CONFIDENCE INTERVAL
FOR 2-SAMPLE MEANS
Case 1: calcium and blood
pressure
Does increasing the amount
of calcium in our diet reduce blood
pressure? Examination of a large sample of people revealed a
relationship between calcium intake and blood pressure. The
relationship was strongest for black men. Such observational
studies do not establish causation. Researchers therefore
designed a randomized comparative experiment.
The subjects in part of the experiment were 21 healthy black
men. A randomly chosen group of 10 of the men received a
calcium supplement for 12 weeks. The control group of 11 men
received a placebo pill that looked identical. The experiment
was double-blind. The response variable is the decrease in
systolic (top number) blood pressure for a subject after 12
weeks, in millimeters of mercury. An increase appears as a
negative response.
Take Group 1 to be the calcium group and Group 2 the
placebo group. Here are the data for the 10 men in Group
1 (calcium),
and for the 11 men in Group 2 (placebo),
From the data, calculate the summary statistics:
Confidence interval
For a 90% confidence interval, Table C shows that the
t(9) critical value is t* = 1.833. We are 90% confident
that the mean advantage of calcium over a placebo, μ1
= μ2, lies in the interval
Warm-up
Teaching reading, I An educator believes that new reading activities in the
classroom will help elementary school pupils improve their reading ability. She
arranges for a third-grade class of 21 students to follow these activities for an 8week period. A control classroom of 23 third-graders follows the same curriculum
without the activities. At the end of the 8 weeks, all students are given the Degree
of Reading Power (DRP) test, which measures the aspects of reading ability that
the treatment is designed to improve. Here are the data:
(a) Is there good evidence that the new activities improve the
mean DRP score? Carry out a test and report your conclusions.
(b) Construct and interpret a 95% confidence interval for the
difference in mean DRP scores.
Answers:
(a) H0: μt = μc vs. Ha: μt > μc.
t = 2.311,
0.01 < P–value < 0.02 with df = 20
(TI calculator gives P–value = 0.0132 with df = 37.86).
(b) (0.97, 18.94) with df = 20; (1.233, 18.68)
on TI calculator with df = 37.86.
With 95% confidence, we estimate the mean
improvement in reading ability using the new reading
activities compared to not using them over an 8–week
period to be between 1.23 and 18.68 points.