Transcript Chapter 11

Business Statistics, 4e
by Ken Black
Chapter 11
D iscrete D istributions
Analysis of
Variance
& Design of
Experiments
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
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, 4e, by Ken Black. © 2003 John Wiley & Sons.
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, 4e, by Ken Black. © 2003 John Wiley & Sons.
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, 4e, by Ken Black. © 2003 John Wiley & Sons.
11-4
Introduction to Design
of Experiments, #3
Dependent Variable
- the response to the different levels of the
independent variables.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
11-5
Three Types
of Experimental Designs
• Completely Randomized Design
• Randomized Block Design
• Factorial Experiments
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
11-6
Completely Randomized Design
1
Machine Operator
2
3
4
Valve Opening
Measurements
.
.
.
.
.
.
.
.
.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
.
.
.
11-7
Valve Openings by Operator
1
2
3
4
6.33
6.26
6.44
6.29
6.26
6.36
6.38
6.23
6.31
6.23
6.58
6.19
6.29
6.27
6.54
6.21
6.4
6.19
6.56
6.5
6.34
6.19
6.58
6.22
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
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, 4e, by Ken Black. © 2003 John Wiley & Sons.
11-9
One-Way ANOVA: Procedural
Overview
H o:  
1

2


3
 

k
H a : A t least one of th e m eans is different from the others
F 
If F >
If F 
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
F
F
c
c
M SC
M SE
, reject H o .
, do not reject H o .
11-10
One-Way ANOVA:
Sums of Squares Definitions
+ between
= error sum of squares
total sum of squares
sum of squares
SST = SSC + SSE
  X ij  X
nj
i =1
C

2
C

n
j 1
j= 1
where :
j
X
j
i  particular
C = number
j
 number
X = grand
X
X
j
ij


i 1
 X
C
j 1
member
ij
X

2
j
of a treatment
of treatment
levels
of observatio
ns in a given trea
level
tment
level
mean
= mean of a treatment
 individual
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
nj
level
j = a treatment
n
X
2
group
or level
value
11-11
Partitioning Total Sum
of Squares of Variation
SST
(Total Sum of Squares)
SSC
(Treatment Sum of Squares)
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
SSE
(Error Sum of Squares)
11-12
One-Way ANOVA:
Computational Formulas
C
j
j 1
SSE 
  X
SST 
nj
C
i 1
j 1
nj
C
j
  X ij  X

MSC 
MSE 

2
ij  X
j 1 i 1
F 

 n X j X
SSC 

df

2
df
 N C
E
2
SSC
df
 C 1
C
C
df
 N 1
T
w here: i = a particu lar m em ber of a treatm ent leve l
j = a treatm ent level
SSE
C = num ber of treatm ent levels
df
n
j
= num ber of observations in a g iven treatm ent level
E
MSC
MSE
X = grand m ean
X
X
ij
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
j
 colum n m e an
= individua l value
11-13
One-Way ANOVA:
Preliminary Calculations
1
2
3
4
6.33
6.26
6.44
6.29
6.26
6.36
6.38
6.23
6.31
6.23
6.58
6.19
6.29
6.27
6.54
6.21
6.4
6.19
6.56
6.5
6.34
6.19
6.58
6.22
Tj
T1 = 31.59
T2 = 50.22
T3 = 45.42
T4 = 24.92
T = 152.15
nj
n1 = 5
n2 = 8
n3 = 7
n4 = 4
N = 24
6.318000
6.277500
6.488571
6.230000
6.339583
Mean
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
11-14
One-Way ANOVA:
Sum of Squares Calculations
C
SSC 

 n X j X
j
j 1
 [ 5 ( 6 . 318
 6 . 339583 )
 7 ( 6 . 488571

SSE 


2
2
 8 ( 6 . 2775
 6 . 339583 )
2
 6 . 339583 )
2
 6 . 339583 )
2
( 6 . 31  6 . 318 )
2
 4 ( 6 . 23
0 . 23658
  X
nj
C
i 1
j 1
ij  X

2
j
( 6 . 33  6 . 318 )

2

( 6 . 4  6 . 318 )
2
( 6 . 26  6 . 318 )

2

( 6 . 26  6 . 2775 )
   ( 6 . 22  6 . 230 )
 0 . 15492
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
2

2


( 6 . 29  6 . 318 )
( 6 . 36  6 . 2775 )
( 6 . 19  6 . 230 )
2
2
11-15
2
One-Way ANOVA:
Sum of Squares Calculations
SST 

nj
C
i 1
j 1

  X ij  X

2
( 6 . 33  6 . 339583 )

