Two Sample Tests TEST FOR EQUAL VARIANCES TEST FOR EQUAL MEANS HHoHHo HHaHHa Population 1 Population 2 Population 1 Population 2 HHoHHo HHaHHa Population

Two Sample TestsTwo Sample TestsTEST FOR EQUAL VARIANCESTEST FOR EQUAL VARIANCES TEST FOR EQUAL MEANSTEST FOR EQUAL MEANS

HHo

HHa

Population 1

Population 2

Population 1

Population 2

HHo

HHa

Population 1

Population 2

Population 1 Population 2

Hypothesis Tests for Two Hypothesis Tests for Two Population VariancesPopulation Variances

Two-Tailed Two-Tailed TestTest

Upper One-Upper One-Tailed TestTailed Test

Lower One-Lower One-Tailed TestTailed Test

22

21

22

210

:

:

AH

H22

21

22

210

:

:

AH

H22

21

22

210

:

:

AH

H


F-TEST STATISTIC FOR TESTING WHETHER F-TEST STATISTIC FOR TESTING WHETHER TWO POPULATIONS HAVE EQUAL TWO POPULATIONS HAVE EQUAL

VARIANCESVARIANCES

where:ni = Sample size from ith population

nj = Sample size from jth population

si2= Sample variance from ith

populationsj

2= Sample variance from jth population

)( 12112

2

jij

i nDandnDdfs

sF


(Example 9-2)(Example 9-2)

F 0

df: Di = 10, Dj =12

Rejection Region /2 = 0.05

76.22/ F

47.1017.0

025.022

21 s

sF

Since F=1.47 F/2= 2.76, do not reject H0

22

21

22

210

:

:

AH

H

= .10

F = 1.47

Independent SamplesIndependent Samples

Independent samplesIndependent samples Selected from two or more populationsValues in one sample have no influence on the values in the other sample(s).

Hypothesis Tests for Two Hypothesis Tests for Two Population MeansPopulation Means

Format 1Format 1Two-Tailed Two-Tailed TestTest



0.0:

0.0:

21

210

AH

H

0.0:

0.0:

21

210

AH

H

0.0:

0.0:

21

210

AH

H





21

210

:

:

AH

H

21

210

:

:

AH

H

21

210

:

:

AH

H


T-TEST STATISTIC T-TEST STATISTIC

(EQUAL POPULATION VARIANCES)(EQUAL POPULATION VARIANCES)

where:Sample means from populations

1 and 2Hypothesized differenceSample sizes from the two

populationsPooled standard deviation

:21 xandx

21̀ :21 nandn

ps

21

2121

11

)()(

nns

xxt

p

221 nndf


POOLED STANDARD DEVIATIONPOOLED STANDARD DEVIATION

Where:

s12 = Sample variance from

population 1s2

2 = Sample variance from population 2n1 and n2 = Sample sizes from populations 1 and 2 respectively

2

)1()1(

21

222

211

nn

snsnsp


t-TEST STATISTICt-TEST STATISTIC

where:s1

2 = Sample variance from population 1

s22 = Sample variance from

population 2

(Unequal Variances)

2

22

1

21

2121 )(

ns

ns

xxt

048.22/ t0

Hypothesis Tests for Two Hypothesis Tests for Two Population Means Population Means



Since t < 2.048, do not reject H0

048.22/ t


465.0

151

151

23.677

)0.0()140.2255.2()(

2

22

1

21

2121

ns

ns

xxt

0.0:

0.0:

21

210

AH

H


DEGREES OF FREEDOM FOR t-TEST DEGREES OF FREEDOM FOR t-TEST STATISTIC WITH UNEQUAL POPULATION STATISTIC WITH UNEQUAL POPULATION

VARIANCESVARIANCES

)1)/(

1)/(

(

)//(

2

22

22

1

21

21

2221

21

nns

nns

nsns

Confidence Interval Confidence Interval Estimates for Estimates for 11 - - 22

STANDARD DEVIATIONS UNKNOWN STANDARD DEVIATIONS UNKNOWN

ANDAND 1122 = = 22

22

where: = Pooled standard

deviation

t/2 = critical value from t-distribution for desired confidence level and degrees of freedom equal to n1 + n2 -2

212/21

11)(

nnstxx p

2

)1()1(

21

222

211

nn

snsnsp



94.563$)45.508,7$39.072,8($21 xx

72.086,1299

)11.813)(19()12.304,1)(19(

2

)1()1( 22

21

222

211

nn

snsnsp

40.894$94.563$9

1

9

1)72.086,1)(7459.1(94.563$

- - $330.4$330.4

66

$1,458.$1,458.3434


STANDARD DEVIATIONS UNKNOWN STANDARD DEVIATIONS UNKNOWN

ANDAND 1122 22

22

where:t/2 = critical value from t-distribution for desired confidence level

and degrees of freedom equal to:

2

22

1

21

2/21 )(n

s

n

stxx

)1)/(

1)/(

(

)//(

2

22

22

1

21

21

2221

21

nns

nns

nsns


LARGE SAMPLE SIZESLARGE SAMPLE SIZES

where:z/2 = critical value from the standard

normal distribution for desired confidence level

2

22

1

21

2/21 )(n

s

n

szxx

Paired Samples Paired Samples Hypothesis Testing and Hypothesis Testing and

EstimationEstimation

Paired samplesPaired samples are samples that selected such that each data value from one sample is related (or matched) with a corresponding data value from the second sample. The sample values from one population have the potential to influence the probability that values will be selected from the second population.



