business research methods william g. zikmund chapter 22: bivariate analysis - tests of differences
TRANSCRIPT
Business Research Methods
William G. Zikmund
Chapter 22:
Bivariate Analysis -
Tests of Differences
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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
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Type of MeasurementDifferences between
two independent groups
Differences amongthree or more
independent groups
OrdinalMann-Whitney U-test
Wilcoxon testKruskal-Wallis test
Common Bivariate Tests
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Type of MeasurementDifferences between
two independent groups
Differences amongthree or more
independent groups
NominalZ-test (two proportions)
Chi-square testChi-square tst
Common Bivariate Tests
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Type ofMeasurement
Differences between two 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
i
ii )²( ²
E
EOx
x² = chi-square statisticsOi = observed frequency in the ith cellEi = expected frequency on the ith cell
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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
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DEGREES OF FREEDOM
D.f.=(R-1)(C-1)
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Awareness of TireManufacturer’sBrand Men WomenTotalAware 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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161.22
643.8654.4762.5102.32
2
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1)12)(12(..
)1)(1(..
fd
CRfd
X2=3.84 with 1 d.f.
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Type ofMeasurement
Differences between two independent groups
Interval andratio
t-test or Z-test
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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
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NULL HYPOTHESIS ABOUT MEAN DIFFERENCES BETWEEN GROUPS
021
21
OR
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t-Test for difference of means
meansrandomofyVariabilit
MeanMeant
21
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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.
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t-Test for the difference of Means
21
21 XXS
t
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X1 = mean for Group 1X2 = mean for Group 2SX1-X2
= the pooled or combined standard error
of difference between means.
t-Test for the difference of Means
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Pooled estimate of the standard error
2121
222
211 11
2
))1(121 nnnn
SnSnS XX
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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
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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
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DEGREES OF FREEDOM
• 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 between two independent groups
Nominal Z-test (two proportions)
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COMPARING TWO GROUPS WHEN COMPARING PROPORTIONS
• PERCENTAGE COMPARISONS
• SAMPLE PROPORTION - P
• POPULATION PROPORTION -
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DIFFERENCES BETWEEN TWO GROUPS WHEN COMPARING PROPORTIONS
The hypothesis is
Ho: 1
may be restated as
Ho: 1
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A Z-test for differences of proportions
21: oHor
0: 21 oH
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Z-Test for differences of proportions
21
2121
ppS
ppZ
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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
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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 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
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Z-Test for differences of proportions- to calculate the standard error of the difference of proportions - weighted average
21
2211
nn
pnpnp
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A Z-test for differences of proportions
100
1
100
1625.375.
21 ppS
068.
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A Z-test for differences of proportions
100100
4.10035.100
p
375.
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Type ofMeasurement
Differences between three or more
independent groups
Interval or ratio One-wayANOVA
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ANALYSIS OF VARIANCE
HYPOTHESIS WHEN COMPARING THREE GROUPS
1
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groupswithinVariance
groupsbetweenVarianceF
Analysis of variance - F-Ratio
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Analysis of variance - Sum of squares
betweenwithintotal SS SS SS
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Analysis of variance - Sum of squares
n
i
c
j1 1
2total )( SS XX ij
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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 ot test units in a groupc = number of jth groups (or columns)
ijX
X
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n
i
c
jj
1 1
2within )( SS XX ij
Analysis of variance - Sum of squares
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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 ot test units in a groupc = number of jth groups (or columns)
ijX
X
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n
jjjn
1
2between )( SS XX
Analysis of variance - Sum of squares
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Analysis of variance - Sum of squares
= 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
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1
c
SSMS between
between
Analysis of variance - Mean Squares Between
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ccn
SSMS within
within
Analysis of variance - Mean Square Within
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within
between
MS
MSF
Analysis of variance - Calculation of F-Ratio
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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
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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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ANOVA Summary table Source of variation
• 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
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ANOVA Summary table Source of variation
• 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
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Type ofMeasurement
Differences between two independent groups
Ordinal Mann-Whitney U-testWilcocon test