ex st 801 statistical methods hypothesis test for two population means: dependent samples
TRANSCRIPT
Ex St 801Statistical Methods
Hypothesis test for
Two Population
Means: DEPENDENT SAMPLES
STATISTICAL INFERENCE FOR TWO POPULATION MEANS
POPULATION 1 POPULATION 2
1, 12 2, 2
2
SAMPLE 1 SAMPLE 2
y1, s12 y2, s2
2
SITUATIONS FOR INDEPENDENT SAMPLES
When all the samples are drawn at random, or independent of each other.
TWO POPULATION MEANS: INDEPENDENT SAMPLES
• Is the mean of the first population larger than the mean of the second population?
• Is the mean of the second population larger than the mean of the first population?
0μμ :H OR μμ :H 21A21A
0μμ :H OR μμ :H 21A21A
TWO POPULATION MEANS: INDEPENDENT SAMPLES
• IS THE MEAN OF THE FIRST POPULATION DIFFERENT FROM THE MEAN OF THE SECOND POPULATION?
0μμ :H OR μμ :H 21A21A
TWO POPULATION MEANS: INDEPENDENT SAMPLES
H D
H DA
0 2 0
2 0
:
:
1
1
021A
0210
Dμμ :H
Dμμ :H
021A
0210
Dμμ :H
Dμμ :H
LOWER TAILTEST
UPPER TAILTEST
TWO-TAILTEST
TEST STATISTIC FOR INDEPENDENT SAMPLES AND EQUAL VARIANCES
t
y y D
sn n
OBS
p
1 2 0
1 2
1 1
DEPENDENT SAMPLES(paired-t)
• When the sample element is a member of both populations.– Before and after tests.
• When the comparison of two populations is made with respect to a particular variable.– Testing two brands of tires by placing one brand
of each tire on the rear wheels of the same car.
EXAMPLE
• Ten plots of land are available to determine if a new genetically-engineered variety of corn will increase the yield.
• The research plans to plant five plots with the new corn and five plots with the same variety but, has not been genetically treated (control group).
1
2
34
5
6
7
8
910
COMPLETELY RANDOMIZED DESIGN
• Briefly discuss how you would assign plots of land if you were conducting a completely randomized design.
• Do you see any problems with using this type of design?
RESULTS FROM THE GENETICALLY-ENGINEERED
CORN EXPERIMENT
HYPOTHESIS TESTING PROCEDURE FOR MATCHED PAIRS
H D
H DA
0 0
0
:
:
d
d
H D
H DA
0 0
0
:
:
d
d
H D
H DA
0 0
0
:
:
d
d
HYPOTHESES
HYPOTHESIS TESTING PROCEDURE FOR MATCHED PAIRS
td D
s nOBSd
0
/
TEST STATISTIC
HYPOTHESIS TESTING PROCEDURE FOR MATCHED PAIRS
REJECTION REGION
A t-DISTRIBUTION WITH df=n-1
WHERE n=number of PAIRS
RESULTS FROM THE GENETICALLY-ENGINEERED
CORN EXPERIMENT
HYPOTHESIS TEST FOR THE GENETICALLY-ENGINEERED
CORN EXPERIMENT
HYPOTHESES
H0: d = 0 HA: d < 0
CALCULATIONS FOR THE GENETICALLY-ENGINEERED
CORN EXPERIMENT
5.757
45
218.188.55
s
3.625
3.25.30.34.25.7d
2d
TEST STATISTIC FOR THE GENETICALLY-ENGINEERED
CORN EXPERIMENT
TEST STATISTIC
tOBS
3 62 0
2 4 53 373
.
. /.
REJECTION REGION FOR THE GENETICALLY-ENGINEERED
CORN EXPERIMENT
REJECTION REGION
Lower tail test with critical value:
t0.05,4 = -2.132.
SUMMARY
• Independent: difference between the averages
• Dependent: average of the differences
the end