Chapter 10 Hypothesis Testing III (ANOVA). Chapter Outline Introduction The Logic of the Analysis...
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Transcript of Chapter 10 Hypothesis Testing III (ANOVA). Chapter Outline Introduction The Logic of the Analysis...
Chapter 10
Hypothesis Testing III (ANOVA)
Chapter Outline
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
Local State National
∑X 441 559 723
∑X220,213 27,607 45,253
Group Mean
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. Use Formula 10.4 to find SSB Find SSW by subtraction
SSW = SST – SSB SSW = 10,612.13 - 3,342.99 SSW= 7269.14
Use Formulas 10.5 and 10.6 to calculate degrees of freedom.
()()222930733647.8693073(82460.87)10612.13SSTXNXSSTSSTSST=−=−=−=∑
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
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.
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.