11.9: BG ANOVA Practice Exercises
- Page ID
- 22119
What are the three pieces of variance analyzed in ANOVA?
- Answer
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Variance between groups (\(SSB\)), variance within groups (\(SSW\)) and total variance (\(SST\)).
What is the purpose of post hoc tests?
- Answer
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Post hoc tests are run if we reject the null hypothesis in ANOVA; they tell us which specific group differences are significant.
Based on the ANOVA table below, do you reject or retain the null hypothesis?
Source | \(SS\) | \(df\) | \(MS\) | \(F\) |
---|---|---|---|---|
Between | 60.72 | 3 | 20.24 | 3.88 |
Within | 213.61 | 41 | 5.21 | |
Total | 274.33 | 44 |
- Answer
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The null hypothesis should be rejected (F(3,41) = 3.88, p < .05 because the calculated F-score is larger (more extreme) than the critical F-score found in the Critical Values of F table with df's of 3 and 40 (Fcrit = 2.23)
Finish filling out the following ANOVA tables:
- \(K = 4\)
Source | \(SS\) | \(df\) | \(MS\) | \(F\) |
---|---|---|---|---|
Between | 87.40 | |||
Within | ||||
Total | 199.22 | 33 |
- \(N=14\)
Source | \(SS\) | \(df\) | \(MS\) | \(F\) |
---|---|---|---|---|
Between | 2 | 14.10 | ||
Within | ||||
Total | 64.65 |
Source | \(SS\) | \(df\) | \(MS\) | \(F\) |
---|---|---|---|---|
Between | 2 | 42.36 | ||
Within | 54 | 2.48 | ||
Total |
- Answer:
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- \(K=4\)
Table \(\PageIndex{5}\)- Completed ANOVA Summary Table for a. Source \(SS\) \(df\) \(MS\) \(F\) Between 87.40 3 29.13 7.81 Within 111.82 30 3.73 leave blank Total 199.22 33 leave blank leave blank - \(N=14\)
Table \(\PageIndex{6}\)- Completed ANOVA Summary Table for b. Source \(SS\) \(df\) \(MS\) \(F\) Between 28.20 2 14.10 4.26 Within 36.45 11 3.31 leave blank Total 64.65 13 leave blank leave blank Table \(\PageIndex{6}\)- Completed ANOVA Summary Table for c. Source \(SS\) \(df\) \(MS\) \(F\) Between 210.10 2 105.05 42.36 Within 133.92 54 2.48 leave blank Total 344.02 56 leave blank leave blank
You and your friend are debating which type of candy is the best. You find data on the average rating for hard candy (e.g. jolly ranchers, \(\overline{\mathrm{X}}\)= 3.60), chewable candy (e.g. starburst, \(\overline{\mathrm{X}}\) = 4.20), and chocolate (e.g. snickers, \(\overline{\mathrm{X}}\)= 4.40); each type of candy was rated by 30 people. Test for differences in average candy rating using SSB = 16.18 and SSW = 28.74 with no research hypothesis (so you don't have to do pairwise comparisons if you reject the null hypotheses).
- Answer
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Step 1: \(H_0: μ_1 = μ_2 = μ_3\) “There is no difference in average rating of candy quality”, \(H_A\): “At least one mean is different.”
Step 2: 3 groups and 90 total observations yields \(df_{num} = 2\) and \(df_{den} = 87\), \(α = 0.05\), \(F^* = 3.11\).
Step 3: based on the given \(SSB\) and \(SSW\) and the computed \(df\) from step 2, is:
Table \(\PageIndex{7}\)- Completed ANOVA Summary Table Source \(SS\) \(df\) \(MS\) \(F\) Between 16.18 2 8.09 24.52 Within 28.74 87 0.33 leave blank Total 44.92 89 leave blank leave blank Step 4: \(F > F^*\), reject \(H_0\). Based on the data in our 3 groups, we can say that there is a statistically significant difference in the quality of different types of candy, \(F(2,87) = 24.52, p < .05\).
You are assigned to run a study comparing a new medication (\(\overline{\mathrm{X}}\)= 17.47, \(n\) = 19), an existing medication (\(\overline{\mathrm{X}}\)= 17.94, \(n\) = 18), and a placebo (\(\overline{\mathrm{X}}\)= 13.70, \(n\) = 20), with higher scores reflecting better outcomes. Use \(SSB = 210.10\) and \(SSW = 133.90\) to test for differences with no research hypothesis (so you don't have to do pairwise comparisons if you reject the null hypotheses).
- Answer
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Step 1: \(H_0: μ_1 = μ_2 = μ_3\) “There is no difference in average outcome based on treatment”, \(H_A\): “At least one mean is different.”
Step 2: 3 groups and 57 total participants yields \(df_{num} = 2\) and \(df_{den} = 54\), \(α = 0.05, F^* = 3.18\).
Step 3: based on the given \(SSB\) and \(SSW\) and the computed \(df\) from step 2, is:
Table \(\PageIndex{8}\)- Completed ANOVA Summary Table Source \(SS\) \(df\) \(MS\) \(F\) Between 210.10 2 105.02 42.36 Within 133.90 54 2.48 leave blank Total 344.00 56 leave blank leave blank Step 4: \(F > F^*\), reject \(H_0\). Based on the data in our 3 groups, we can say that there is a statistically significant difference in the effectiveness of the treatments, \(F(2,54) = 42.36, p < .05\).