050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi

8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi

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JURUS N TEKNIK SIPIL

UNIVERSIT S ND L S

oleh :Purnawan, PhD ----- Kuliah ke 9 -----

ST TISTIKdan

PROB BILIT S


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Chap 9-2Chap 9-2

Chapter 9 Estimation and Hypothesis Testing

for Two Population Parameters


3/37

Chap 9-3

Chapter Goals

After completing this chapter, you should beable to:

Test hypotheses or form interval estimates fortwo independent population means

Standard deviations knownStandard deviations unknown

the difference between two populationproportions


4/37

Chap 9-4

Estimation for Two Populations

Estimating twopopulation values

Populationmeans,

independentsamples

Populationproportions

Group 1 vs.independentGroup 2

Proportion 1 vs.Proportion 2

Examples:


5/37

Chap 9-5

Difference Between Two Means

Population means,independent

samples

1 and 2 known

1 and 2 unknown,n1 and n 2 30

1 and 2 unknown,n1 or n 2 < 30

Goal: Form a confidenceinterval for the difference

between two populationmeans, 1 2

The point estimate for the

difference is

x1 x2

*


6/37

Chap 9-6

Independent Samples


samples

1 and 2 known



Different data sourcesUnrelatedIndependent

Sample selected fromone population has noeffect on the sampleselected from the otherpopulation

Use the difference between 2sample meansUse z test or pooled variancet test

*


7/37Chap 9-7


samples

1 and 2 known



1 and 2 known

Assumptions:

Samples are randomly andindependently drawn

population distributions arenormal or both sample sizes

are 30

Population standarddeviations are known

*


8/37

Chap 9-8


samples

1 and 2 known



and the standard error ofx1 x2 is

When 1 and 2 are known andboth populations are normal orboth sample sizes are at least 30,

the test statistic is a z- value

2

22

1

21

xx n

n

21

(continued)

1 and 2 known

*


9/37

Chap 9-9


samples

1 and 2 known

1 and

2 unknown,

n1 and n 2 30


2

22

1

21

/221

n

n

zxx

The confidence interval for1 2 is:

1 and 2 known(continued)

*


10/37

Chap 9-10


samples

1 and 2 known

1 and

2 unknown,

n1 and n 2 30


1 and 2 unknown, large samples

Assumptions:

Samples are randomly andindependently drawn

both sample sizesare 30

Population standarddeviations are unknown*


11/37

Chap 9-11


samples

1 and 2 known

1 and

2 unknown,

n1 and n 2 30


1 and 2 unknown, large samples

Forming intervalestimates:

use sample standarddeviation s to estimate

the test statistic is a z value

(continued)

*


12/37

Chap 9-12


samples

1 and 2 known

1 and

2 unknown,

n1 and n 2 30


2

22

1

21

/221

n

s

n

szxx


1 and 2 unknown, large samples(continued)

*


13/37

Chap 9-13


samples

1 and 2 known

1 and

2 unknown,

n1 and n 2 30


1 and 2 unknown, small samples

Assumptions:

populations are normally

distributed

the populations have equalvariances

samples are independent

*


14/37

Chap 9-14


samples

1 and 2 known

1 and

2 unknown,

n1 and n 2 30



Forming intervalestimates:

The population variancesare assumed equal, so usethe two sample standarddeviations and pool them to

estimate the test statistic is a t value

with (n 1 + n 2 2) degreesof freedom

(continued)

*


15/37

Chap 9-15


samples

1 and 2 known

1 and

2 unknown,

n1 and n 2 30



The pooled standarddeviation is

(continued)

2nn

s1ns1ns

21

222

211

p

*


16/37

Chap 9-16


samples

1 and 2 known

1 and

2 unknown,

n1 and n 2 30


21

p/221 n1

n1

stxx


1 and 2 unknown, small samples(continued)

Where t /2 has (n 1 + n 2 2) d.f.,

and

2nns1ns1n

s21

222

211

p*


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Chap 9-17

Hypothesis Tests for the DifferenceBetween Two Means

Testing Hypotheses about 1 2

Use the same situations discussed already:Standard deviations known or unknown

Sample sizes 30 or < 30


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Chap 9-18

Hypothesis Tests forTwo Population Proportions

Lower tail test:

H0: 1 2 H A: 1 < 2

i.e.,

H0: 1 2 0H A: 1 2 < 0

Upper tail test:

H0: 1 2H A: 1 > 2

i.e.,

H0: 1 2 0H A: 1 2 > 0

Two-tailed test:

H0: 1 = 2H A: 1 2

i.e.,

H0: 1 2 = 0H A: 1 2 0

Two Population Means, Independent Samples


19/37

Chap 9-19

Hypothesis tests for 1 2

Population means, independent samples

1 and 2 known



Use a z test statistic

Use s to estimate unknown , approximate with a z test statistic

Use s to estimate unknown , use a t test statistic andpooled standard deviation


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Chap 9-20


samples

1 and 2 known

1 and

2 unknown,

n1 and n 2 30


2

2

2

1

2

1

2121

n

n

xxz

The test statistic for1 2 is:

1 and 2 known

*


21/37


22/37

Chap 9-22


samples

1 and 2 known

1 and

2 unknown,

n1 and n 2 30



Where t /2 has (n 1 + n 2 2) d.f.,

and

2nn

s1ns1ns

21

222

211

p

21

p

2121

n1

n1

s

xxt

The test statistic for1 2 is:

*


23/37

Chap 9-23

Two Population Means, Independent Samples

Lower tail test:

H0: 1 2 0

H A: 1 2 < 0

Upper tail test:

H0: 1 2 0

H A: 1 2 > 0

Two-tailed test:

H0: 1 2 = 0

H A: 1 2 0

/2 /2

-z -z /2 z z /2 Reject H 0 if z < -z Reject H 0 if z > z Reject H 0 if z < -z /2

or z > z /2

Hypothesis tests for 1 2


24/37

Chap 9-24

Pooled s p t Test: Example

To select the best stabilisation material. Is there a differencein the result of analysis on soil cement stabilisation and limestabilisation? The result of laboratory analysis is:

Cement LimeNumber 21 25Sample mean 3.27 2.53Sample std dev 1.30 1.16

Assuming equal variances, isthere a difference in averageyield ( = 0.05 )?


25/37

Chap 9-25

Calculating the Test Statistic

1.225622521

1.161251.301212nn

s1ns1ns

22

21

222

211

p

2.040

251

2111.2256

02.533.27

n1

n1s

xxz

21p

2121

The test statistic is:


26/37

Chap 9-26

Solution

H0: 1 - 2 = 0 i.e. ( 1 = 2)HA: 1 - 2 0 i.e. ( 1 2)

= 0.05

df = 21 + 25 - 2 = 44Critical Values: t = 2.0154

Test Statistic: Decision:

Conclusion:Reject H 0 at = 0.05

There is evidence of adifference in means.

t0 2.0154-2.0154

.025

Reject H 0 Reject H 0

.025

2.040

040.2

251

211

2256.153.227.3z


27/37

Chap 9-27

Two Population Proportions

Goal: Form a confidence interval foror test a hypothesis about thedifference between two population

proportions, p 1 p 2

The point estimate forthe difference is p1 p 2


Assumptions: n1p1 5 , n 1(1-p 1) 5

n2p2 5 , n 2(1-p 2) 5


28/37

Chap 9-28

Confidence Interval forTwo Population Proportions


222

1

11/221 n

)p(1pn

)p(1pzpp

The confidence interval forp1 p 2 is:


29/37

Chap 9-29


Population proportions

Lower tail test:

H0: p 1 p 2 H A: p 1 < p 2

i.e.,

H0: p 1 p 2 0H A: p 1 p 2 < 0

Upper tail test:

H0: p 1 p 2H A: p 1 > p 2

i.e.,

H0: p 1 p 2 0H A: p 1 p 2 > 0

Two-tailed test:

H0: p 1 = p 2H A: p 1 p 2

i.e.,

H0: p 1 p 2 = 0H A: p 1 p 2 0


30/37

Chap 9-30



21

21

21

2211

nnxx

nnpnpn

p

The pooled estimate for theoverall proportion is:

where x 1 and x 2 are the numbers fromsamples 1 and 2 with the characteristic of interest

Since we begin by assuming the nullhypothesis is true, we assume p 1 = p 2 and pool the two p estimates


31/37

Chap 9-31



21

2121

n1

n1)p1(p

ppppz

The test statistic forp1 p 2 is:

(continued)


32/37

Chap 9-32


Population proportions

Lower tail test:

H0: p 1 p 2 0

H A: p 1 p 2 < 0

Upper tail test:

H0: p 1 p 2 0

H A: p 1 p 2 > 0

Two-tailed test:

H0: p 1 p 2 = 0

H A: p 1 p 2 0

/2 /2

-z -z /2 z z /2 Reject H 0 if z < -z Reject H 0 if z > z Reject H 0 if z < -z /2

or z > z /2


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34/37

Chap 9-34

The hypothesis test is:

H0: p 1 p 2 = 0 (the two proportions are equal)H A: p 1 p 2 0 (there is a significant difference between proportions)

The sample proportions are:Men: p 1 = 36/72 = .50

Women: p 2 = 31/50 = .62

.54912267

50723136

nnxx

p21

21

The pooled estimate for the overall proportion is:

Example:Two population Proportions

(continued)


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Chap 9-35

The test statistic for p1 p 2 is:

Example:Two population Proportions

(continued)

.025

-1.96 1.96

.025

-1.31

Decision: Do not reject H 0

Conclusion: There is notsignificant evidence of adifference in proportionswho will vote yes between

men and women.

1.31

50

1

72

1.549)(1.549

0.62.50

n1n1p)(1p

ppppz

21

2121

Reject H 0 Reject H 0

Critical Values = 1.96 For = .05


36/37

Chap 9-36

Two Sample Tests in EXCEL

For independent samples:

Independent sample Z test with variances known:Tools | data analysis | z-test: two sample for means

Independent sample Z test with large sampleTools | data analysis | z-test: two sample for means

If the population variances are unknown, use sample variances

For paired samples (t test): Tools | data analysis | t -test: paired two sample for means


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Chapter Summary

Compared two independent samplesFormed confidence intervals for the differences between twomeansPerformed Z test for the differences in two means

Performed t test for the differences in two means

Compared two population proportionsFormed confidence intervals for the difference between twopopulation proportions

Performed Z-test for two population proportions

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050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi