Matakuliah
Tahun
Versi
: I0284 - Statistika
: 2008
: Revisi
Pertemuan 20
Analisis Varians Klasifikasi Dua arah
1
Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa
akan mampu :
• Mahasiswa akan dapat menerapkan
analisis varians dua arah.
2
Outline Materi
•
•
•
•
Analisis varians rancangan kelompok
Partisi jumlah kuadrat perlakuan
Prosedur uji F
Pembandingan ganda perlakuan
3
Analysis of Variance and
Experimental Design
• An Introduction to Analysis of Variance
• Analysis of Variance: Testing for the
Equality of
k Population Means
• Multiple Comparison Procedures
• An Introduction to Experimental Design
• Completely Randomized Designs
• Randomized Block Design
4
An Introduction to Analysis of
Variance
• Analysis of Variance (ANOVA) can be used to test for the
equality of three or more population means using data
obtained from observational or experimental studies.
• We want to use the sample results to test the following
hypotheses.
H0: 1 = 2 = 3 = . . . = k
Ha: Not all population means are equal
• If H0 is rejected, we cannot conclude that all population
means are different.
• Rejecting H0 means that at least two population means have
different values.
5
Assumptions for Analysis of
Variance
• For each population, the response
variable is normally distributed.
• The variance of the response variable,
denoted 2, is the same for all of the
populations.
• The observations must be independent.
6
Analysis of Variance:
Testing for the Equality of K Population
Means
• Between-Samples Estimate of Population
Variance
• Within-Samples Estimate of Population
Variance
• Comparing the Variance Estimates: The F
Test
• The ANOVA Table
7
Randomized Block Design
• The ANOVA Procedure
• Computations and Conclusions
8
The ANOVA Procedure
• The ANOVA procedure for the randomized block
design requires us to partition the sum of squares
total (SST) into three groups: sum of squares due to
treatments, sum of squares due to blocks, and sum
of squares due to error.
• The formula for this partitioning is
•
SST = SSTR + SSBL + SSE
• The total degrees of freedom, nT - 1, are partitioned
such that k - 1 degrees of freedom go to treatments,
b - 1 go to blocks, and (k - 1)(b - 1) go to the error
term.
9
ANOVA Table for a
Randomized Block Design
Source of
Variation
F
Sum of
Squares
Degrees of
Freedom
Treatments
SSTR
k-1
Blocks
SSBL
b-1
Error
SSE
(k - 1)(b - 1)
Total
SST
nT - 1
Mean
Squares
SSTR MSTR
k - 1 MSE
SSBL
MSBL 
b-1
MSTR 
SSE
MSE 
( k  1)(b  1)
10
Randomized Block Design
• Example: Crescent Oil Co.
Crescent Oil has developed three
new blends of gasoline and must
decide which blend or blends to
produce and distribute. A study
of the miles per gallon ratings of the
three blends is being conducted to determine if the
mean ratings are the same for the three blends.
Randomized Block Design

Example: Crescent Oil Co.
Five automobiles have been
tested using each of the three
gasoline blends and the miles
per gallon ratings are shown on
the next slide.
Randomized Block Design
Automobile
(Block)
Type of Gasoline (Treatment)
Blend X
Blend Y
Blend Z
Block
Means
30.333
29.333
28.667
31.000
25.667
1
2
3
4
5
31
30
29
33
26
30
29
29
31
25
30
29
28
29
26
Treatment
Means
29.8
28.8
28.4
Randomized Block Design

Mean Square Due to Treatments
The overall sample mean is 29. Thus,
SSTR = 5[(29.8 - 29)2 + (28.8 - 29)2 + (28.4 - 29)2] = 5.2
MSTR = 5.2/(3 - 1) = 2.6

Mean Square Due to Blocks
SSBL = 3[(30.333 - 29)2 + . . . + (25.667 - 29)2] = 51.33
MSBL = 51.33/(5 - 1) = 12.8
• Mean Square Due to Error
SSE = 62 - 5.2 - 51.33 = 5.47
MSE = 5.47/[(3 - 1)(5 - 1)] = .68
Randomized Block Design

ANOVA Table
Source of
Variation
Degrees of
Freedom
Mean
Squares
F
5.20
2
2.60
3.82
51.33
4
12.80
Error
5.47
8
.68
Total
62.00
14
Treatments
Blocks
Sum of
Squares
Randomized Block Design
• Rejection Rule
p-Value Approach:
Reject H0 if p-value < .05
Critical Value Approach:
Reject H0 if F > 4.46
For  = .05, F.05 = 4.46
(2 d.f. numerator and 8 d.f. denominator)
Randomized Block Design
Test Statistic
F = MSTR/MSE = 2.6/.68 = 3.82
• Conclusion
The p-value is greater than .05 (where F = 4.46)
and less than .10 (where F = 3.11). (Excel provides
a p-value of .07). Therefore, we cannot reject H0.

There is insufficient evidence to conclude that
the miles per gallon ratings differ for the three
gasoline blends.
• Selamat Belajar Semoga Sukses.
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