Your SlideShare is downloading. ×
Business
Research Methods
William G. Zikmund
Chapter 22:
Bivariate Analysis -
Tests of Differences
Type of Measurement
Differences between
two independent groups
Differences among
three or more
independent groups
Interval...
Type of Measurement
Differences between
two independent groups
Differences among
three or more
independent groups
Ordinal
...
Type of Measurement
Differences between
two independent groups
Differences among
three or more
independent groups
Nominal
...
Type of
Measurement
Differences between
two independent groups
Nominal Chi-square test
Differences Between Groups
• Contingency Tables
• Cross-Tabulation
• Chi-Square allows testing for significant
differences...
Chi-Square Test
∑
−
=
i
ii )²(
²
E
EO
x
x² = chi-square statistics
Oi = observed frequency in the ith
cell
Ei = expected f...
n
CR
E
ji
ij =
Ri = total observed frequency in the ith
row
Cj = total observed frequency in the jth
column
n = sample siz...
Degrees of Freedom
(R-1)(C-1)=(2-1)(2-1)=1
d.f.=(R-1)(C-1)
Degrees of Freedom
Men Women Total
Aware 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( 22
2 −
+
−
=X
14
)1425(
26
)2615( 22
−
+
−
+
161.22
643.8654.4762.5102.3
2
2
=
=+++=
χ
χ
1)12)(12(..
)1)(1(..
=−−=
−−=
fd
CRfd
X2
=3.84 with 1 d.f.
Type of
Measurement
Differences between
two independent groups
Interval and
ratio
t-test or
Z-test
Differences Between Groups
when Comparing Means
• Ratio scaled dependent variables
• t-test
– When groups are small
– When...
021
21
=−
−
µµ
µµ
OR
Null Hypothesis About Mean
Differences Between Groups
meansrandomofyVariabilit
2mean-1mean
t =
t-Test for Difference of Means
21
21
XX
S
t
−
Χ−Χ
=
X1 = mean for Group 1
X2 = mean for Group 2
SX1-X2 = the pooled or combined standard error
of differe...
21
21
XX
S
t
−
Χ−Χ
=
t-Test for Difference of Means
X1 = mean for Group 1
X2 = mean for Group 2
SX1-X2
= the pooled or combined standard error
of difference between means.
t-...
Pooled Estimate of the
Standard Error
( )






+





−+
−+−
=−
2121
2
22
2
11 11
2
))1(1
21
nnnn
SnSn
S ...
S1
2
= the variance of Group 1
S2
2
= the variance of Group 2
n1 = the sample size of Group 1
n2 = the sample size of Grou...
Pooled Estimate of the Standard Error
t-test for the Difference of Means
( )






+





−+
−+−
=−
2121
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 XX
S
797.=
t-Test for Difference of Means
Ex...
797.
2.125.16 −
=t
797.
3.4
=
395.5=
Type of
Measurement
Differences between
two independent groups
Nominal Z-test (two proportions)
Π
Comparing Two Groups when
Comparing Proportions
• Percentage Comparisons
• Sample Proportion - P
• Population Proportion...
Differences Between Two Groups
when Comparing Proportions
The hypothesis is:
Ho: Π1 = Π2
may be restated as:
Ho: Π1 − Π2 =...
21: ππ =oH
or
0: 21 =− ππoH
Z-Test for Differences of
Proportions
Z-Test for Differences of
Proportions
( ) ( )
21
2121
ppS
pp
Z
−
−−−
=
ππ
p1 = sample portion of successes in Group 1
p2 = sample portion of successes in Group 2
(π1 − π1) = hypothesized populatio...
Z-Test for Differences of
Proportions






−=−
21
11
21
nn
qpS pp
p = pooled estimate of proportion of success in a
sample of both groups
p = (1- p) or a pooled estimate of proportion of
f...
Z-Test for Differences of
Proportions
21
2211
nn
pnpn
p
+
+
=
( )( ) 





