Unformatted text preview: − µC , µ A − µD ,
µB − µC , µ B − µD , µ C − µD .
Calculate s = 907.1/36 = 5.02, use α = 0.01, α/c = 0.05/6 = 0.0167,
tα/(2c) = t0.0167 ≈ 2.42.
The Bonferroni intervals are (with the probability P = 1−α/c ≈ 0.9833)
µA − µB ∈ [−15.2, −4.4], µA − µC ∈ [−24.1, −13.2], µA − µD ∈ [−3.5, 7.3], µB − µC ∈ [−14.3, −3.5], µB − µD ∈ [6.3, 17.1], µC − µD ∈ [15.2, 26.0]. All intervals contain respective diﬀerences of µ with α = 0.1.
We see that only interval for µA − µD contains 0 and does not support
the conclusion that the brand’s mean distances diﬀer. One factor model as
well as multifactor models are usually analyzed on a computer using, say,
SP SS software.
Randomized block design.
Oneway ANOVA tests for diﬀerence in the treatments, splitting the
variation into two parts: due to treatments and due to errors. However,
6 A. Zhensykbaev ANOVA there are likely to be sources of variation other than the treatments. If
another source of variation is removed, perhaps the treatments would now
be seen to have a signiﬁcant eﬀect. This question is for twoway ANOVA.
It allows a source of variation to be isolated before testing for treatments.
First source of variation is referred to as the treatments, the second one as
the block. The matched sets of experimental units are called blocks. As
early we present these data in the table
Treatment 1 Treatment 2 . . . Treatment k Mean
Block 1
x11
x21
...
xk 1
xB1
¯
Block 2
x12
x22
...
xk 2
xB2
¯
.
.
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Block b
x1b
x2b
...
xkb
xBb
¯
Mean
xT1
¯
xT2
¯
...
xTk
¯
x
¯
ANOVA F test to compare treatment means for randomized
block design.
Assumptions:
1. The probability distribution of observations associated with all blocktreatment combinations are approximately normal with mean =0.
2. The variances of blocktreatment distributions are equal.
3. The blocks are randomly selected and all treatments are applied to
each block.
Like before, the null hypothesis is
H0 = {µ1 = · · · = µk },
alternative hypothesis is
Ha = {at least two means diﬀer},
test statistics is
F=
M ST = SST
,
k−1 M ST
,
M SE M SE =
7 S...
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 Spring '13
 ChristopherStocker
 Math, Variance, A. Zhensykbaev

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