Transcript Analysis of Variance - Mr. Bautista's Website

Analysis of Variance
One-Way ANOVA
Post-Hoc Tests
Two-Way ANOVA
Objective
 We wish to test the equality of several means
 H1: At least one pair of means is not equal.
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Notations
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Why analysis of variance?
 Explain the variation in the data through two
components:
 Experimental error
 Error due to treatments
 Variance:
 Sum of Squares Identity:
SST
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=
SSR +
SSE
Computational Formulas
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The ANOVA Table
 We summarize our computations in the following
table:
Source of
Variation
Treatment
Error
Total
Sum of
Squares
SSR
SSE
SST
 Reject Ho if
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Degrees of
Freedom
k-1
k(n - 1)
nk - 1
Mean Square
F
MSR
MSE
f = MSR/MSE
Example 1
 The following data represent the number of hours of
pain relief provided by 5 different brands of headache
tablets administered to 25 subjects. The 25 subjects
were randomly divided into 5 groups and each group
was treated with a different brand.
A
5
4
8
6
3
B
9
7
8
6
9
Tablet
C
3
5
2
3
7
D
2
3
4
1
4
E
7
6
9
4
7
 Perform the analysis of variance, and test the
hypothesis at the 0.05 level of significance that the
mean number of hours of relief provided by the
tablets is the same for all five brands
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Example 2
 Six different machines are being considered for use in
manufacturing rubber seals. The machines are being
compared with respect to tensile strength of the product.
A random sample of 4 seals from each machine is used to
determine whether the mean tensile strength varies from
machine to machine. The following are the tensile strength
measurements in kilograms per square centimeter.
Machine
1
17.5
16.9
15.8
18.6
2
16.4
19.2
17.7
15.4
3
20.3
15.7
17.8
18.9
4
14.6
16.7
20.8
18.9
5
17.5
19.2
16.5
20.5
6
18.3
16.2
17.5
20.1
 Perform the analysis of variance at the 0.05 level of
significance and indicate whether or not the mean tensile
strengths differ significantly for the 6 machines.
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Unequal Sample Sizes
 Degrees of freedom: SST  N - 1, SSR  k – 1, SSE 
N-k
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Example 3
 It is suspected that higher priced automobiles are assembled
with greater care than lower-priced automobiles. To investigate
this, a large luxury model A, a medium size sedan B, and a
subcompact hunchback C were compared for defects when they
arrived at the dealer’s showroom. All cars were manufactured by
the same company. The number of defects for several of the
three models were recorded and are shown below:
A
4
7
6
6
Model
B
5
1
3
5
3
4
C
8
6
8
9
5
 Test the hypothesis at the 0.05 level of significance that the
average number of defects is the same for the three models.
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Post-Hoc Tests
 We use these to determine which of the treatments
are significantly different from one another.
 Steps:
 Arrange the treatments in increasing order of their
means.
 Compute the critical value:

𝑄 = 𝑄𝛼 (𝑘, 𝑁 − 𝑘)
 Compute the differences among all pairs.
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Two-Way ANOVA (with Replication)
 We wish to test the following hypotheses:
 Ho: The column means are all equal
 H1: The column means are significantly different
 Ho: The row means are all equal
 H1: The row means are significantly different
 Ho: There is no significant interaction effect.
 H1: There is a significant interaction effect.
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