2

( 6 . 26  6 . 339583 )
( 6 . 31  6 . 339583 )
2
( 6 . 19  6 . 339583 )
 0 . 39150

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
 
2
( 6 . 22  6 . 339583 )
2
2
11-16
One-Way ANOVA:
Mean Square
and F Calculations
 C 1  4 1  3
df
C
df
E
df
T
 N  C  24  4  20
 N  1  24  1  23
MSC 
MSE 
F 
SSC
df
MSC
MSE
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
 . 078860
3
C
SSE
df

. 23658

. 15492
 . 007746
20
E

. 078860
 10 . 18
. 007746
11-17
Analysis of Variance
for Valve Openings
Source of Variance
df
Between
Error
Total
3
20
23
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
SS
MS
0.23658 0.078860
0.15492 0.007746
0.39150
F
10.18
11-18
A Portion of the F Table for  = 0.05
F
. 05 , 3 , 20
df1
df 2
1
2
3
4
5
6
7
8
9
1
161.45
199.50
215.71
224.58
230.16
233.99
236.77
238.88
240.54
…
…
…
…
…
…
…
…
…
…
18
4.41
3.55
3.16
2.93
2.77
2.66
2.58
2.51
2.46
19
4.38
3.52
3.13
2.90
2.74
2.63
2.54
2.48
2.42
20
4.35
3.49
3.10
2.87
2.71
2.60
2.51
2.45
2.39
21
4.32
3.47
3.07
2.84
2.68
2.57
2.49
2.42
2.37
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
11-19
One-Way ANOVA:
Procedural Summary
Ho :  
1


2


3

4
H a : At least one of the means
is different
If F >
If
F
FF
c
c
from the others


1
Rejection Region
 3
2
 20
 3 . 10 , reject H o .
 3 . 10 , do reject H o .
Since F = 10.18 >
F
c
 3 . 10 , reject H o .
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
Non rejection
Region

F
. 05 , 9 ,11
 3 . 10
Critical Value
11-20
Excel Output
for the Valve Opening Example
Anova: Single Factor
SUMMARY
Groups
Count
Sum
Average
Variance
Operator 1
5
31.59
6.318
0.00277
Operator 2
8
50.22
6.2775
0.0110786
Operator 3
7
45.42
6.488571429
0.0101143
Operator 4
4
24.92
6.23
0.0018667
ANOVA
Source of Variation
SS
df
MS
Between Groups
0.236580119
3
0.07886004
Within Groups
0.154915714
20
0.007745786
Total
0.391495833
23
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
F
P-value
F crit
10.181025
0.00028
3.09839
11-21
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, 4e, by Ken Black. © 2003 John Wiley & Sons.
11-22
Tukey’s Honestly Significant
Difference (HSD) Test
H SD 
q
M SE
,C ,N -C
n
w here: M S E = m ean square error
n = sam ple size
q
,C ,N -C
= critical value of the studentized range distribution from T able A .10
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
11-23
Data from Demonstration Problem 11.1
PLANT (Employee Age)
Group Means
nj
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
C=3
dfE = N - C = 12
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
MSE = 1.63
11-24
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
. 01 , 3 ,12
 5.04
.
.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
11-25
Tukey’s HSD Test
for the Employee Age Data
H SD 
X
1
X
1
X
2

X
2

X
3

X
3
q
M SE
,C , N  C
n
 5.0 4
1.6 3
5
 2 .8 8
 2 8 .2  3 2 .0  3.8
 2 8 .2  2 4 .8  3.4
 3 2 .0  2 4 .8  7 .2
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
11-26
Tukey-Kramer Procedure:
The Case of Unequal Sample Sizes
H SD 
q
M SE
,C ,N -C
2
(
1
n

r
1
n
)
s
w here: M S E = m ean square error
n
n
q
r
s
,C ,N -C
= sam ple size for
= sam ple size for
r
s
th
th
sam ple
sam ple
= critical value of the studentized range distribution from T able A .10
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
11-27
Freighter Example: Means and
Sample Sizes for the Four Operators
Operator
1
2
3
4
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
Sample Size
5
8
7
4
Mean
6.3180
6.2775
6.4886
6.2300
11-28
Tukey-Kramer Results
for the Four Operators
Pair
1 and 2
Critical
Difference
.1405
|Actual
Differences|
.0405
1 and 3
.1443
.1706*
1 and 4
.1653
.0880
2 and 3
.1275
.2111*
2 and 4
.1509
.0475
3 and 4
.1545
.2586*
*denotes significant at .05
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
11-29
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, 4e, by Ken Black. © 2003 John Wiley & Sons.
SSE’
(Sum of Squares
Error)
11-30
A Randomized Block Design
Single Independent Variable
.
Individual
observations
.
Blocking
Variable
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
11-31
Randomized Block Design Treatment
Effects: Procedural Overview
Ho :  
1


