10.3. Inference about the Difference:
Match Samples
Example 4:
Objective: we want to compare two
production methods. The original data are
|
Worker |
1 |
2 |
3 |
4 |
5 |
6 |
|
Method 1 |
6.0 |
5.0 |
7.0 |
6.2 |
6.0 |
6.4 |
|
Method 2 |
5.4 |
5.2 |
6.5 |
5.9 |
6.0 |
5.8 |
: the mean completion time for production method 1
: the mean completion time for production method 2
We
want to test 
vs. 
with
.
Two
different designs can be considered:
A simple random sample of
worker using method 1 is selected.
A second simple random
sample of worker using method 2 is selected.
The methods in 10.1 and
10.2 can be used.
Disadvantage:
the variation between
workers is not considered. The effect of the different production methods
might not be distinguished
with the effect of the capability of different workers,
especially in small
sample.
(Matched sample design):
only one simple random sample of workers is selected.
Each worker use both
methods. Each worker provides a pair of data values,
one value for method 1 and
the other for method 2.
Advantage:
Two production methods are
tested under similar conditions (i.e., with the same workers);
Variation between workers
is eliminated.
Let 
the matched
sample from two populations,
where 
are from
population 1 and
are from
population 2. In this example,
Let
.
General Case:
and level of
significance 
, 
is the standard error of 
.
(I): 
vs. 
Then,
In
addition,
(II):
vs.
Then,
In
addition,
(III):
vs.
Then,
In
addition,
.
confidence
interval for 
is
and level of
significance
,
(I): vs.
Then,
In
addition,
(II):
vs.
Then,
In addition,
(III):
vs.
Then,
In
addition,
.
confidence
interval for is
Example 4 (continue):
|
|
0.6 |
-0.2 |
0.5 |
0.3 |
0.0 |
0.6 |
.
Then,
.
Thus,
we do not reject 
. In addition,
.
Therefore,
we also do not reject based on p-value.
A
95% confidence interval for is
.
Since
, we do not reject .
JavaStatSoft:
Z
test:
Statistics
-> Tests -> Matched Samples -> Z Test
T
test:
Statistics
-> Tests -> Matched Samples -> T Test