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