2

 
3

k
H a : At least one of the means is different
F 
If F >
If F 
from the others
M SC
M SE
F
F
c
c
, reject H o .
, do not reject H o .
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
11-32
Randomized Block Design:
Computational Formulas
C
SSC  n
j 1
n
( X j X )
2
( X i X )
2
SSR  C 
i 1
n
n
j 1 i 1
n
n
  ( X ij  X )
SST 
j 1 i 1
M SC 
SSC
C 1
SSR
M SR 
SSE
N nC 1
trea tm en ts
b lo cks


C
R
2
df
E
 C 1
 n1
  C  1 n  1  N  n  C  1
2
df
E
 N 1
w here: i = block group (row )
j = a treatm ent level (colum n)
C = num ber of treatm ent levels (colum ns)
n = num ber of observations in each treatm en t level (num ber of blocks - row s)
n1
M SE 
F
df
  ( X ij  X i  X i  X )
SSE 
F
df
M SC
M SE
M SR
M SE
X
ij
X
j
X
i
 individual observation
SSC  sum of squares colum ns (treatm ent)
 treatm ent (colum n) m ean
SSR = sum of squares row s (blocking)
 block (row ) m ean
SSE = sum of squares error
SST = sum of squares total
X = grand m ean
N = total num ber of observations
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
11-33
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, 4e, by Ken Black. © 2003 John Wiley & Sons.
11-34
Randomized Block Design:
Sum of Squares Calculations (Part 1)
C
SSC  n 
j 1
( X j X )
2
 5[ (3.5 4  3.5 6 ) 
2

(4.1 6  3.5 6 )
2

(2.9 8  3.5 6 )
2

(3.5 3  3.5 6 )
2
3.4 8 4
n
SSR  C 
i 1
( X i X )
2
 3[ (3.7 7  3.5 6 ) 
2
(3.3 7  3.5 6 )
2

(3.1 0  3.5 6 )
2

(4.0 3  3.5 6 )
 1.5 4 9
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
11-35
2
]
Randomized Block Design:
Sum of Squares Calculations (Part 2)
SSE 

n
C
i 1
j 1
  ( X ij  X j  X i  X )
2
(3 . 7  3 . 54  3 . 77  3 . 56 )
2

( 2 . 6  2 . 98  3 . 10  3 . 56 )
 0 . 143
SST 
n
C
i 1
j 1
  ( X ij  X )
(3 . 7  3 . 56 )
 5 . 176

2

2
(3 . 4  3 . 54  3 . 37  3 . 56 )

2
 
(3 . 4  2 . 98  4 . 03  3 . 56 )
2
2
(3 . 4  3 . 56 )
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
2
 
( 2 . 6  3 . 56 )
2

(3 . 4  3 . 56 )
2
11-36
Randomized Block Design:
Mean Square Calculations
M SC 
M SR 
SSC
C 1
SSR
n1
M SE 


3.484
2
1.549
SSE
4
 1.742
 0 .387

0 .143
N nC 1
8
M SC
1.742
F 

 96 .78
M SE
0 .018
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
 0 .018
11-37
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, 4e, by Ken Black. © 2003 John Wiley & Sons.
MS
2
4
8
14
F
1.742
0.387
0.018
96.78
21.50
11-38
Randomized Block Design Treatment
Effects: Procedural Summary
H o:  
1

2


3
H a : A t least one of th e m eans is different from the others
F 
MSC
MSE

1 . 742
 96 . 78
0 . 018
F = 96.78 >
F
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
.01,2,8
= 8.65, reject H o .
11-39
Randomized Block Design Blocking
Effects: Procedural Overview
H o:  
1

