim that the oxygen treatment is effective. Find the test statistic Z to two or more decimals and the P-value to three or more.

Answer 1

p1cap = X1/N1 = 114/154 = 0.7403
p1cap = X2/N2 = 30/155 = 0.1935
pcap = (X1 + X2)/(N1 + N2) = (114+30)/(154+155) = 0.466

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 = (0.7403-0.1935)/sqrt(0.466*(1-0.466)*(1/154 + 1/155))
z = 9.63

P-value Approach
P-value = 0
As P-value < 0.05, reject the null hypothesis.

There is sufficient evidence to conclude that the oxygen treatment is effective