22 bivariate stats

7/30/2019 22 Bivariate Stats

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Exploring

Marketing Research

William G. Zikmund

Chapter 22:

Bivariate Statistics-

Tests of Differences


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Copyright 2000 by Harcourt, Inc.

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Copyright 2000 by Harcourt, Inc. All rights reserved.

Common Bivariate Tests

Type of Measurement Differences betweentwo independent groups

Differences amongthree or more

independent groups

Interval and ratioIndependent groups:

t-test orZ-test

One-way

ANOVA


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Type of MeasurementDifferences between

two independent groups

Differences among

three or moreindependent groups

OrdinalMann-Whitney U-test

Wilcoxon test

Kruskal-Wallis test



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Type of MeasurementDifferences between


Differences among


NominalZ-test (two proportions)

Chi-square test

Chi-square tst



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Type ofMeasurement Differences betweentwo independent groups

Nominal Chi-square test


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Differences Between Groups

Contingency Tables

Cross-Tabulation

Chi-Square allows testing for significant

differences between groups

Goodness of Fit


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Chi-Square Test

iii )(

EEOx

x = chi-square statistics

Oi = observed frequency in the ith cell

Ei = expected frequency on the ith cell


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n

CR

Eji

ij Ri = total observed frequency in the i

th row

Cj = total observed frequency in thejth column

n = sample size

Chi-Square Test


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


Degrees of F reedom


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Awareness of Tire

Manufacturers

Brand Men Women TotalAware 50 10 60

Unaware 15 25 40

65 35 100


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Chi-Square Test: Differences Among

Groups Example

21

)2110(

39

)3950( 222

X

14

)1425(

26

)2615( 22


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161.22643.8654.4762.5102.3

2

2


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1)12)(12(..

)1)(1(..

fd

CRfd

X2=3.84 with 1 d.f.


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Type of

Measurement

Differences between


Interval and

ratio

t-test or

Z-test


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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


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Null Hypothesis About Mean


021

21

OR


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t-Test for Difference of Means

meansrandomofyVariabilitMeanMeant

21


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21

21

XX

St

X1 = mean for Group 1

X2 = mean for Group 2

SX1-X2 = the pooled or combined standard error

of difference between means.

t-Test for Difference of Means


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21

21XXS

t

t-Test for Dif ference of Means


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

SX1-X2= the pooled or combined standard error

of difference between means.

t-Test for Dif ference of Means


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Pooled estimate of the

standard error

2121

2

22

2

11 112

))1(121 nnnn

SnSnSXX


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S12 = the variance of Group 1

S22 = the variance of Group 2

n1 = the sample size of Group 1


Pooled estimate of the

standard error


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Pooled estimate of the Standard Error

t-Test for the difference of Means

2121

2

22

2

11 112

))1(121 nnnn

SnSnS XX

S12 = the variance of Group 1S2

2 = the variance of Group 2




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Degrees of F reedom

d.f. = n - k

where:

n = n1 + n2

k = number of groups


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t-Test for Difference of Means-example

14

1

21

1

33

6.2131.220 22

21 XXS

797.


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797.

2.125.16

t 797.

3.4

395.5


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Nominal Z-test (two proportions)


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Comparing Two Groups when

Compar ing Proportions

Percentage Comparisons

Sample Proportion - P

Population Proportion -


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Differences Between Two Groups

when Comparing Proportions

The hypothesis is:

Ho: 1 2

may be restated as:

Ho: 1 2 0


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A Z-Test for Differences of Proportions

21: oH

or

0: 21 oH


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Z-Test for Dif ferences of Proportions

21

2121

pp

SppZ


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p1= sample portion of successes in Group 1

p2

= sample portion of successes in Group 2

1 1)= hypothesized population proportion 1

minus hypothesized population

proportion 1 minus

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

Z-Test for Dif ferences of Proportions


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

21

1121 nn

qpS pp


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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 groups

n1= sample size for group 1

n2= sample size for group 2

p

q p

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


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Z-Test for Dif ferences of Proportions- to calculate the standard error of the difference of proportions

- weighted average

21

2211

nnpnpnp


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100

1

100

1625.375.21 ppS

068.


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1001004.10035.100

p

375.



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Type ofMeasurement

Differences between


Interval or ratioOne-wayANOVA


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Analysis of Variance

Hypothesis when comparing three groups

1 2 3


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groupswithinVariancegroupsbetweenVarianceF

Analysis of Var iance- F-Ratio


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Analysis of Var iance- Sum of squares

betweenwithintotal SSSSSS


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Analysis of Variance- Sum of squares

n

i

c

j1 1

2total )(SS XXi j


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pi= individual scores, i.e., the ith observation or

test unit in thejth group

pi= grand mean

n = number of all observations or test units in a

group

c = number ofjth groups (or columns)

i jX

X


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n

i

c

j

j

1 1

2within )(SS XXi j



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Analysis of var iance- Sum of squares

pi= individual scores, i.e., the ith observation or


pi= grand mean

n = number of all observations or test units in a

group

c = number ofjth groups (or columns)

i jX

X


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n

j

jjn1

2between )(SS XX



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Analysis of Variance- Sum of squares

= individual scores, i.e., the ith observation or


= grand mean

nj= number of all observations or test units in a

group

jX

X


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1

cSSMS betweenbetween

Analysis of Variance- Mean Squares Between


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ccnSSMS withinwithin

Analysis of Var iance- Mean Square Within


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within

between

MSMSF

Analysis of Variance- Calculation of F-Ratio


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A Test Market Exper iment 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 C

Test Market D, E, or F

Test Market G, H, or I

Test Market J, K, or L

MeanGrand Mean


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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


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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

ANOVA Summary Table

Source of Variation


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Total

Sum of Squares

SSTOTAL

Degrees of Freedom

cn-1 where c=number of groups, n= number of

observations in a group

WITHIN

BETWEEN

MS

MSF

ANOVA Summary Table

Source of Variation


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Ordinal Mann-Whitney U-testWilcocon test

22 bivariate stats

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