Business Statistics, 3e
by Ken Black
Chapter 11
Discrete Distributions
Analysis of
Variance
& Design of
Experiments
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-1
Learning Objectives
• Understand the differences between various
experimental designs and when to use them.
• Compute and interpret the results of a one-way
ANOVA.
• Compute and interpret the results of a random
block design.
• Compute and interpret the results of a two-way
ANOVA.
• Understand and interpret interaction.
• Know when and how to use multiple comparison
techniques.
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-2
Introduction to Design
of Experiments, #1
Experimental Design
- a plan and a structure to test hypotheses in
which the researcher controls or manipulates
one or more variables.
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-3
Introduction to Design of Experiments, #2
Independent Variable
• Treatment variable is one that the experimenter
controls or modifies in the experiment.
• Classification variable is a characteristic of the
experimental subjects that was present prior to the
experiment, and is not a result of the
experimenter’s manipulations or control.
• Levels or Classifications are the subcategories of
the independent variable used by the researcher in
the experimental design.
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-4
Introduction to Design
of Experiments, #3
Dependent Variable
- the response to the different levels of the
independent variables.
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-5
Three Types
of Experimental Designs
• Completely Randomized Design
• Randomized Block Design
• Factorial Experiments
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-6
Completely Randomized Design
1
Machine Operator
2
3
Valve Opening
Measurements
.
.
.
.
.
.
.
.
.
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-7
Example: Number of Foreign
Freighters Docking in each Port per
Day
Long
Beach
Houston
New York
New
Orleans
5
2
8
3
7
3
4
5
4
5
6
3
2
4
7
4
6
9
2
8
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-8
Analysis of Variance:
Assumptions
• Observations are drawn from normally
distributed populations.
• Observations represent random samples
from the populations.
• Variances of the populations are equal.
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-9
One-Way ANOVA: Procedural
Overview
H :       
o
1
2
3
k
Ha: At least one of the means is different from the others
MSC
F
MSE
If F >
If F 
F , reject H .
F , do not reject H .
c
o
c
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
o
11-10
One-Way ANOVA:
Sums of Squares Definitions
total sum of squares = error sum of squares + between sum of squares
SST = SSC + SSE
C
nj

  X ji  X
j=1 i=1
where:
   n j X j  X     X ij  X j
2
C
2
j 1
C
nj
2
j 1 i 1
i  particular member of a treatment level
j = a treatment level
C = number of treatment levels
n
j
 number of observations in a given treatment level
X = grand mean
X
X
j
ij
= mean of a treatment group or level
 individual value
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-11
Partitioning Total Sum
of Squares of Variation
SST
(Total Sum of Squares)
SSC
(Treatment Sum of Squares)
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
SSE
(Error Sum of Squares)
11-12
One-Way ANOVA:
Computational Formulas
X  X
  X  X 
2
C
SSC   n j
j
j 1
C
SSE 
nj
nj
SST   
j 1 i 1
MSC 
ij
MSE 
X
SSC
df
C
 C 1
2
j 1 i 1
C
df
C
ij  X
j

df
E
 N C
2
df
T
 N 1
where: i = a particular member of a treatment level
j = a treatment level
SSE
C = number of treatment levels
df
n=
MSC
F
MSE
j
E
number of observations in a given treatment level
X = grand mean
X
X =
j
ij
column mean
individual value
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-13
One-Way ANOVA:
Preliminary Calculations
New
Orleans
Long Beach
Houston
New York
5
7
4
2
2
3
5
4
6
8
4
6
7
9
8
3
5
3
4
2
T1 = 18
T2 = 20
T3 = 42
T4 = 17
n1= 4
n2 = 5
n3 = 6
n4 = 5
T = 97
N = 20
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-14
One-Way ANOVA:
Sum of Squares Calculations
1
 18
1
4
T: T
n: n
X : X
j
j
j
C
1
 4.5
2
 20
2
5
T
n
X
2
3
 42
3
6
T
n
X
 4.0
3
 7.0
4
 42
T  97
4
5
N  20
 3.4
X  4.85
T
n
X
4
nj
 X
j 1 i 1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
 5  7  4  2  2  3 5  4  6 8  4
ji
2
2
2
2
2
2
 6  7  9  8  3  5  3  4  2  557.00
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-15
One-Way ANOVA:
Sum of Squares Calculations
C
SSC   n j
j 1
X

