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11.9: BG ANOVA Practice Exercises

  • Page ID
    22119
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    Exercise \(\PageIndex{1}\)

    What are the three pieces of variance analyzed in ANOVA?

    Answer

    Variance between groups (\(SSB\)), variance within groups (\(SSW\)) and total variance (\(SST\)).

    Exercise \(\PageIndex{2}\)

    What is the purpose of post hoc tests?

    Answer

    Post hoc tests are run if we reject the null hypothesis in ANOVA; they tell us which specific group differences are significant.

    Exercise \(\PageIndex{3}\)

    Based on the ANOVA table below, do you reject or retain the null hypothesis?

    Table \(\PageIndex{1}\)- ANOVA Summary Table
    Source \(SS\) \(df\) \(MS\) \(F\)
    Between 60.72 3 20.24 3.88
    Within 213.61 41 5.21  
    Total 274.33 44    
    Answer

    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)

    Exercise \(\PageIndex{4}\)

    Finish filling out the following ANOVA tables:

    1. \(K = 4\)
    Table \(\PageIndex{2}\)- ANOVA Summary Table with Missing Results
    Source \(SS\) \(df\) \(MS\) \(F\)
    Between 87.40      
    Within        
    Total 199.22 33  
    1. \(N=14\)
    Table \(\PageIndex{3}\)- ANOVA Summary Table with Missing Numbers
    Source \(SS\) \(df\) \(MS\) \(F\)
    Between   2 14.10  
    Within        
    Total 64.65    
    Table \(\PageIndex{4}\)- ANOVA Summary Table with Missing Results
    Source \(SS\) \(df\) \(MS\) \(F\)
    Between   2   42.36
    Within   54 2.48  
    Total      
    Answer:
    1. \(K=4\)
    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
    1. \(N=14\)
    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

    Exercise \(\PageIndex{5}\)

    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

    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, \(F(2,87) = 24.52, p < .05\).

    Exercise \(\PageIndex{6}\)

    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

    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, \(F(2,54) = 42.36, p < .05\).


    This page titled 11.9: BG ANOVA Practice Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Michelle Oja.