+=−
100
1
100
1
625.375.21 ppS
068.=
Z-Test for Differences of
Proportions
( )( ) ( )( )
100100
4.10035.100
+
+
=p
375.=
A Z-Test for Differences of
Proportions
Type of
Measurement
Differences between
three or more
independent groups
Interval or ratio One-way
ANOVA
Analysis of Variance
Hypothesis when comparing three groups
µ1 = µ2 = µ3
groupswithinVariance
groupsbetweenVariance
F
−−
−−
=
Analysis of Variance
F-Ratio
Analysis of Variance
Sum of Squares
betweenwithintotal SSSSSS +=
∑∑= =
−=
n
i
c
j1 1
2
total )(SS XXij
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
group
pi = g...
∑∑= =
−=
n
i
c
j
j
1 1
2
within )(SS XXij
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
group
...
∑=
−=
n
j
jjn
1
2
between )(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
=...
1−
=
c
SS
MS between
between
Analysis of Variance
Mean Squares Between
ccn
SS
MS within
within
−
=
Analysis of Variance
Mean Square Within
within
between
MS
MS
F =
Analysis of Variance
F-Ratio
Sales in Units (thousands)
Regular Price
$.99
130
118
87
84
X1=104.75
X=119.58
Reduced Price
$.89
145
143
120
131
X2=134.7...
ANOVA Summary Table
Source of Variation
• Between groups
• Sum of squares
– SSbetween
• Degrees of freedom
– c-1 where c=n...
ANOVA Summary Table
Source of Variation
• Within groups
• Sum of squares
– SSwithin
• Degrees of freedom
– cn-c where c=nu...
WITHIN
BETWEEN
MS
MS
F =
ANOVA Summary Table
Source of Variation
• Total
• Sum of Squares
– SStotal
• Degrees of Freedom
–...
Business Research Methods, 9th ed.Chapter 22
Upcoming SlideShare
Loading in...5
×

Business Research Methods, 9th ed.Chapter 22

426

Published on

William G. Zikmund, Barry J. Babin, Jon C. Carr, Mitch Griffin

Published in: Education
0 Comments
1 Like
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total Views
426
On Slideshare
0
From Embeds
0
Number of Embeds
3
Actions
Shares
0
Downloads
0
Comments
0
Likes
1
Embeds 0
No embeds

No notes for slide

Transcript of "Business Research Methods, 9th ed.Chapter 22"

