Question

In a clinical trial, 401,974 adults were randomly assigned to two groups. The treatment group consisted of 201,229 adults given a vaccine and the other 200,745 adults were given a placebo. Among the adults in the treatment group, 33 adults developed the disease and among the placebo group, 115 adults developed the disease. The doctors' claim that the rate for the group receiving the vaccine is less than the group receiving the placebo. Answer the following questions:

a. If w idenitfy the symbolic null and alternative hypothesis.

b. If the P-value for this test is reported as "less than 0.001", what is your decision? What would you conclude about the original claim?

c. Assume that we want to use a 0.05 significance level to test the claim that p1 < p2. If we want to test that claim by using a confidence interval, what confidence level should we use?

d. If we test the original claim, we get the confidence interval -0.000508 < p1 - p2 < −0.000309 , what does this confidence interval suggest about the claim? e. In general, when dealing with inferences for two population proportions, which two of the following are equivalent: Confidence Interval method; P-value method; Critical Value method? Explain.

Answer 1

(a)

Let be the probability that a randomly selected person from the treatment group will develop the disease.

Let be the probability that a randomly selected person from the control group (those who were given placebo) will develop the disease.

The null hypothesis for testing would be;

whereas the alternative hypothesis would be; .

(b) If the p-value is reported to be less than 0.001, then it means that if the null hypothesis is true, then obtaining a value of the test statistic as extreme as the one observed is just 0.1%. Hence, with a significance level of 0.001, we reject the null hypothesis and conclude in support of the doctors, i.e.  the disease rate for the group receiving the vaccine is less than the group receiving the placebo.

(c) If we want to perform a 0.05 significance level test, then we shall use (100 * (1 - 0.05))% = 95% confidence interval for the difference in proportions (p1 - p2).

(d) We obtain the confidence interval as; -0.000508 < (p1 - p2) < −0.000309. Note that, the confidence interval does not contain the value 0, which is asserted by the null hypothesis. Hence, inverting the confidence interval, we can say that we reject the null hypothesis at 0.05 significance level, and conclude in support of the doctor's claims.