Business Research Methods William G. Zikmund Chapter 22: Bivariate Analysis - Tests of Differences.

Business Research Methods

William G. Zikmund

Chapter 22:

Bivariate Analysis -

Tests of Differences

Copyright © 2000 by Harcourt, Inc.

All rights reserved. Requests for permission to make copies of any part

of the work should be mailed to the following address: Permissions

Department, Harcourt, Inc., 6277 Sea Harbor Drive, Orlando, Florida

32887-6777.

Copyright © 2000 by Harcourt, Inc. All rights reserved.

Common Bivariate Tests

Type of MeasurementDifferences between

two independent groups

Differences amongthree or more

independent groups

Interval and ratioIndependent groups:

t-test or Z-testOne-wayANOVA





independent groups

OrdinalMann-Whitney U-test

Wilcoxon testKruskal-Wallis test






independent groups

NominalZ-test (two proportions)

Chi-square testChi-square tst



Type ofMeasurement

Differences between two independent groups

Nominal Chi-square test


DIFFERENCES BETWEEN GROUPS

• CONTINGENCY TABLES

• CROSS-TABULATION

• CHI-SQUARE ALLOWS TESTING FOR SIGNIFICANT DIFFERENCES BETWEEN GROUPS

• “GOODNESS OF FIT”


Chi-Square Test

i

ii )²( ²

E

EOx

x² = chi-square statisticsOi = observed frequency in the ith cellEi = expected frequency on the ith cell


Chi-Square Test

n

CRE ji

ij Ri = total observed frequency in the ith rowCj = total observed frequency in the jth columnn = sample size

DEGREES OF FREEDOM

(R-1)(C-1)=(2-1)(2-1)=1


DEGREES OF FREEDOM

D.f.=(R-1)(C-1)



Awareness of TireManufacturer’sBrand Men WomenTotalAware 50 10 60

Unaware 15 25 40

65 35 100


Chi-Square test: Differences among groups example

21

)2110(

39

)3950( 222

X

14

)1425(

26

)2615( 22


161.22

643.8654.4762.5102.32

2


1)12)(12(..

)1)(1(..

fd

CRfd

X2=3.84 with 1 d.f.


Type ofMeasurement


Interval andratio

t-test or Z-test


DIFFERENCE BETWEEN GROUPS WHEN COMPARING MEANS

• RATIO SCALED DEPENDENT VARIABLES

• t-test – WHEN GROUPS ARE SMALL– WHEN POPULATION STANDARD

DEVIATION IS UNKNOWN

• z-test – WHEN GROUPS ARE LARGE


NULL HYPOTHESIS ABOUT MEAN DIFFERENCES BETWEEN GROUPS

021

21

OR


t-Test for difference of means

meansrandomofyVariabilit

MeanMeant

21


t-Test for the difference of Means

21

21 XXS

t

X1 = mean for Group 1X2 = mean for Group 2SX1-X2 = the pooled or combined standard error of difference between means.



21

21 XXS

t


X1 = mean for Group 1X2 = mean for Group 2SX1-X2

= the pooled or combined standard error

of difference between means.



Pooled estimate of the standard error

2121

222

211 11

2

))1(121 nnnn

SnSnS XX


Pooled estimate of the standard error

S12 = the variance of Group 1

S22

= the variance of Group 2n1 = the sample size of Group 1n2 = the sample size of Group 2


Pooled estimate of the standard error t-Test for the difference of Means

2121

222

211 11

2

))1(121 nnnn

SnSnS XX

S12 = the variance of Group 1

S22

= the variance of Group 2n1 = the sample size of Group 1n2 = the sample size of Group 2


DEGREES OF FREEDOM

• d.f. = n - k

• where:

– n = n1 + n2

– k = number of groups


t-Test for difference of means-example

14

1

21

1

33

6.2131.220 22

21 XXS

797.


797.

2.125.16 t

797.