  1. 1. Business Research Methods William G. Zikmund Chapter 22: Bivariate Analysis - Tests of Differences
  2. 2. Type of Measurement Differences between two independent groups Differences among three or more independent groups Interval and ratio Independent groups: t-test or Z-test One-way ANOVA Common Bivariate Tests
  3. 3. Type of Measurement Differences between two independent groups Differences among three or more independent groups Ordinal Mann-Whitney U-test Wilcoxon test Kruskal-Wallis test Common Bivariate Tests
  4. 4. Type of Measurement Differences between two independent groups Differences among three or more independent groups Nominal Z-test (two proportions) Chi-square test Chi-square test Common Bivariate Tests
  5. 5. Type of Measurement Differences between two independent groups Nominal Chi-square test
  6. 6. Differences Between Groups • Contingency Tables • Cross-Tabulation • Chi-Square allows testing for significant differences between groups • “Goodness of Fit”
  7. 7. Chi-Square Test ∑ − = i ii )²( ² E EO x x² = chi-square statistics Oi = observed frequency in the ith cell Ei = expected frequency on the ith cell
  8. 8. n CR E ji ij = Ri = total observed frequency in the ith row Cj = total observed frequency in the jth column n = sample size Chi-Square Test
  9. 9. Degrees of Freedom (R-1)(C-1)=(2-1)(2-1)=1
  10. 10. d.f.=(R-1)(C-1) Degrees of Freedom
  11. 11. Men Women Total Aware 50 10 60 Unaware 15 25 40 65 35 100 Awareness of Tire Manufacturer’s Brand
  12. 12. Chi-Square Test: Differences Among Groups Example 21 )2110( 39 )3950( 22 2 − + − =X 14 )1425( 26 )2615( 22 − + − +
  13. 13. 161.22 643.8654.4762.5102.3 2 2 = =+++= χ χ
  14. 14. 1)12)(12(.. )1)(1(.. =−−= −−= fd CRfd X2 =3.84 with 1 d.f.
  15. 15. Type of Measurement Differences between two independent groups Interval and ratio t-test or Z-test
  16. 16. 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
  17. 17. 021 21 =− − µµ µµ OR Null Hypothesis About Mean Differences Between Groups
  18. 18. meansrandomofyVariabilit 2mean-1mean t = t-Test for Difference of Means
  19. 19. 21 21 XX S t − Χ−Χ = 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
  20. 20. 21 21 XX S t − Χ−Χ = t-Test for Difference of Means
  21. 21. 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
  22. 22. Pooled Estimate of the Standard Error ( )       +      −+ −+− =− 2121 2 22 2 11 11 2 ))1(1 21 nnnn SnSn S XX
  23. 23. S1 2 = the variance of Group 1 S2 2 = 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
  24. 24. Pooled Estimate of the Standard Error t-test for the Difference of Means ( )       +      −+ −+− =− 2121 2 22 2 11 11 2 ))1(1 21 nnnn SnSn S XX S1 2 = the variance of Group 1 S2 2 = the variance of Group 2 n1 = the sample size of Group 1 n2 = the sample size of Group 2
  25. 25. Degrees of Freedom • d.f. = n - k • where: –n = n1+n2 –k = number of groups
  26. 26. ( )( ) ( )( )       +        + =− 14 1 21 1 33 6.2131.220 22 21 XX S 797.= t-Test for Difference of Means Example
  27. 27. 797. 2.125.16 − =t 797. 3.4 = 395.5=
  28. 28. Type of Measurement Differences between two independent groups Nominal Z-test (two proportions)
  29. 29. Π Comparing Two Groups when Comparing Proportions • Percentage Comparisons • Sample Proportion - P • Population Proportion -
  30. 30. Differences Between Two Groups when Comparing Proportions The hypothesis is: Ho: Π1 = Π2 may be restated as: Ho: Π1 − Π2 = 0
  31. 31. 21: ππ =oH or 0: 21 =− ππoH Z-Test for Differences of Proportions
  32. 32. Z-Test for Differences of Proportions ( ) ( ) 21 2121 ppS pp Z − −−− = ππ
  33. 33. p1 = sample portion of successes in Group 1 p2 = sample portion of successes in Group 2 (π1 − π1) = hypothesized population proportion 1 minus hypothesized population proportion 1 minus Sp1-p2 = pooled estimate of the standard errors of difference of proportions Z-Test for Differences of Proportions
  34. 34. Z-Test for Differences of Proportions       −=− 21 11 21 nn qpS pp
  35. 35. p = pooled estimate of proportion of success in a sample of both groups p = (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
  36. 36. Z-Test for Differences of Proportions 21 2211 nn pnpn p + + =
  37. 37. ( )( )       +=− 100 1 100 1 625.375.21 ppS 068.= Z-Test for Differences of Proportions
  38. 38. ( )( ) ( )( ) 100100 4.10035.100 + + =p 375.= A Z-Test for Differences of Proportions
  39. 39. Type of Measurement Differences between three or more independent groups Interval or ratio One-way ANOVA
  40. 40. Analysis of Variance Hypothesis when comparing three groups µ1 = µ2 = µ3
  41. 41. groupswithinVariance groupsbetweenVariance F −− −− = Analysis of Variance F-Ratio
  42. 42. Analysis of Variance Sum of Squares betweenwithintotal SSSSSS +=
  43. 43. ∑∑= = −= n i c j1 1 2 total )(SS XXij Analysis of Variance Sum of SquaresTotal
  44. 44. Analysis of Variance Sum of Squares pi = individual scores, i.e., the ith observation or test unit in the jth group pi = grand mean n = number of all observations or test units in a group c = number of jth groups (or columns) ijX X
  45. 45. ∑∑= = −= n i c j j 1 1 2 within )(SS XXij Analysis of Variance Sum of SquaresWithin
  46. 46. Analysis of Variance Sum of SquaresWithin pi = individual scores, i.e., the ith observation or test unit in the jth group pi = grand mean n = number of all observations or test units in a group c = number of jth groups (or columns) ijX X
  47. 47. ∑= −= n j jjn 1 2 between )(SS XX Analysis of Variance Sum of Squares Between
  48. 48. Analysis of Variance Sum of squares Between = individual scores, i.e., the ith observation or test unit in the jth group = grand mean nj = number of all observations or test units in a group jX X
  49. 49. 1− = c SS MS between between Analysis of Variance Mean Squares Between
  50. 50. ccn SS MS within within − = Analysis of Variance Mean Square Within
  51. 51. within between MS MS F = Analysis of Variance F-Ratio
  52. 52. Sales in Units (thousands) Regular Price $.99 130 118 87 84 X1=104.75 X=119.58 Reduced Price $.89 145 143 120 131 X2=134.75 Cents-Off Coupon Regular Price 153 129 96 99 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 Mean Grand Mean A Test Market Experiment on Pricing
  53. 53. 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
  54. 54. 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
  55. 55. WITHIN BETWEEN MS MS F = 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

×