# Class_15 - One-Way Analysis of Variance Recapitulation 1...

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One-Way Analysis of Variance

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Recapitulation 1. Comparing differences among three or more subsamples requires a different statistical test than either z-tests or t-tests. 2. The solution is to perform an analysis of variance (ANOVA). 3. ANOVA involves the comparison of two estimates for the population variance . 4. One variance estimate captures only the random differences among sampled units, the other these random differences plus the effects of being in the different subsamples . 5. The ratio between the two estimated variances is evaluated using the F-statistic sampling distributions.
Y ij = μ + α X ij + ε ij

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As an example, consider an experiment on worker productivity in an introductory psychology class. Thirty students were randomly selected for the experiment from PSYCH 100 and randomly assigned to one of three subgroups. The productivity measure (Y ij ) was the number of puzzles that these students solved in a fixed period of time. The three experimental conditions (treatments, X ij ) were: left alone to solve puzzles; solving puzzles in the presence of the other nine group members (so that each subject could observe her or his own rate of puzzle solving); and solving puzzles in the presence of other subjects AND in the presence of a monitor, meant to simulate a supervisor. The results look like this:
————————————————————————————————————— Not Monitored Alone Not Monitored Together Monitored Subject Y i,1 Subject Y i,2 Subject Y i,3 ————————————————————————————————————— 1 13 1 9 1 8 2 14 2 11 2 6 3 10 3 10 3 9 4 11 4 8 4 7 5 12 5 10 5 8 6 10 6 12 6 10 7 12 7 11 7 8 8 12 8 10 8 9 9 13 9 9 9 6 10 11 10 10 10 11 ————————————————————————————————————— N 1 = 10 N 2 = 10 N 3 = 10 Σ 1 = 118 Σ 2 = 100 Σ 3 = 82 _ _ _ Y 1 = 11.8 Y 2 = 10.0 Y 3 = 8.2 _ Y = 10.0

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Our hypothesis (H 1 ) is that working conditions affect worker performance in ways that we do not fully understand: H 1 : μ 1 μ 2 μ 3 Our null hypothesis (H 0 ) is that worker performance is unaffected by working conditions: H 0 : μ 1 = μ 2 = μ 3 Since a comparison of THREE subgroup means is required, t-tests are inappropriate. The approach known generically as the analysis of variance must be used.
————————————————————————————————————— Not Monitored Alone Not Monitored Together Monitored Subject Y i,1 Subject Y i,2 Subject Y i,3 ————————————————————————————————————— 1 13 1 9 1 8 2 14 2 11 2 6 3 10 3 10 3 9 4 11 4 8 4 7 5 12 5 10 5 8 6 10 6 12 6 10 7 12 7 11 7 8 8 12 8 10 8 9 9 13 9 9 9 6 10 11 10 10 10 11

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• Fall '07
• Velez
• General Linear Models, Minimum Significant Difference, Linear Models Procedure, Tests General Linear, Comparison Tests General

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