Question

Researchers want to test the effectiveness of a new anti-anxiety medication. Three hundred eighty patients were recruited to participate in the clinical trial and were randomly assigned to two groups, of which one received the medication, and the other, a placebo. In clinical testing, 64 out of 200 people taking the medication reported symptoms of anxiety. Of the other 180 people receiving a placebo, 88 reported symptoms of anxiety. Do you think if the patients in the two groups performed significantly different with respect to symptoms of anxiety? Please use 0.05 as the level of significance. (Hint: use 1.96 and -196 as the critical z-values).

Group of answer choices

Yes, the proportions were significantly different.

No, the proportions were not significantly different.

We don't have sufficient information to conduct the test.

The data do not meet the independence requirement so we cannot use a z-test.

Answer 1

p1cap = X1/N1 = 64/200 = 0.32
p1cap = X2/N2 = 88/180 = 0.4889
pcap = (X1 + X2)/(N1 + N2) = (64+88)/(200+180) = 0.4

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

Rejection Region
This is two tailed test, for α = 0.05
Critical value of z are -1.96 and 1.96.
Hence reject H0 if z < -1.96 or z > 1.96

Test statistic
z = (p1cap - p2cap)/sqrt(pcap * (1-pcap) * (1/N1 + 1/N2))
z = (0.32-0.4889)/sqrt(0.4*(1-0.4)*(1/200 + 1/180))
z = -3.3557

Reject H0
Yes, the proportions were significantly different.