11.9: BG ANOVA Practice Exercises
 Page ID
 18095
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?
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 Fscore is larger (more extreme) than the critical Fscore found in the Critical Values of F table with df's of 3 and 40 (F_{crit} = 2.23)
Exercise \(\PageIndex{4}\)
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:

 \(K=4\)
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 
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, \(F(2,54) = 42.36, p < .05\).
Contributors and Attributions
Foster et al. (University of MissouriSt. Louis, Rice University, & University of Houston, Downtown Campus)