Section 10.3
Inference about the Difference Between
Two Population Proportions
LAST HANDOUT FOR EXAM 1
Goals for CH 10.3:
1)
Hypothesis test for two proportions
2)
Confidence interval for two proportions
Comparing
two populations containing
qualitative (
categorical)
data
When data are qualitative, we can only count the number of times each value of the variable occurs.
From
these
counts
, we compute
proportions
(or percentages).
Parameter of interest is the
difference
between two population proportions:
(p
1
– p
2
)
.
Hypothesis Testing for the Difference Between Two Population Proportions
:
Test statistic
:
Z
=
n
1
= # of observations in sample 1
p
1
= pop. proportion in population 1
1
ˆ
p
= sample proportion in sample 1
n
2
= # of observations in sample 2
p
2
= pop. proportion in population 2
2
ˆ
p
= sample proportion in sample 2
Hypotheses
:
I.
H
o
:
(
p
1
–
p
2
)
= 0
vs.
H
A
:
(
p
1
–
p
2
)
0
II .
H
o
:
(
p
1
–
p
2
)
= 0
vs.
H
A
:
(
p
1
–
p
2
)
> 0
which p is larger in this Ha? __P
1
___
III .
H
o
:
(
p
1
–
p
2
)
= 0
vs.
H
A
:
(
p
1
–
p
2
)
< 0
Q:
After completing your analyses, will you know the true values of p
1
and p
2
?
NO (FALSE)
Q:
The H
o
above implies that the two proportions are ______
EQUAL
_______.
Q:
If you reject H
o
and go with the alternative: H
A
:
p
1
–
p
2
< 0. Which proportion is larger? _P2___
Since the initial values of
p
1
and
p
2
are unknown we will define n
1
and n
2
as
sufficiently large if n
1
1
ˆ
p
,
n
1
(1-
1
ˆ
p
)
,
n
2
2
ˆ
p
,
n
2
(1-
2
ˆ
p
)
are all greater than or equal to 5.
Conditions for inference on P
(Assumptions)
The data are from
SRS’s
from the populations of interest
1
2
2
2
1
1
1
2
1
2
1
)
ˆ
1
(
ˆ
)
ˆ
1
(
ˆ
)
(
)
ˆ
ˆ
(
n
p
p
n
p
p
p
p
p
p

The sample sizes
n
1
and n
2
are
large enough
(i.e. sufficiently large)

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- Spring '17
- mr. smith