Copyright © 2012 by Nelson Education Limited. Chapter 9 Hypothesis Testing III: The Analysis of Variance 9-1

Copyright © 2012 by Nelson Education Limited.

Chapter 9Hypothesis Testing III:

The Analysis of Variance

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• The basic logic of hypothesis testing as applied to analysis of variance (ANOVA)

• Perform the ANOVA test using the five-step model

• Limitations of ANOVA

In this presentation you will learn about:

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• ANOVA (analysis of variance) can be used in situations where the researcher is interested in the differences in sample means across three or more categories.

• Examples:◦How do urban, suburban, and rural families vary in

terms of number of children?◦How do people with less than high school, high school,

and post-secondary education vary in terms of income?◦How do younger, middle-aged, and older people vary in

terms of frequency of religious service attendance?

Basic Logic

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• Can think of ANOVA as extension of t test for more than two groups.

• ANOVA asks “are the differences between the samples large enough to reject the null hypothesis and justify the conclusion that the populations represented by the samples are different?”

– The H0 is that the population means are the same:

H0: μ1= μ2= μ3 = … = μk

– If the H0 is true, the sample means should be about the same value.

Basic Logic (continued)

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• If H0 is false, there should be substantial differences

between the sample means of the categories, combined with relatively little difference within (sample standard deviations should be low in value) categories.

• When we reject the H0, we are saying there are

differences between the populations represented by the samples.

Basic Logic (continued)

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• Students taking introductory biology at a large university were randomly assigned to one of three sections:

1. the first section was taught by traditional “lecture-lab” method

2. the second section by “all-lab” method

3. the third section by “videotaped lectures and labs” method. At the end of the semester, random samples of final exam scores were

collected from each section.

Basic Logic: An Example

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In Scenario 1 (Table 9.1)

Means and standard deviations of the groups are very similar. These results would be quite consistent with the null hypothesis of no difference.

Basic Logic: An Example (continued)

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In Scenario 2 (Table 9.2)

There are large differences in scores between groups (means) but small differences in scores within each group (standard deviations). These results would contradict the null hypothesis, and support the notion that final exam scores do vary by teaching method.

Basic Logic: An Example (continued)

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1. Calculate total sum of squares (SST):

OR

Highlighted formula provides a quicker way to calculate the statistic.

Six Steps in Computation of ANOVA

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2. Calculate sum of squares between (SSB):

Six Steps in Computation of ANOVA (continued)

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3. Calculate sum of squares within (SSW):

OR

Highlighted formula provides a quicker way to calculate the statistic.


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4. Calculate degrees of freedom (Formulas 9.5 and 9.6):


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5. Calculate the mean squares (Formulas 9.7 and 9.8):


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6. Calculate F ratio (Formula 9.9):


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The computational routine for ANOVA can be summarized as:

Six Steps in Computation of ANOVA: Summary

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The grade point average of students in three (co-ed, all-male, and all-female) residences has been monitored by the administration of a university. The GPA from random samples of 14 students from each residence was collected.

Computation of ANOVA: An Example

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Does GPA vary significantly by type of residence?

Computation of ANOVA: An Example (continued)

Co-Ed All-Male All-Female 3.5 2.0 2.0 4.0 1.6 4.0 3.0 3.0 3.5 3.0 2.0 3.5 2.7 3.2 3.2 3.1 3.4 3.1 3.2 2.1 3.0 3.3 2.6 1.8 3.5 2.8 2.5 3.2 1.8 2.8 4.0 2.4 3.1 3.8 3.1 2.8 3.6 1.0 2.6 2.8 0.8 2.5

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Co-Ed All-Male All-Female 3.5 2.0 2.0 4.0 1.6 4.0 3.0 3.0 3.5 3.0 2.0 3.5 2.7 3.2 3.2 3.1 3.4 3.1 3.2 2.1 3.0 3.3 2.6 1.8 3.5 2.8 2.5 3.2 1.8 2.8 4.0 2.4 3.1 3.8 3.1 2.8 3.6 1.0 2.6 2.8 0.8 2.5

ΣX = 46.7 31.8 40.4 = 3.34 2.27 2.89ΣX2= 158.01 80.62 121.14

= 2.83

kX

X


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• The difference in the means suggests that GPA does vary by type of residence.

• GPA seems to be highest in co-ed residence and lowest in all-male residence.

• Are these differences statistically significant?


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Six Steps in Computation of ANOVA:

1. SST (Formula 9.10)

= (158.01+80.62+121.14)-(42)(2.83)2

= 359.77 -(42)(8.01)

= 359.77 -336.42

= 22.35


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2. SSB (Formula 9.4)

= 14(3.34-2.83)2 + 14(2.27-2.83)2 + 14(2.89- 2.83)2

= 14(0.26) + 14(0.31) + 14(0.0036)= 3.64 + 4.34 + 0.05 = 8.03


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3. SSW (Formula 9.11)= 22.35– 8.03= 14.32

4. Degrees of freedom (Formulas 9.5 and 9.6)dfw = n - k = 42 - 3 = 39dfb = k - 1 = 3 - 1 = 2


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5. Mean Squares (Formulas 9.7 and 9.8) MSW = SSW/dfw

=14.32/39 = 0.37

MSB = SSB/dfb = 8.03 /2

= 4.02


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6. F ratio (Formula 9.9) = 4.02 / 0.37 = 10.86

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• Independent Random Samples

• Level of Measurement is Interval-Ratio– The dependent variable (e.g., GPA) should be I-R to justify

computation of the mean.

• Populations are normally distributed.

• Population variances are equal.*ANOVA will tolerate some deviation from its assumptions as long as sample sizes are

roughly equal.

Performing the ANOVA Test Using the Five-Step Model

Step 1: Make Assumptions and Meet Test Requirements*

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• H0: μ1 = μ2= μ3

– The H0 states that the population

means are the same.

• H1: At least one population mean is different.

Step 2: State the Null Hypothesis

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• Sampling Distribution = F distribution

• Alpha = 0.05

• dfw = (n – k) = 39

• dfb = k – 1 = 2

• F(critical) = 3.32 (Note, the exact dfw (39) is not in the table but

dfw = 30 and dfw = 40 are. Choose the larger F ratio as F critical).

Step 3: Select Sampling Distribution and Establish the Critical Region

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• F (obtained) = 10.86

Step 4: Calculate the Test Statistic

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• F (obtained) = 10.86

F (critical) = 3.32

– The test statistic, F (obtained), falls in the

critical region.

• We reject the null hypothesis, H0, of no difference.

• At least one of these residences is significantly different than the other residences.

Step 5: Make Decision and Interpret Results

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1. Requires interval-ratio level measurement of the dependent variable

2. Statistically significant differences are not necessarily important.

Limitations of ANOVA

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3. The alternative (research) hypothesis, H1, is not specific. It only asserts that at least one of the population means differs from the others.

– Thus, we must use other (e.g., post hoc) statistical techniques for more specific differences.

Limitations of ANOVA(continued)

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Copyright © 2012 by Nelson Education Limited. Chapter 9 Hypothesis Testing III: The Analysis of Variance 9-1