Question

P5. Researchers investigated whether swimming with dolphins improves clinical depression. The table shows results for a group that swam with dolphins compared to a control group that swam without dolphins.

Group with Dolphins	Group without Dolphins
x=10 patients improved n=15 patients	x=3 patients improved n=15 patients

At the 1% level of significance, test the claim that the percentage showing improvement with dolphins was significantly greater than that without dolphins.

Answer 1

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

p1cap = X1/N1 = 10/15 = 0.6667
p1cap = X2/N2 = 3/15 = 0.2
pcap = (X1 + X2)/(N1 + N2) = (10+3)/(15+15) = 0.4333

Test statistic
z = (p1cap - p2cap)/sqrt(pcap * (1-pcap) * (1/N1 + 1/N2))
z = (0.6667-0.2)/sqrt(0.4333*(1-0.4333)*(1/15 + 1/15))
z = 2.58

P-value Approach
P-value = 0.0049
As P-value < 0.01, reject the null hypothesis.