Hypothesis testing for matched sample.
A pharmaceutical company wants to test the effectiveness of an experimental sleeping pill. An experiment is conducted over two nights, with 10 volunteers who have difficulties sleeping at night. On the first night, each of the ten persons is given the sleeping pill one hour before bedtime and the time needed to fall asleep is recorded. On the second night, the same persons are given a placebo in the form of a pill that looks exactly as the real one (without being told about it) and the time needed to fall asleep is again recorded. The times needed to fall asleep are assumed to follow a normal distribution. The results for the two nights for the ten volunteers are shown below:
Individual

A

B

C

D

E

F

G

H

I

J

Pill

4

7

10

15

6

13

6

11

1

11

Placebo

7

9

8

16

10

15

8

15

4

10

a. work out the mean and variance of the sample data.
b. Do the data indicate that the sleeping pill is effective at a 1 percent level of significance?
c. Estimate with 95 percent confidence the mean reduction in time it takes for someone who has taken the sleeping pill to fall asleep.