Matakuliah
Tahun
: L0104 / Statistika Psikologi
: 2008
Analisis Varians Klasifikasi Satu
Arah
Pertemuan 17
Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa
akan mampu :
• Mahasiswa akan dapat menyusun simpulan
tentang sumber variasi, jumlah kuadrat, derajat
bebas dan kuadrat tengah dan uji F.
3
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Outline Materi
•
•
•
•
•
Konsep dasar analisis varians
Klasifikasi satu arah ulangan sama
Klasifikasi satu arah ulangan tidak sama
Prosedur uji F
Pembandingan perlakuan
4
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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
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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: μ1 = μ2 = μ3 = . . . = μk
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.
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Assumptions for Analysis of Variance
• For each population, the response variable is
normally distributed.
• The variance of the response variable, denoted σ2, is
the same for all of the populations.
• The observations must be independent.
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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
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Between-Samples Estimate
of Population Variance
• A between-samples estimate of
_ =
square between (MSB).
k
MSB 
2 is called the mean
2
 n j ( x j  x) 2
j1
k1
• The numerator of MSB is called the sum of squares
between (SSB).
• The denominator of MSB represents the degrees of
freedom associated with SSB.
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Within-Samples Estimate
of Population Variance
• The estimate of σ 2 based on the variation of the
sample observations within each sample is called the
mean square within (MSW).
k
MSW 
2
 (n j  1) s 2j
j1
nT  k
• The numerator of MSW is called the sum of squares
within (SSW).
• The denominator of MSW represents the degrees of
freedom associated with SSW.
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Comparing the Variance Estimates: The F
Test
• If the null hypothesis is true and the ANOVA
assumptions are valid, the sampling
distribution of MSB/MSW is an F distribution
with MSB d.f. equal to k - 1 and MSW d.f.
equal to nT - k.
• If the means of the k populations are not
equal, the value of MSB/MSW will be inflated
because MSB overestimates σ2.
• Hence, we will reject H0 if the resulting value
of MSB/MSW appears to be too large to have
been selected at random from the appropriate
F distribution.
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Test for the Equality of k Population
Means
• Hypotheses
H0: μ1 = μ2 = μ3 = . . . = μk
Ha: Not all population means are equal
• Test Statistic
F = MSB/MSW
• Rejection Rule
Reject H0 if F > Fα
where the value of Fα is based on an F distribution
with k - 1 numerator degrees of freedom and nT - 1
denominator degrees of freedom.
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Sampling Distribution of MSTR/MSE
• The figure below shows the
rejection region associated
with a level of significance
equal to α where Fα denotes
the critical value.
Do Not Reject H0
Reject H0
F
Critical Value
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MSTR/MSE
The ANOVA Table
Source of
Sum of
Variation Squares
Treatment SSTR
Error
SSE
Total
SST
Degrees of
Freedom
k-1
nT - k
nT - 1
Mean
Squares
F
MSTR MSTR/MSE
MSE
SST divided by its degrees of freedom nT - 1 is simply the
overall sample variance that would be obtained if we
treated the entire nT observations as one data set.
k
nj
SST   ( xij  x) 2  SSTR  SSE
j 1 i 1
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ANOVA Table for a
Completely Randomized Design
Source of
Variation
Sum of
Squares
Degrees of
Freedom
Treatments
SSTR
k-1
Error
SSE
nT - k
Total
SST
nT - 1
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Mean
Squares
MSTR 
SSTR
k-1
SSE
MSE 
nT - k
F
MSTR
MSE
Selamat Belajar
Semoga Sukses
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