Lecture Notes Page 99
Stats 250 Lecture Notes
6: Learning about the Difference in Population Proportions
Part 1: Distribution for a Difference in Sample Proportions
The Independent Samples Scenario
Two samples are said to be
independent samples
when the measurements in one sample are
not related to the measurements in the other sample.
Independent samples are generated in a
variety of ways.
Some common ways:
x
Random samples are taken separately from two populations
and the same
response variable is recorded for each individual.
x
One random sample
is taken and a variable is recorded for each individual, but
then
units are categorized as belonging to one population or another
, e.g. old/young.
x
Participants are randomly assigned to one of two treatment conditions
, and the
same response variable, such as weight loss, is recorded for each individual unit.
If the
response variable is categorical
, a researcher might compare two independent groups by
looking at the
difference between the two proportions
There are usually two questions of interest about a difference in two population proportions.
First, we want to estimate the value of the difference. Second, often we want to test the
hypothesis that the difference is 0, which would indicate that the two proportions are equal.
In
either case, we will need to know about the sampling distribution for the difference in two
sample proportions (from independent samples).
Sampling Distribution for the Difference in Two Sample Proportions
Example: Do Older People Snore More than Younger?
Question of interest:
How much of a difference is there between older adults and younger
adults with regard to the proportion who snore?
Study:
Researchers at the National Sleep Foundation were interested in comparing the
proportion of people who snore for two age populations (1 = older adults defined as over 50
years old and 2 = younger adults defined as between 18 and 30 years old).
Let
p
1
be the
population proportion
of all
older adults
who snore.
Let
p
2
be the
population proportion
of all
younger adults
who snore.
We want to learn about
p
1
and
p
2
and how they compare to each other. We could estimate the
difference
p
1
–
p
2
with the corresponding difference in the sample proportions
Will it be a good estimate? How close can we expect the difference in sample proportions to be
to the true difference in population proportions (on average)?
2
1
ˆ
ˆ
p
p
±
.
.

Lecture Notes Page 100