Question

1. Suppose that a company is separately testing 100 ineffective drugs and one effective drug, but they don't know which is effective. For each drug, they perform a hypothesis test of H0: The drug is ineffective vs. H1: The drug is effective. The test has a 0.05 significance level and power of 0.5.

(a) If the null hypothesis is rejected for a randomly selected drug, what is the probability that this drug is actually effective?
(b) If we fail to reject the null for a randomly selected drug, what is the probability that this drug is actually effective?
(c) What is the probability that at least one of the drugs in the sample will be incorrectly identified by this test?

Answer 1

a)

P(null rejected)=P(effective)*P(null rejected|effective)+P(not effective)*P(null rejected|not effective)
	=0.01*0.5+0.99*0.05=0.0545
therefore P(effective|null rejected)=P(effective)*P(null rejected|effective)/P(null rejected)=0.0917

b)

P(actually effective given null not rejected )=0.01*0.5/(1-0.0545)=0.0053

c)P(incorrectly identified) =P(effective and fail to reject null)+P(no effective and reject null)

=0.01*0.5+0.99*0.05=0.0545

P(at least one will be incorrectly indentified) =1-(1-0.0545)^100 =0.9963