Research Methods William G. Zikmund, Ch22

Business Research Methods

William G. Zikmund

Chapter 22:

Bivariate Analysis -

Tests of Differences

Type of MeasurementDifferences between

two independent groups

Differences amongthree or more

independent groups

Interval and ratioIndependent groups:

t-test or Z-testOne-wayANOVA

Common Bivariate Tests




independent groups

OrdinalMann-Whitney U-test

Wilcoxon testKruskal-Wallis test





independent groups

NominalZ-test (two proportions)

Chi-square testChi-square test


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

n

CRE ji

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

Chi-Square Test

Degrees of Freedom

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

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

Degrees of Freedom

Men WomenTotalAware 50 10 60

Unaware 15 25 40

65 35 100

Awareness of Tire Manufacturer’s Brand

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

Differences 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

021

21

OR

Null Hypothesis About Mean Differences Between Groups

means random ofy Variabilit2mean - 1mean

t

t-Test for 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

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

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

14

1

21

1

33

6.2131.220 22

21 XXS

797.

t-Test for Difference of Means Example

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

21: oHor

0: 21 oH

Z-Test for Differences of Proportions


21

2121

ppS

ppZ

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



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



21

2211

nn

pnpnp

100

1

100

1625.375.

21 ppS

068.


100100

4.10035.100

p

375.

A Z-Test for Differences of Proportions

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

Analysis of Variance Sum of SquaresTotal

Analysis of Variance Sum of Squares

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

ijX

X

n

i

c

jj

1 1

2within )( SS XX ij

Analysis of Variance Sum of SquaresWithin

Analysis of Variance Sum of SquaresWithin

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

ijX

X

n

jjjn

1

2between )( SS XX

Analysis of Variance Sum of Squares Between

Analysis of Variance Sum of squares Between

= individual scores, i.e., the ith observation or test unit in the jth group = grand meannj = number of all observations or 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 F-Ratio

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

A Test Market Experiment on Pricing

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

WITHIN

BETWEEN

MSMS

F


• Total

• Sum of Squares – SStotal

• Degrees of Freedom– cn-1 where c=number of groups, n= number of

observations in a group