2


3


4


5
H a: A t least one of the blocking m eans is different from the others
F 
MSR

MSE
F = 2 1 .5 >
. 387
 21 . 5
. 018
F
.0 1, 4 ,8
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
= 7 .0 1 , reject H o .
11-40
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
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
Slow
Medium
Fast
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
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
14
11-41
Two-Way Factorial Design
Column Treatment
.
.
Row
Treatment
Cells
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
11-42
Two-Way ANOVA: Hypotheses
R ow E ffects:
H o : R ow M eans are all equal.
H a : A t least one row m ean is different from the others.
C olum ns E ffects:
H o : C olum n M eans are all equal.
H a : A t least one colum n m ean is different from the others.
Interaction E ffects: H o : T he interaction effects are zero.
H a : T here is an interaction effect.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
11-43
Formulas for Computing
a Two-Way ANOVA
R
SSR  nC 
i 1
C
SSC  nR 
R
( X i X )
( X j X )
(X
SST 
(X
c 1 r 1 a 1
M SR 
M SC 
M SI 
M SE 
ijk
j 1 k 1
R
n
ijk
R
 R 1
2
df
 ( X ij  X i  X j  X )
i 1 j 1
R
C
n
i 1
C
df
j 1
C
SSI  n 
SSE 
2
 X ij )
X)
C
2
df
I
w h ere :
 C 1
n = n u m b er o f o b serv atio n s p er cell
  R  1 C  1
C = n u m b er o f co lu m n treatm en ts
R = n u m b er o f ro w treatm en ts
2
df
E
 R C  n  1
i = ro w treatm en t lev el
j = co lu m n treatm en t le v el
2
SSR
R 1
SSC
C 1
SSI
 R  1 C  1
df
F
F
F
T
R
C
I
 N 1



M SR
M SE
M SC
M SE
M SI
M SE
k = cell m em b er
X
X
X
X
ijk
ij
i
j
= in d iv id u a l o b serv atio n
= cell m ean
= ro w m ean
= co lu m n m ean
X = gran d m ean
SSE
R C  n  1
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
11-44
A 2  3 Factorial Design
with Interaction
Row effects
Cell
Means
R1
R2
C1
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
C2
Column
C3
11-45
A 2  3 Factorial Design
with Some Interaction
Row effects
Cell
Means
R1
R2
C1
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
C2
Column
C3
11-46
A 2  3 Factorial Design
with No Interaction
Row effects
Cell
Means
R1
R2
C1
C2
C3
Column
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
11-47
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
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
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
11-48
A 2  3 Factorial Design: Calculations
for the CEO Dividend Example (Part 1)
R
SSR
 nC 
( X i X )
i 1
2
 ( 4 )( 3 )[( 2 .5  2 .7 0 8 3 )  ( 2 .9 1 6 7  2 .7 0 8 3 ) ]
2
2
 1.0 4 1 8
C
SSC
 nR 
j 1
( X j X )
2
 ( 4 )( 2 )[(1.7 5  2 .7 0 8 3 )  ( 2 .7 5  2 .7 0 8 3 )  ( 3.6 2 5  2 .7 0 8 3 ) ]
2
2
2
 1 4 .0 8 3 3
R
SSI
 n
i 1
C
 ( X ij  X i  X j  X )
2
j 1
 4 [(1.5  2 .5  1.7 5  2 .7 0 8 3 )  ( 2 .5  2 .5  2 .7 5  2 .7 0 8 3 )
2
2
 ( 3.5  2 .5  3.6 2 5  2 .7 0 8 3 )  ( 2 .0  2 .9 1 6 7  1.7 5  2 .7 0 8 3 )
2
2
 ( 3.0  2 .9 1 6 7  2 .7 5  2 .7 0 8 3 )  ( 3.7 5  2 .9 1 6 7  3.6 2 5  2 .7 0 8 3 ) ]
2

2
0.0 8 3 3
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
11-49
A 2  3 Factorial Design: Calculations
for the CEO Dividend Example (Part 2)
R
SSE

n
(X
ijk  X ij )
( 2  1.5)
(1  1.5)
i 1

C
j 1 k 1
2

2
2
   ( 3  3.7 5) 
2
( 4  3.7 5)
2
 7 .7 5 0 0
C
SST

R
n
(X
c 1 r 1 a 1
ijk
( 2  2 .7 0 8 3)
 2 2 .9 5 8 3

X)
2

2
(1  2 .7 0 8 3)
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
2
   ( 3  2 .7 0 8 3) 
2
( 4  2 .7 0 8 3)
2
11-50
A 2  3 Factorial Design: Calculations
for the CEO Dividend Example (Part 3)
M SR 
SSR
 1.0418
1
14 .0833
M SC 

 7 .0417
C 1
2
SSI
0 .0833
M SI 

 0 .0417
 R  1 C  1
2
SSE
7 .7500
M SE 

 0 .4306
R C  n  1
18
R 1
SSC

1.0418
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
F
R
F
C
F
I



M SR
M SE
M SC
M SE
M SI
M SE



1.0418
0 .4306
7 .0417
0 .4306
0 .0417
0 .4306
 2 .42
 16 .35
 0 .10
11-51
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, 4e, by Ken Black. © 2003 John Wiley & Sons.
11-52
Excel
Output
for the
CEO
Dividend
Example
(Part 1)
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
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
11-53
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
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
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
11-54