Transcript Chapter 10 Analysis of Variance (Hypothesis Testing III)

Chapter 10 (1e), Ch. 9 (2/3e)
Hypothesis Testing Using
Analysis of Variance (ANOVA)
Basic Logic
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ANOVA can be used in situations where the
researcher is interested in the differences in
sample means across three or more
categories.
Outline
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The basic logic of ANOVA
A sample problem applying ANOVA
The Five Step Model
Logic (cont.)
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Examples:
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How do Protestants, Catholics and Jews vary in
terms of number of children?
How do Liberals, Conservatives, and NDP
supporters vary in terms of income?
How do older, middle-aged, and younger people
vary in terms of frequency of church attendance?
Logic (cont.)
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ANOVA asks “are the differences between
the sample means so large that we can
conclude that the populations represented by
the samples are different?”
The H0 is that the population means are the
same:
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H0: μ1= μ2= μ3 = … = μk
Logic (cont.)
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If the H0 is true, the sample means should be
about the same value.
If the H0 is false, there should be substantial
differences between categories, combined
with relatively little difference within
categories.
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The sample standard deviations should be low in
value.
Logic (cont.)
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If the H0 is true, there will be little difference
between sample means.
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If the H0 is false, there will be big difference
between sample means combined with small
values for s.
Logic (cont.)
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The larger the differences between the sample
means, the more likely the H0 is false.-- especially
when there is little difference within categories.
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When we reject the H0, we are saying there are
differences between the populations represented by
the sample.
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The ANOVA test uses the F-statistic and the Fdistribution (Appendix D). Table uses degrees of
freedom for the between number dfb = k-1 and the
within number dfw = n – k.
Steps in Computation of ANOVA
1.Find SST (Formula 10.10
or 9.10):
2. Find SSB (Formula 10.4
or 9.4):
3. Find SSW by subtraction
(Formula 10.11 or 9.11):
SST    n
2
2

SSB  nk  k  

SSW    i   k

2

2
Steps in Computation of ANOVA
4. Calculate the degrees of freedom (see 10.5,10.6
or 9.5, 9.6): dfb = k-1 and dfw = n – k.
5. Construct the mean square estimates by dividing
SSB and SSW by their degrees of freedom
(10.7,10.8 or 9.7, 9.8):
MSw = SSW / dfw
MSb = SSB / dfb
6. Find F ratio by Formula 10.9:
F = MSb / MSw
Example: 1e #10.6, 2/3e #9.6
Does voter turnout vary by type of election? Calculate the
data for 3 types of elections and make a table:
∑Xi
∑X2
Group
Means
Grand mean:
Municipal
Provincial
Federal
441
559
723
20,213
27,607
45,253
36.75
46.58
60.25
  47.86
Example (cont.)
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The difference in the means suggests that
turnout does vary by type of election.
Turnout seems to increase as the scope
of the election increases.
Are these differences significant?
Step 1 Make Assumptions and Meet
Test Requirements
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Independent Random Samples
LOM is I-R
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The dependent variable (e.g., voter turnout) should be I-R to justify
computation of the mean. ANOVA is often used with ordinal variables
with wide ranges.
Populations are normally distributed.
Population variances are equal.
Step 2: State the Null Hypothesis
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H0: μ1 = μ2= μ3
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The H0 states that the population means are the
same.
H1: At least one population mean is different.
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If we reject the H0, the test does not specify which
population mean is different from the others.
Step 3: Select the Sampling Distribution
and Determine the Critical Region
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Sampling Distribution = F distribution
Alpha = 0.05
dfw = (n – k) = 33
dfb = k – 1 = 2
F(critical) = 3.32
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The exact dfw (33) is not in the table but dfw = 30
and dfw = 40 are. Choose the larger F ratio as F
critical.
Step 4: Calculate the Test Statistic
1.SST:
SST   X  NX
2
2
SST  93073   36   47.86 
2
SST  93073  (82460.87)
SST  10612.13
2. SSB =

nk  k  

2
 12(36.75  47.86) 2  12(46.58  47.86) 2  12(60.25  47.86) 2
 3342.99
3. SSW by subtraction:
SSW = SST – SSB = 10,612.13 - 3,342.99 = 7269.14
Step 4: Calculate the Test Statistic (cont.)
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Calculate degrees of freedom:
dfw = n-k = 33 and dfb = k-1 = 2
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Find the Mean Square Estimates:
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MSW = SSW/dfw
MSW =7269.14/33
MSW = 220.28
MSB = SSB/dfb
MSB = 3342.99/2
MSB = 1671.50
Step 4: Calculate the Test Statistic (cont.)
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Find the F ratio by Formula 10.9 (9.9):
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F = MSB/MSW
F = 1671.95/220.28
F = 7.59
Step 5 Making a Decision and
Interpreting the Test Results
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F (obtained) = 7.59
F (critical) = 3.32
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The test statistic is in the critical region. Reject H0.
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Voter turnout varies significantly by type of election.
Now…
Work with a partner and try 1e #10.5 (2/3e # 9.5)