2
j X
 [ 4 (4.5 4.85)  5 (4.0 4.85)  6 (4.7  4.85)  5 (3.4  4.85)
2
2
 42.35
C
nj
SSE   
j 1 i 1
X
ij  X j
2
2

2
 (5 4.5)  (7  4.5)  (4  4.5)  (2  4.5)
2
2
2
2
 (2  4.0)  (3 4.0)  (4  34
. )  (2  34
. )
2
2
2
2
 44.20
C
nj
SST   
j 1 i 1
X
ij  X

2
 (5 4.85)  (7  4.85)  (4  4.85)  (4  4.85)  (2  4.85)
2
2
2
2
2
 8655
.
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-16
One-Way ANOVA:
Mean Square
and F Calculations
df
df
df
C
 C 1  4 1  3
 N  C  20  4  16
E
T
 N  1  20  1  19
MSC 
MSE 
SSC
df
C
SSE
df
42.35

 14.12
3
44.20

 2.76
16
E
MSC 14.12
F

 512
.
MSE
2.76
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-17
Analysis of Variance
for the Freighter Example
Source of Variancedf
SS
MS
F
Factor
Error
Total
3
16
19
42.35
44.20
86.55
14.12
2.76
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
5.12
11-18
A Portion of the F Table for  = 0.05
F
Denominator
Degrees of Freedom
1
...
15
16
17
.05, 3,16
Numerator Degrees of Freedom
1
2
3
4
5
6
7
8
9
161.45 199.50 215.71 224.58 230.16 233.99 236.77 238.88 240.54
...
...
...
...
...
...
...
...
...
4.54
3.68
3.29
3.06
2.90
2.79
2.71
2.64
2.59
4.49
3.63
3.24
3.01
2.85
2.74
2.66
2.59
2.54
4.45
3.59
3.20
2.96
2.81
2.70
2.61
2.55
2.49
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-19
One-Way ANOVA:
Procedural Summary
Ho :       
1
2
3
4


Ha : At least one of the means
is different from the others
If F >
If F 
F  3.24, reject H .
F  3.24, do reject H .
3
2
 16
o
c
o
c
Since F = 5.12 >
1
Rejection Region
Non rejection
Region
F  3.24, reject H .
c
o
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning

F
.05,9 ,11
 3.24
Critical Value
11-20
MINITAB Output
for the Freighter Example
ANALYSIS OF VARIANCE
Source
df
Factor
3
Error
16
Total
19
SS
42.35
44.20
86.55
MS
14.12
2.76
LEVEL
Long B
Houston
New York
NewOrlns
N
4
5
6
5
Mean
4.500
4.000
7.000
3.400
StDev
2.082
1.581
1.789
1.140
Pooled StDev =
1.662
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
F
5.11
p
0.011
11-21
Excel Output
for the Freighter Example
Anova: Single Factor
SUMMARY
Groups
Long Beach
Houston
New York
New Orleans
Count
4
5
6
5
ANOVA
Source of Variation
Between Groups
Within Groups
SS
42.35
44.2
Total
86.55
Sum Average Variance
18
4.5 4.3333
20
4
2.5
42
7
3.2
17
3.4
1.3
df
3
16
MS
14.117
2.7625
F
P-value
5.1101 0.0114
F crit
3.2389
19
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-22
Multiple Comparison Tests
An analysis of variance (ANOVA) test is an
overall test of differences among groups.
Multiple Comparison techniques are used to
identify which pairs of means are
significantly different given that the
ANOVA test reveals overall significance.
• Tukey’s honestly significant difference
(HSD) test requires equal sample sizes
• Tukey-Kramer Procedure is used when
sample sizes are unequal.
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-23
Tukey’s Honestly Significant
Difference (HSD) Test
MSE
HSD  q ,C,N-C
n
where: MSE = mean square error
n = sample size
q
,C,N-C
= critical value of the studentized range distribution from Table A.10
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-24
Data for Demonstration Problem 11.1
PLANT (Employee Age)
Group Means
nj
C=3
dfE = N - C = 12
1
29
27
30
27
28
2
32
33
31
34
30
3
25
24
24
25
26
28.2
5
32.0
5
24.8
5
MSE = 1.63
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-25
q Values for  = .01
Number of Populations
Degrees of
Freedom
1
2
3
4
5
90
135
164
186
2
14
19
22.3
24.7
3
8.26
10.6
12.2
13.3
4
6.51
8.12
9.17
9.96
11
4.39
5.14
5.62
5.97
12
4.32
5.04
5.50
5.84
...
q

504
.
.01,3,12
.
.
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-26
Tukey’s HSD Test
for the Employee Age Data
HSD  q ,C , N  C
XX
XX
X X
MSE
163
.
 5.04
 2.88
n
5
1
2
 28.2  32.0  38
.
1
3
 28.2  24.8  3.4
2
3
 32.0  24.8  7.2
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-27
Tukey-Kramer Procedure:
The Case of Unequal Sample Sizes
HSD  q ,C,N-C
MSE 1 1
(  )
2 nr ns
where: MSE = mean square error
n
n
q
r
sample size for s
r
= sample size for
s
=
,C,N-C
th
th
sample
sample
= critical value of the studentized range distribution from Table A.10
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-28
Freighter Example: Means and
Sample Sizes for the Four Ports
Port
Long Beach
Houston
New York
New Orleans
Sample Size
4
5
6
5
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
Mean
4.50
4.00
7.00
3.40
11-29
TukeyKramer
Results
for the
Freighter
Example
Pair
1 and 2
Critical
|Actual
Difference Differences|
3.19
0.50
(Long Beach and Houston)
1 and 3
3.07
2.50
3.19
0.35
2.88
3.00 *
3.01
0.60
2.88
3.60 *
(Long Beach and New York)
1 and 4
(Long Beach and New Orleans)
2 and 3
(Houston and New York)
2 and 4
(Houston and New Orleans)
3 and 4
(New York and New Orleans)
*denotes significant at .05
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-30
Partitioning the Total Sum of Squares
in the Randomized Block Design
SST
(Total Sum of Squares)
SSE
(Error Sum of Squares)
SSC
(Treatment
Sum of Squares)
SSR
(Sum of Squares
Blocks)
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
SSE’
(Sum of Squares
Error)
11-31
A Randomized Block Design
Single Independent Variable
.
Individual
observations
.
Blocking
Variable
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-32
Randomized Block Design Treatment
Effects: Procedural Overview
Ho :         
1
2
3
k
Ha : At least one of the means is different from the others
MSC
F
MSE
If F >
If F 
F , reject H .
F , do not reject H .
c
c
o
o
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-33
Randomized Block Design:
Computational Formulas
C
SSC  n ( X j  X )
j 1
n
SSR  C  ( X
i 1
n
n
i X )
2
2
SSE    ( X ij  X i  X i  X )
j 1 i 1
n
n
SST    ( X ij  X )
j 1 i 1
SSC
MSC 
C 1
SSR
MSR 
n 1
SSE
MSE 
N  n  C 1
MSC
F treatments  MSE
MSR

F blocks MSE
2
2
df
C
df
R
df
E
df
E
 C 1
 n 1
  C  1 n  1  N  n  C  1
 N 1
where: i = block group (row)
j = a treatment level (column)
C = number of treatment levels (columns)
n = number of observations in each treatment level (number of blocks - rows)
X  individual observation
X  treatment (column) mean
X  block (row) mean
ij
j
i
SSC  sum of squares columns (treatment)
SSR = sum of squares rows (blocking)
SSE = sum of squares error
SST = sum of squares total
X = grand mean
N = total number of observations
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-34
Randomized Block Design:
Tread-Wear Example
Speed
Supplier
Slow
Medium
Fast
Block
Means
( X )
i
n=5
1
3.7
4.5
3.1
3.77
2
3.4
3.9
2.8
3.37
3
3.5
4.1
3.0
3.53
4
3.2
3.5
2.6
3.10
5
3.9
4.8
3.4
4.03
3.54
4.16
2.98
3.56
Treatment
Means( X )
j
N = 15
X
C=3
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-35
Randomized Block Design:
Sum of Squares Calculations (Part 1)
C
SSC  n ( X j  X )
j 1
2
 5[(3.54  356
. )  (4.16 356
. )  (2.98 356
. )
2
2
2
 3484
.
n
SSR  C  ( X
i 1
i X )
2
 3[(3.77  356
. )  (3.37  356
. )  (3.53 356
. )  (3.10 356
. )  (4.03 356
. )]
2
2
2
2
2
 1549
.
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-36
Randomized Block Design:
Sum of Squares Calculations (Part 2)
C
n
SSE    ( X ij  X j  X i  X )
j 1 i 1
2
 (3.7  354
.  377
.  356
. )  (3.4  354
.  337
.  356
. ) 
2
2
(2.6 2.98 310
.  356
. )  (3.4  2.98 4.03 356
. )
 0143
.
2
C
n
SST    ( X ij  X )
2
2
j 1 i 1
 (3.7  356
. )  (3.4  356
. )  (2.6 3.56)  (3.4  356
. )
2
2
2
2
 5176
.
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-37
Randomized Block Design:
Mean Square Calculations
SSC 3.484
MSC 

 1742
.
C 1
2
SSR 1549
.
MSR 

 0.387
n 1
4
SSE
0143
.
MSE 

 0.018
N  n  C 1
8
MSC 1742
.
F

 96.78
MSE 0.018
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-38
Analysis of Variance
for the Tread-Wear Example
Source of VarianceSS
df
Treatment
3.484
Block
1.549
Error
0.143
Total
5.176
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
MS
2
4
8
14
F
1.742
0.387
0.018
96.78
21.50
11-39
Randomized Block Design Treatment
Effects: Procedural Summary
Ho: 1   2   3
Ha: At least one of the means is different from the others
MSC 1742
.
F
 96.78
MSE 0.018
F = 96.78 >
F
.01,2,8
= 8.65, reject Ho.
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-40
Randomized Block Design
Blocking Effects: Procedural
Overview
Ho: 1   2   3   4   5
Ha: At least one of the blocking means is different from the others
MSR .387
F

 215
.
MSE .015
F = 21.5 >
F
.01,4 ,8
= 7.01, reject Ho.
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-41
Excel Output for Tread-Wear
Example: Randomized Block Design
Anova: Two-Factor Without Replication
SUMMARY
Suplier 1
Suplier 2
Suplier 3
Suplier 4
Suplier 5
Slow
Medium
Fast
Count
Sum
11.3
10.1
10.6
9.3
12.1
Average
3.7666667
3.3666667
3.5333333
3.1
4.0333333
Variance
0.4933333
0.3033333
0.3033333
0.21
0.5033333
5 17.7
5 20.8
5 14.9
3.54
4.16
2.98
0.073
0.258
0.092
3
3
3
3
3
ANOVA
Source of Variation
SS
df
MS
F
P-value
F crit
Rows
1.5493333
4 0.3873333 21.719626 0.0002357 7.0060651
Columns
3.484
2
1.742 97.682243 2.395E-06 8.6490672
Error
0.1426667
8 0.0178333
Total
5.176
14
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-42
Two-Way Factorial Design
Column Treatment
.
.
Row
Treatment
Cells
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-43
Two-Way ANOVA: Hypotheses
Row Effects:
Ho: Row Means are all equal.
Ha: At least one row mean is different from the others.
Columns Effects:
Ho: Column Means are all equal.
Ha: At least one column mean is different from the others.
Interaction Effects: Ho: The interaction effects are zero.
Ha: There is an interaction effect.
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-44
Formulas for Computing
a Two-Way ANOVA
R
SSR  nC  ( X
i 1
C
i X )
2
SSC  nR  ( X j  X )
j 1
R
df
2
C
SSI  n   ( X ij  X i  X j  X )
i 1 j 1
SSE     ( X ijk  X ij )
R
C
n
i 1 j 1 k 1
C
R
n
SST     ( X ijk  X )
c 1 r 1 a 1
SSR
R 1
SSC
MSC 
C 1
SSI
MSI 
 R  1 C  1
SSE
MSE 
RC n  1
MSR 
2
2
2
R
df
C
df
I
df
df
E
T
 R 1
 C 1
where:
n = number of observations per cell
  R  1 C  1
C = number of column treatments
 RC n  1
 N 1
MSR
MSE
MSC