PAIRED DIFFERENCEPAIRED DIFFERENCE

where: d = Paired difference

x1 and x2 = Values from sample 1 and 2, respectively

21 xxd



MEAN PAIRED DIFFERENCEMEAN PAIRED DIFFERENCE

where: di = ith paired difference

n = Number of paired differences

n

iidd

1



STANDARD DEVIATION FOR PAIRED STANDARD DEVIATION FOR PAIRED DIFFERENCESDIFFERENCES

where: di = ith paired difference

= Mean paired difference

1

)(1

2

n

dds

n

ii

d

d



t-TEST STATISTIC FOR PAIRED t-TEST STATISTIC FOR PAIRED DIFFERENCESDIFFERENCES

where: = Mean paired difference

d = Hypothesized paired difference

sd = Sample standard deviation of paired differences

n = Number of paired differences

d

1

ndf

n

sd

td

d

833.1t0


EstimationEstimation(Example 9-6)(Example 9-6)

Rejection Region = 0.05

Since t=0.9165 < 1.833, do not reject H0

05.0

0.1:

0.1:0

dA

d

H

H

9165.0

10

382.40.127.2

n

sd

td

d



PAIRED CONFIDENCE INTERVAL PAIRED CONFIDENCE INTERVAL ESTIMATEESTIMATE

n

std d

2/


EstimationEstimation(Example 9-7)(Example 9-7)

95% Confidence Interval

173.210.7100

95.109842.110.7

2/

n

std d

4.9274.927 9.2739.273

Hypothesis Tests for Two Hypothesis Tests for Two Population ProportionsPopulation Proportions




0.0:

0.0:

21

210

AH

H

0.0:

0.0:

21

210

AH

H

0.0:

0.0:

21

210

AH

H





21

210

:

:

AH

H

21

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AH

H

21

210

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H


POOLED ESTIMATOR FOR OVERALL POOLED ESTIMATOR FOR OVERALL PROPORTIONPROPORTION

where: x1 and x2 = number from samples 1 and 2 with desired characteristic.

21

21

21

2211

nn

xx

nn

pnpnp


TEST STATISTIC FOR DIFFERENCE IN TEST STATISTIC FOR DIFFERENCE IN POPULATION PROPORTIONSPOPULATION PROPORTIONS

where: (1 - 2) = Hypothesized difference in proportions

from populations 1 and 2, respectivelyp1 and p2 = Sample proportions for samples selected from population 1 and 2

= Pooled estimator for the overall proportion for both populations combined

)11

)(1(

)()(

21

2121

nnpp

ppz

p

645.1z 0

Hypothesis Tests for Two Hypothesis Tests for Two Population Proportions Population Proportions


Rejection Region = 0.05

Since z =-2.04 < -1.645, reject H0

75,55

250,250

0:

0:

21

21

21

210

xx

nn

H

H

A

04.2

)2501

2501

)(26.01(26.0

0)30.022.0(

)11

)(1(

)()(

21

2121

nnpp

ppz

Confidence Intervals for Confidence Intervals for Two Population Two Population

ProportionsProportions

CONFIDENCE INTERVAL ESTIMATE CONFIDENCE INTERVAL ESTIMATE

FOR FOR 11- - 22

where:p1 = Sample proportion from population 1

p2 = Sample proportion from population 2

z = Critical value from the standard normal table

2

22

1

112/21

)1(()1(()(

n

pp

n

ppzpp

Confidence Intervals for Confidence Intervals for Two Population Two Population

ProportionsProportions(Example 9-10)(Example 9-10)

w

ww

m

mmwm n

pp

n

ppzpp

)1(()1(()( 2/

530.0370

196565.0

425

240 wm pandp

069.0035.0370

)530.01)(530.0(

425

)565.01)(565.0(96.1)530.0565.0(

--0.030.0344

0.100.1044

Key TermsKey Terms

Independent Samples

Paired Samples

Documents

Two Sample Tests TEST FOR EQUAL VARIANCES TEST FOR EQUAL MEANS HHoHHo HHaHHa Population 1 Population 2 Population 1 Population 2 HHoHHo HHaHHa Population