11.9: BG ANOVA Practice Exercises
- Page ID
- 18095
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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
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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.
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 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:
- \(K = 4\)
| 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
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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, \(F(2,87) = 24.52, p < .05\).
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 Missouri-St. Louis, Rice University, & University of Houston, Downtown Campus)