10.2. Hypothesis Test about the Difference
Large Sample Case (
):
General Case: as and level of significance 
As 
are known, let
.
As are unknown, let
.
(I): 
vs. 
Then,
In
addition,
(II):
vs. 
Then,
In
addition,
(III):
vs. 
Then,
In
addition,
Example 2:
Consider
the following results for two samples randomly taken from two populations.
|
|
Sample 1 |
Sample 2 |
|
Sample size |
64 |
49 |
|
Mean |
1150 |
921 |
|
Standard deviation |
90 |
65 |
Let 
and 
be the population
means.
(a) For 
, test 
using the
classical hypothesis test.
(b) For 
, please use p-value to test .
(c) For , please use the confidence interval method to test the
hypothesis
.
[solution:]
(a)
.
Then,
Therefore,
we reject 
.
(b)
.
Therefore,
we do not reject .
(c)
A
95% confidence interval for 
is
.
Since 
, we reject .
Small Sample Case (
):
Similar
to 10.1, two assumptions are made:
1.
Both populations have normal distribution.
2.
The variances of the populations are equal (
)
General Case: as and level of significance
.
(I): vs.
Then,
In
addition,
(II):
vs.
Then,
In
addition,
(III):
vs.
Then,
In
addition,
Example 3:
Consider
the following results for two samples randomly taken from two
normal populations with equal variance
|
|
Sample 1 |
Sample 2 |
|
Sample size |
10 |
12 |
|
Mean |
48 |
44 |
|
Standard deviation |
9 |
8 |
(a) Test 
vs. 
at
using
the classical hypothesis
test.
(b)
Test 
vs. 
at using p-value.
(c) Test 
vs. 
at using
the confidence interval
method.
[solution:]
(a) 
.
Then,
Thus,
Therefore,
we reject .
(b) 
Therefore,
we reject .
(c) A 95% confidence interval for 
is
Since 
, we do not reject .
JavaStatSoft:
Z
test:
Statistics
-> Tests -> Two Samples -> Means -> Z Test
T
test:
Statistics
-> Tests -> Two Samples -> Means -> T Test