Suppose we are developing a new drug that is supposed to improve the recovery time of an illness.

We want to investigate if this is true. We conduct an experiment: we randomize twenty participants

into two groups of size n = 10 each. In one group, we give the participants the drug. In the other

group, we give them a placebo. We measure the recovery time of the participants in days. We

observe the following data:

In the group given the drug, we observe recovery times (y1,...,y10 ) = (15, 10, 13, 7, 9, 8, 21, 9,

14, 8), with y1 = 11.4.

In the group given the placebo, we observe recovery times (y11,...,y20 ) = (15, 14, 12, 8, 14, 7, 16,

10, 15, 12) with y2 = 12.3.

Assume that the data in each group follow Normal(1,2 ) and Normal(2,2 ) distributions respectively

for the drug and placebo groups, 2 = 15. We want to test the following hypothesis:

H0:12=0 versus HA:12<0

using the test statistic T(Y) =Y1Y2

i) Define the rejection region as {y:y1y2<c1 } where c1 is some real value.

Calculate the Type I error rate.

ii) Propose a way to select c1

iii) Find c1 such that the Type I error rate is at most 0.05

iv) What is the conclusion about the ability of the drug to reduce recovery time?