3.4

395.5


Type ofMeasurement


Nominal Z-test (two proportions)


COMPARING TWO GROUPS WHEN COMPARING PROPORTIONS

• PERCENTAGE COMPARISONS

• SAMPLE PROPORTION - P

• POPULATION PROPORTION -


DIFFERENCES BETWEEN TWO GROUPS WHEN COMPARING PROPORTIONS

The hypothesis is

Ho: 1

may be restated as

Ho: 1


A Z-test for differences of proportions

21: oHor

0: 21 oH


Z-Test for differences of proportions

21

2121

ppS

ppZ


Z-Test for differences of proportions

p1 = sample portion of successes in Group 1p2 = sample portion of successes in Group 21 1)= hypothesized population proportion 1

minus hypothesized populationproportion 1 minus

Sp1-p2 = pooled estimate of the standard errors of difference of proportions


Z-Test for differences of proportions- to calculate the standard error of the difference of proportions

21

1121 nn

qpS pp


p = pooled estimate of proportion of success in a sample of both groupsp = (1- p) or a pooled estimate of proportion of failures in a sample of both groupsn= sample size for group 1 n= sample size for group 2

p

q p

Z-Test for differences of proportions- to calculate the standard error of the difference of proportions


Z-Test for differences of proportions- to calculate the standard error of the difference of proportions - weighted average

21

2211

nn

pnpnp



100

1

100

1625.375.

21 ppS

068.



100100

4.10035.100

p

375.


Type ofMeasurement

Differences between three or more

independent groups

Interval or ratio One-wayANOVA


ANALYSIS OF VARIANCE

HYPOTHESIS WHEN COMPARING THREE GROUPS

1


groupswithinVariance

groupsbetweenVarianceF

Analysis of variance - F-Ratio


Analysis of variance - Sum of squares

betweenwithintotal SS SS SS



n

i

c

j1 1

2total )( SS XX ij



pi = individual scores, i.e., the ith observation or test unit in the jth grouppi = grand meann = number of all observations ot test units in a groupc = number of jth groups (or columns)

ijX

X


n

i

c

jj

1 1

2within )( SS XX ij




pi = individual scores, i.e., the ith observation or test unit in the jth grouppi = grand meann = number of all observations ot test units in a groupc = number of jth groups (or columns)

ijX

X


n

jjjn

1

2between )( SS XX




= individual scores, i.e., the ith observation or test unit in the jth group = grand meannj = number of all observations ot test units in a group

jX

X


1

c

SSMS between

between

Analysis of variance - Mean Squares Between


ccn

SSMS within

within

Analysis of variance - Mean Square Within


within

between

MS

MSF

Analysis of variance - Calculation of F-Ratio


A Test Market Experiment on Pricing

Sales in Units (thousands)

Regular Price$.99

1301188784

X1=104.75X=119.58

Reduced Price$.89

145143120131

X2=134.75

Cents-Off CouponRegular Price

1531299699

X1=119.25

Test Market A, B, or CTest Market D, E, or FTest Market G, H, or ITest Market J, K, or L

MeanGrand Mean


ANOVA Summary table Source of variation

• BETWEEN GROUPS

• SUM OF SQUARES – SSBETWEEN

• DEGREES OF FREEDOM– c-1 where c=number of groups

• MEAN SQUARED-MSBETWEEN

– SSBETWEEN/c-1



• WITHIN GROUPS

• SUM OF SQUARES – SSWITHIN

• DEGREES OF FREEDOM– cn-c where c=number of groups, n= number of

observations in a group

• MEAN SQUARED-MSWITHIN

– SSWITHIN/cn-c



• TOTAL

• SUM OF SQUARES – SSTOTAL

• DEGREES OF FREEDOM– cn-1 where c=number of groups, n= number of

observations in a group

•

WITHIN

BETWEEN

MSMS

F


Type ofMeasurement


Ordinal Mann-Whitney U-testWilcocon test

Business Research Methods William G. Zikmund Chapter 22: Bivariate Analysis - Tests of Differences.

Documents

cop yright

ztest slide

independent groups differences

chisquare test x

large slide

groups goodness of fit

ratio independent groups

independent groups interval