# 11.9: BG ANOVA Practice Exercises


## Exercise $$\PageIndex{1}$$

What are the three pieces of variance analyzed in ANOVA?

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

## Exercise $$\PageIndex{2}$$

What is the purpose of post hoc tests?

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

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
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).

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).

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

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.