Confidence interval for difference in population proportions:
( ˆ
p
1

ˆ
p
2
)
±
z
α/
2
·
s
ˆ
p
1
(1

ˆ
p
1
)
n
1
+
ˆ
p
2
(1

ˆ
p
2
)
n
2
Example:
A study on sideeffects of a new medication is being con
ducted.
80
people are given a placebo, and
120
people are given the new
medication. Of those receiving a placebo,
10
have dry mouth. Of those
receiving the medication,
18
have dry mouth. Let
p
1
denote the proportion
of placebo patients who have dry mouth, and
p
2
denote the proportion of
medicated patients who have dry mouth. Find a
90%
confidence interval
for
p
1

p
2
.
23
Hypothesis tests on the difference in population proportions:
We
use the following as our test statistic.
z
=
( ˆ
p
1

ˆ
p
2
)

(
p
1

p
2
)
r
ˆ
p
1
(1

ˆ
p
1
)
n
1
+
ˆ
p
2
(1

ˆ
p
2
)
n
2
Example:
Is there evidence that the drymouth proportion of those on
the medication is larger than the drymouth proportion for those on the
placebo?
24
Example:
Researchers wish to investigate the impact of nutritional coun
seling on birth weight for firsttime mothers. Two groups were formed,
with one group receiving nutritional advising. The table below shows the
number of underweight babies and sample sizes for the two groups.
Is
there significant evidence to show that advising reduced the proportion
of underweight babies by at least
4
percentage points? Test at the level
α
= 0
.
05
.
No Advising
Advising
Underweight
29
24
Total
200
300
25
Example:
Using this as a pilot study, estimate the common sample size
needed to create a
95%
confidence interval no more than
2%
away from
the true difference in proportions.
Hint:
Let
n
1
=
n
2
=
n
.
26
Sets 25 and 26
In the next two sets, we will deal with comparing the means of two samples
from two populations.
Parameter of interest:
μ
1

μ
2
, the difference in the means.
Point estimate:
x
1

x
2
.
In the handout, (“Test Statistics & Confidence Intervals (from two inde
pendent samples)” on the course page), there are four different scenarios.
•
The first scenario requires
σ
1
and
σ
2
to be known; this scenario
is highly unlikely to occur, and is presented only for completeness
purposes.
•
The second scenario has no restrictions, other than a large sample
size (
40
or more) is taken independently from two populations.
27
Example:
The Hamilton Depression Scale (HAMD) is used by psychia
trists to assess the severity of depression in patients, on a scale from
0
to
54
. Two groups of patients are selected for study, and randomly assigned
to antidepressant medication
A
or
B
.
Of the
80
patients taking medication
A
, their HAMD rating decreased
by an average of
7
.
6
points, with a standard deviation of
1
.
3
.
Of the
100
patients taking medication
B
, their HAMD rating decreased by an
average of
8
.
1
points, with a standard deviation of
1
.
1
.
Is their significant evidence to suggest that medication
B
has a larger
HAMD score decrease than medication
A
? Test the hypotheses at the
significance level
α
= 0
.
05
.
Let
μ
1
and
μ
2
be the mean decrease in the HAMD score for patients
taking medication A, B (respectively).
Example:
Construct a
98%
confidence interval for the difference in the
mean decrease, and interpret.