pasw & hand calculations for...
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PASW & Hand Calculationsfor ANOVA
Gravetter & WallnauChapter 13, Problem 26
One possible reason that some birds migrate and others don’t is intelligence. Birds with small brains relative to their body size might not be smart enough to find food during the winter. Birds with larger brains might be creative enough to find food even when the weather turns harsh. The next slide has the relative brain size for individual birds in each sample. Are their significant difference among the groups? Compute η2. If appropriate, calculate Tukey’s HSD to determine which groups are sig. different.
Gravetter & WallnauChapter 13, Problem 26
Non-MigratingShort-Distance
Migrants Long-Distance Migrants
181319121612
61179813
495657
Step 1: H0, H1 & α
H0: µNon-Migrating= µShort-Distance Migrants= µLong-Distance Migrants
H1: not H0
α = .05
Step 2: Critical Value
Skip – use the p value from the ANOVA output
F(2, 15) = 3.68From table of critical F values
dfbetween= # levels of factor – 1 = 3 – 1 = 2
dfwithin = Σ(# of people in a condition – 1) = (6 – 1) + (6 – 1) + (6 – 1) = 15
Step 3: Calculate Fobserved
Load PASW
Create two variablesOne indicates the level of the factor
Name: migrate, Label: Migration Status, Values: 1 = Non-Migrating, 2 = Short-Distance Migrant, 3 = Long-Distance Migrant, Measure: Nominal
One is the DVName: brain, Label: Relative Brain Size, Measure: Scale
Type in the data
Step 3: Calculate Fobserved
Click on Analyze | General Linear Model | Univariate
Move the DV (brain size) into the Dependent Variable box
Move the factor (migrate) into the Fixed Factor(s) box
Step 3: Calculate Fobserved
Click on Post HocMove the factor (migrate) into the Post Hoc Tests for boxSelect Tukey as the desired testClick Continue
Click OptionsSelect Descriptive statistics, Estimates of effect sizeClick Continue
Click OK
Step 3: Calculate Fobserved
F(2, 15) = 19.286, p = .000, MSerror = 6.533, η2
= .720
Step 3: Calculate Fobserved
Non-MigratingShort-Distance
MigrantsLong-Distance
Migrants
181319121612
61179813
495657
N = 18(total number of scores)
G = 180(sum of all scores = 90 + 54 +
36)
ΣX2 = 2150(square each score and sum
the squares)
T = 90 = 18+13+19+12+16+12
M = 15= 90 / 6
SS = 48 = (18 - 15)2 + (13 - 15)2 +
(19 - 15)2 + (12 - 15)2 + (16 -15)2 + (12 - 15)2
T = 54 = 6 + 11 + 7 + 9 + 8 + 13
M = 9= 54 / 6
SS = 34=(6 - 9)2 + (11 - 9)2 + (7 - 9)2 +
(9 - 9)2 + (8 - 9)2 + (13 - 9)2
T = 36= 4 + 9 + 5 + 6 + 5 + 7
M = 6= 36 / 6
SS = 16=(4 - 6)2 + (9 - 6)2 + (5 - 6)2 +
(6 - 6)2 + (5 - 6)2 + (7 - 6)2
Step 3: Calculate Fobserved
35018
1802150
N
GXSS
2
22
total
=
−=
−∑=
98
163448
SSSS menteach treat insidewithin
=++=
∑=
252
1800216486135018
180
6
36
6
54
6
90
N
G
n
TSS
2222
22
between
=−++=
−++=
−∑=
17
118
1Ndf total
=−=−=
15
318
kNdfwithin
=−=−=
2
13
1kdfbetween
=−=−=
1262
252
df
SSMS
between
betweenbetweeen
=
=
=
6.5315
98
df
SSMS
within
withinwithin
=
=
=
Step 3: Calculate Fobserved
19.2866.533
126
MS
MSF
within
between
=
=
=
Step 4: Decide
If p ≤ α then reject H0 otherwise fail to reject H0p = .000, α = .05 therefore, reject H0It is unlikely that all the means in the population are equal
If Fobserved≥ Fcritical then reject H0 otherwise fail to reject H0
Fobserved= 19.286, Fcritical = 3.68 therefore, reject H0
η2
η2 = SSbetween/ SStotal = 252 / 350 = 0.720
TukeyHSD
H0: µNon-Migrating= µShort-Distance Migrant
H1: µNon-Migrating≠ µShort-Distance Migrant
H0: µNon-Migrating= µLong-Distance Migrant
H1: µNon-Migrating≠ µLong-Distance Migrant
H0: µLong-Distance Migrant= µShort-Distance Migrant
H1: µLong-Distance Migrant≠ µShort-Distance Migrant
α = .05
TukeyHSD
Pairs that are reliably different (non-migrating vs. short-distance migrant and non-migrating vs. long-distance migrant)p ≤α
Pair that is not reliably different (short-distance migrant vs. long-distance migrant) p > α
TukeyHSD
835.36
6.5333.675
n
MSHSD treatmentwithin
dfk,α,q
treatmentwithin
=
⋅=
⋅= −
−
Any pair of means that is at least 3.835 apart will be reliably different.
Means are likely different
Insufficient evidence to state that means are likely different
From the table of the Studentized Range Statistic with α= .05, k = 3 (three levels) and dfwithin = 15
Writing
The means and standard deviations are presented in Table 1. The analysis of variance revealed a significant effect of migration style on relative brain size, F(2, 15) = 19.286, p = .000, MSerror = 6.533, η2
= .720. Tukey’s post hoc tests revealed that non-migrating birds had a different relative brain size than short-distance migrants (p = .003) and long-distance migrants (p = .000). There was insufficient evidence to suggest that the long- and short-distance migrants had different brain sizes (p = .138).
Writing
Table 1
Relative Brain Size of Three Types of Birds
Type of Bird M SD
Non-migrating 15.00 3.10
Short-distance migrant 9.00 2.61
Long-distance migrant 6.00 1.79