22 bivariate stats
TRANSCRIPT
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Exploring
Marketing Research
William G. Zikmund
Chapter 22:
Bivariate Statistics-
Tests of Differences
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All rights reserved. Requests for
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of the work should be mailed to the
following address: Permissions
Department, Harcourt, Inc., 6277 Sea
Harbor Drive, Orlando Florida32887-6777
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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
Common Bivariate Tests
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Type of MeasurementDifferences between
two independent groups
Differences among
three or moreindependent groups
NominalZ-test (two proportions)
Chi-square test
Chi-square tst
Common Bivariate Tests
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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)
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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
two independent groups
Interval and
ratio
t-test or
Z-test
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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
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Null Hypothesis About Mean
Differences Between Groups
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
n2 = the sample size of Group 2
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
n1 = the sample size of Group 1
n2 = the sample size 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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Type ofMeasurement Differences betweentwo independent groups
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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A Z-Test for Differences of Proportions
100
1
100
1625.375.21 ppS
068.
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1001004.10035.100
p
375.
A Z-Test for Differences of Proportions
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Type ofMeasurement
Differences between
three or moreindependent groups
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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Analysis of Var iance- Sum of squares
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
Analysis of Var iance- Sum of squares
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Analysis of var iance- Sum of squares
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
j
jjn1
2between )(SS XX
Analysis of Var iance- Sum of squares
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Analysis of Variance- Sum of squares
= individual scores, i.e., the ith observation or
test unit in thejth group
= 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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Type ofMeasurement Differences betweentwo independent groups
Ordinal Mann-Whitney U-testWilcocon test