11.9: BG ANOVA Practice Exercises

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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 (F_crit = 2.23)

Exercise \(\PageIndex{4}\)

Finish filling out the following ANOVA tables:

\(K = 4\)

Table \(\PageIndex{2}\)- ANOVA Summary Table with Missing Results
Source	\(SS\)	\(df\)
Between	87.40
Within
Total	199.22	33

\(N=14\)

Table \(\PageIndex{3}\)- ANOVA Summary Table with Missing Numbers
Source	\(SS\)	\(df\)	\(MS\)
Between		2	14.10
Within
Total	64.65

Table \(\PageIndex{4}\)- ANOVA Summary Table with Missing Results
Source	\(df\)	\(MS\)	\(F\)
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, \(F(2,54) = 42.36, p < .05\).