MSE
MSI

MSE
FR 
F
F
C
I
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
R = number of row treatments
i = row treatment level
j = column treatment level
k = cell member
X
X
X
X
ijk
ij
i
j
= individual observation
= cell mean
= row mean
= column mean
X = grand mean
11-45
A 2  3 Factorial Design
with Interaction
Row effects
Cell
Means
R1
R2
C1
C2
Column
C3
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-46
A 2  3 Factorial Design
with Some Interaction
Row effects
Cell
Means
R1
R2
C1
C2
Column
C3
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-47
A 2  3 Factorial Design
with No Interaction
Row effects
Cell
Means
R1
R2
C1
C2
C3
Column
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-48
A 2  3 Factorial Design: Data and
Measurements for CEO Dividend Example
Location Where Company
Stock is Traded
How Stockholders
are Informed of
Dividends
Annual/Quarterly
Reports
Presentations to
Analysts
Xj
NYSE
AMEX
2
1
2
1
X11=1.5
2
3
1
2
X21=2.0
2
3
3
2
X12=2.5
3
3
2
4
X22=3.0
1.75
2.75
OTC
Xi
4
3
4
2.5
3
X13=3.5
4
4
3
2.9167
4
X23=3.75
X=2.7083
N = 24
n=4
3.625
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-49
A 2  3 Factorial Design: Calculations
for the CEO Dividend Example (Part 1)
R
SSR  nC  ( X i  X )
2
i 1
 ( 4)( 3)[( 2.5  2.7083) 2  (2.9167  2.7083) 2 ]
 10418
.
C
SSC  nR ( X j  X )
2
j 1
 ( 4)( 2)[(1.75  2.7083) 2  ( 2.75  2.7083) 2  ( 3.625  2.7083) 2 ]
 14.0833
R
C
SSI  n  ( X ij  X i  X j  X )
2
i 1 j 1
 4[(15
.  2.5  1.75  2.7083) 2  ( 2.5  2.5  2.75  2.7083) 2
 ( 3.5  2.5  3.625  2.7083) 2  ( 2.0  2.9167  1.75  2.7083) 2
 ( 3.0  2.9167  2.75  2.7083) 2  (3.75  2.9167  3.625  2.7083) 2 ]
 0.0833
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-50
A 2  3 Factorial Design: Calculations
for the CEO Dividend Example (Part 2)
SSE     ( X ijk  X ij)
R
C
n
2
i 1 j 1 k  1
 (2 15
. )  (115
. )  (3 375
. )  (4  375
. )
2
2
2
2
 7.7500
C
R
n
SST     ( X ijk  X )
2
c 1 r 1 a  1
 (2  2.7083)  (1 2.7083)  (3 2.7083)  (4  2.7083)
2
2
2
2
 22.9583
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-51
A 2  3 Factorial Design: Calculations
for the CEO Dividend Example (Part 3)
SSR 10418
.
MSR 

 10418
.
R 1
1
SSC 14.0833
MSC 

 7.0417
C 1
2
SSI
0.0833
MSI 

 0.0417
 R  1 C  1
2
SSE
7.7500
MSE 

 0.4306
RC n  1
18
MSR 10418
.
F R  MSE  0.4306  2.42
MSC 7.0417

F C MSE  0.4306  16.35
MSI 0.0417
.
F I  MSE  0.4306  010
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-52
Analysis of Variance
for the CEO Dividend Problem
Source of VarianceSS
df
Row
1.0418
Column
14.0833
Interaction
0.0833
Error
7.7500
Total
22.9583
*Denotes
MS
1
2
2
18
23
F
1.0418 2.42
7.0417 16.35*
0.0417 0.10
0.4306
significance at = .01.
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-53
Excel
Output
for the
CEO
Dividend
Example
(Part 1)
Anova: Two-Factor With Replication
SUMMARY
NYSE
ASE
OTC
Total
AQReport
Count
4
4
4
12
Sum
6
10
14
30
Average
1.5
2.5
3.5
2.5
Variance
0.3333 0.3333 0.3333
1
Presentation
Count
Sum
Average
Variance
4
8
2
0.6667
4
12
3
0.6667
4
15
3.75
0.25
8
14
1.75
0.5
8
22
2.75
0.5
8
29
3.625
0.2679
12
35
2.9167
0.9924
Total
Count
Sum
Average
Variance
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-54
Excel Output for the
CEO Dividend Example (Part 2)
ANOVA
Source of Variation
Sample
Columns
Interaction
Within
SS
1.0417
14.083
0.0833
7.75
Total
22.958
df
1
2
2
18
MS
1.0417
7.0417
0.0417
0.4306
F
P-value F crit
2.4194 0.1373 4.4139
16.355
9E-05 3.5546
0.0968 0.9082 3.5546
23
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning
11-55