Transcript Chapter 10 Analysis of Variance (Hypothesis Testing III)

Chapter 10
Hypothesis Testing III (ANOVA)
Chapter Outline
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Introduction
The Logic of the Analysis of Variance
The Computation of ANOVA
Computational Shortcut
A Computational Example
Chapter Outline
 A Test of Significance for ANOVA
 An Additional Example for Computing
and Testing the Analysis of Variance
 The Limitations of the Test
 Interpreting Statistics: Does Sexual
Activity Vary by Marital Status?
In This Presentation
 The basic logic of ANOVA
 A sample problem applying ANOVA
 The Five Step Model
Basic Logic
 ANOVA can be used in situations
where the researcher is interested in
the differences in sample means
across three or more categories.
Basic Logic
 Examples:
 How do Protestants, Catholics and Jews
vary in terms of number of children?
 How do Republicans, Democrats, and
Independents vary in terms of income?
 How do older, middle-aged, and younger
people vary in terms of frequency of
church attendance?
Basic Logic
 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:
 H0: μ1= μ2= μ3 = … = μk
Basic Logic
 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.
 The sample standard deviations should
be low in value.
Basic Logic
 If the H0 is true, there will be little
difference between sample means.
 If the H0 is false, there will be big
difference between sample means
combined with small values for s.
Basic Logic
 The larger the differences between the
sample means, the more likely the H0 is
false.-- especially when there is little
difference within categories.
 When we reject the H0, we are saying there
are differences between the populations
represented by the sample.
Steps in Com putation of
ANOVA
1. Find SST by Formula 10.10.
2. Find SSB by Formula 10.4.
3. Find SSW by subtraction (Formula
10.11).
Steps in Computation of ANOVA
4. Calculate the degrees of freedom
(Formulas 0.5 and 10.6).
5. Construct the mean square estimates
by dividing SSB and SSW by their
degrees of freedom. (Formulas 10.7
and 10.8).
6. Find F ratio by Formula 10.9.
Example of Computation of
ANOVA
 Problem 10.6
 Does voter turnout vary by type of
election? Data are presented for local,
state, and national elections.
Example of Computation of
ANOVA
∑X
∑X2
Group
Mean
Local
State
National
441
559
723
20,213
27,607
45,253
36.75
46.58
60.25
Example of Computation of
ANOVA
 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?
Example of Computation of
ANOVA
 Use Formula 10.10 to find SST.2
SST   X  NX 2
 Use Formula 10.4 to find SSB
2
SST  93073   36  47.86 
 Find SSW by subtraction
SST  93073  (82460.87)
 SSW = SST – SSB
 SSW = 10,612.13 - 3,342.99
SST  10612.13
 SSW= 7269.14
 Use Formulas 10.5 and 10.6 to
calculate degrees of freedom.
Example of Computation of
ANOVA
 Use Formulas 10.7 and 10.8 to find
the Mean Square Estimates:
 MSW = SSW/dfw
 MSW =7269.14/33
 MSW = 220.28
 MSB = SSB/dfb
 MSB = 3342.99/2
 MSB = 1671.50
Example of Computation of
ANOVA
 Find the F ratio by Formula 10.9:
 F = MSB/MSW
 F = 1671.95/220.28
 F = 7.59
Step 1 Make Assumptions and
Meet Test Requirements
 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
 H0: μ1 = μ2= μ3
 The H0 states that the population
means are the same.
 H1: At least one population mean is
different.
 If we reject the H0, the test does not
specify which population mean is
different from the others.
Step 3 Select the S.D.
and Determine the C.R.
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Sampling Distribution = F distribution
Alpha = 0.05
dfw = (N – k) = 33
dfb = k – 1 = 2
F(critical) = 3.32
 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
 F (obtained) = 7.59
Step 5 Making a Decision and
Interpreting the Test Results
 F (obtained) = 7.59
 F (critical) = 3.32
 The test statistic is in the critical region.
Reject the H0.
 Voter turnout varies significantly by type
of election.