n_groups = random.randint(2,5)
sample_size = random.randint(5,15)
RandomData(groups = n_groups, n = sample_size).anova(test = "one-way")7 One-Way ANOVA
Sample Problems
7.1 About
the one-way ANOVA problems use the RamdomData class which requires:
- an integer value for the number of groups
- the sample size per group - all groups will have the same sample size
- a call to the
anova()method withtest = "one-way"
Each of the problems below sets factor levels using a random integer between 2 and 5. Per-group sample size is randomly set between 5 and 15. An example funciton call is included below
7.2 Problem 1
Given the following between-subjects data, use a one-way ANOVA with \(\alpha = {0.01}\)
| A | B | C | D | E |
|---|---|---|---|---|
| 46 | 49 | 58 | 42 | 65 |
| 40 | 57 | 45 | 31 | 59 |
| 49 | 67 | 51 | 36 | 61 |
| 41 | 65 | 63 | 40 | 60 |
| 36 | 50 | 54 | 47 | 51 |
| 25 | 52 | 45 | 49 | 61 |
| 35 | 58 | 48 | 44 | 50 |
| 45 | 65 | 49 | 45 | 55 |
| 55 | 55 | 45 | 36 | 57 |
| 36 | 46 | 70 | 26 | 56 |
| 39 | 57 | 59 | 48 | 65 |
| 39 | 67 | 35 | 55 | 58 |
| 26 | 52 | 67 | 40 | 63 |
Summary statistics for these data:
\(G = {3241} \quad \Sigma{X^2} = {169383} \quad k = {5} \quad N = {65}\)
\(T_{A} = {512} \quad SS_{A} = {843.08}\)
\(T_{B} = {740} \quad SS_{B} = {616.92}\)
\(T_{C} = {689} \quad SS_{C} = {1208.0}\)
\(T_{D} = {539} \quad SS_{D} = {745.23}\)
\(T_{E} = {761} \quad SS_{E} = {269.23}\)
State the Hypotheses
\(H_0: \mu_A = \mu_B = \mu_C = \mu_D = \mu_E\)
\(H_1:\) At least one mean is different
The decision criteria:
\(F_{crit} = {3.65}, \alpha = {0.01}\)
Calculate the Degrees of Freedom
\[df_{total} = N - 1\]
\[df_{total} = {65} - 1\]
\[df_{total} = {64}\]
\[df_{between} = k - 1\]
\[df_{between} = {5} - 1\]
\[df_{between} = {4}\]
\[df_{within} = N - K\]
\[df_{within} = {65} - {5}\]
\[df_{within} = {60}\]
Calculate the Sum of Squares
\[SS_{total} = \Sigma X^2 - \frac{G^2}{N}\]
\[SS_{total} = {169383} - \frac{3241^2}{65}\]
\[SS_{total} = {169383} - \frac{10504081}{65}\]
\[SS_{total} = {169383} - {161601.25}\]
\[SS_{total} = {7781.75}\]
\[SS_{within} = \Sigma SS_{inside\_each\_condition}\]
\[SS_{within} = {843.08 + 616.92 + 1208.0 + 745.23 + 269.23}\]
\[SS_{within} = {3682.46}\]
\[SS_{between} = SS_{total} - SS_{within}\]
\[SS_{between} = {7781.75} - {3682.46}\]
\[SS_{between} = {4099.29}\]
note: the other way to calculate \(SS_{betwen}\) is:
\[SS_{between} = \Sigma{\frac{T^2}{n}} - \frac{G^2}{N}\]
Calculate the Mean Squares
\[MS_{between} = \frac{SS_{between}}{df_{between}}\]
\[MS_{between} = \frac{4099.29}{4}\]
\[MS_{between} = {1024.82}\]
\[MS_{within} = \frac{SS_{within}}{df_{within}}\]
\[MS_{within} = \frac{3682.46}{60}\]
\[MS_{within} = {61.37}\]
Calculate the F-Ratio
\[F_{obt} = \frac{MS_{between}}{MS_{within}}\]
\[F_{obt} = \frac{1024.82}{61.37}\]
\[F_{obt} = {16.7}\]
Calculate \(\eta^2\)
\[\eta^2 = \frac{SS{between}}{SS_{total}}\]
\[\eta^2 = \frac{4099.29}{7781.75}\]
\[\eta^2 = {0.53}\]
The results:
reject the null hypothesis, results are significant,
\(F({4}, {60}) = {16.7}, p < {0.01}, \eta^2 = {0.53}\)
7.3 Problem 2
Given the following between-subjects data, use a one-way ANOVA with \(\alpha = {0.05}\)
| A | B | C | D |
|---|---|---|---|
| 58 | 85 | 50 | 32 |
| 75 | 84 | 50 | 67 |
| 75 | 62 | 64 | 67 |
| 69 | 72 | 57 | 73 |
| 69 | 78 | 47 | 66 |
| 60 | 61 | 38 | 66 |
| 42 | 45 | 57 | 43 |
| 64 | 84 | 71 | 75 |
| 71 | 68 | 36 | 67 |
| 70 | 85 | 51 | 53 |
Summary statistics for these data:
\(G = {2507} \quad \Sigma{X^2} = {164581} \quad k = {4} \quad N = {40}\)
\(T_{A} = {653} \quad SS_{A} = {896.1}\)
\(T_{B} = {724} \quad SS_{B} = {1626.4}\)
\(T_{C} = {521} \quad SS_{C} = {1040.9}\)
\(T_{D} = {609} \quad SS_{D} = {1726.9}\)
State the Hypotheses
\(H_0: \mu_A = \mu_B = \mu_C = \mu_D\)
\(H_1:\) At least one mean is different
The decision criteria:
\(F_{crit} = {2.87}, \alpha = {0.05}\)
Calculate the Degrees of Freedom
\[df_{total} = N - 1\]
\[df_{total} = {40} - 1\]
\[df_{total} = {39}\]
\[df_{between} = k - 1\]
\[df_{between} = {4} - 1\]
\[df_{between} = {3}\]
\[df_{within} = N - K\]
\[df_{within} = {40} - {4}\]
\[df_{within} = {36}\]
Calculate the Sum of Squares
\[SS_{total} = \Sigma X^2 - \frac{G^2}{N}\]
\[SS_{total} = {164581} - \frac{2507^2}{40}\]
\[SS_{total} = {164581} - \frac{6285049}{40}\]
\[SS_{total} = {164581} - {157126.22}\]
\[SS_{total} = {7454.78}\]
\[SS_{within} = \Sigma SS_{inside\_each\_condition}\]
\[SS_{within} = {896.1 + 1626.4 + 1040.9 + 1726.9}\]
\[SS_{within} = {5290.3}\]
\[SS_{between} = SS_{total} - SS_{within}\]
\[SS_{between} = {7454.78} - {5290.3}\]
\[SS_{between} = {2164.48}\]
note: the other way to calculate \(SS_{betwen}\) is:
\[SS_{between} = \Sigma{\frac{T^2}{n}} - \frac{G^2}{N}\]
Calculate the Mean Squares
\[MS_{between} = \frac{SS_{between}}{df_{between}}\]
\[MS_{between} = \frac{2164.48}{3}\]
\[MS_{between} = {721.49}\]
\[MS_{within} = \frac{SS_{within}}{df_{within}}\]
\[MS_{within} = \frac{5290.3}{36}\]
\[MS_{within} = {146.95}\]
Calculate the F-Ratio
\[F_{obt} = \frac{MS_{between}}{MS_{within}}\]
\[F_{obt} = \frac{721.49}{146.95}\]
\[F_{obt} = {4.91}\]
Calculate \(\eta^2\)
\[\eta^2 = \frac{SS{between}}{SS_{total}}\]
\[\eta^2 = \frac{2164.48}{7454.78}\]
\[\eta^2 = {0.29}\]
The results:
reject the null hypothesis, results are significant,
\(F({3}, {36}) = {4.91}, p < {0.05}, \eta^2 = {0.29}\)
7.4 Problem 3
Given the following between-subjects data, use a one-way ANOVA with \(\alpha = {0.01}\)
| A | B |
|---|---|
| 29 | 45 |
| 50 | 45 |
| 48 | 40 |
| 38 | 46 |
| 39 | 57 |
| 35 | 44 |
| 38 | 50 |
| 55 | 33 |
| 17 | 46 |
| 55 | 44 |
| 46 | 25 |
| 42 | 34 |
| 41 | 40 |
| 39 | 64 |
| 35 | 29 |
Summary statistics for these data:
\(G = {1249} \quad \Sigma{X^2} = {54835} \quad k = {2} \quad N = {30}\)
\(T_{A} = {607} \quad SS_{A} = {1361.73}\)
\(T_{B} = {642} \quad SS_{B} = {1432.4}\)
State the Hypotheses
\(H_0: \mu_A = \mu_B\)
\(H_1:\) At least one mean is different
The decision criteria:
\(F_{crit} = {7.64}, \alpha = {0.01}\)
Calculate the Degrees of Freedom
\[df_{total} = N - 1\]
\[df_{total} = {30} - 1\]
\[df_{total} = {29}\]
\[df_{between} = k - 1\]
\[df_{between} = {2} - 1\]
\[df_{between} = {1}\]
\[df_{within} = N - K\]
\[df_{within} = {30} - {2}\]
\[df_{within} = {28}\]
Calculate the Sum of Squares
\[SS_{total} = \Sigma X^2 - \frac{G^2}{N}\]
\[SS_{total} = {54835} - \frac{1249^2}{30}\]
\[SS_{total} = {54835} - \frac{1560001}{30}\]
\[SS_{total} = {54835} - {52000.03}\]
\[SS_{total} = {2834.97}\]
\[SS_{within} = \Sigma SS_{inside\_each\_condition}\]
\[SS_{within} = {1361.73 + 1432.4}\]
\[SS_{within} = {2794.13}\]
\[SS_{between} = SS_{total} - SS_{within}\]
\[SS_{between} = {2834.97} - {2794.13}\]
\[SS_{between} = {40.84}\]
note: the other way to calculate \(SS_{betwen}\) is:
\[SS_{between} = \Sigma{\frac{T^2}{n}} - \frac{G^2}{N}\]
Calculate the Mean Squares
\[MS_{between} = \frac{SS_{between}}{df_{between}}\]
\[MS_{between} = \frac{40.84}{1}\]
\[MS_{between} = {40.84}\]
\[MS_{within} = \frac{SS_{within}}{df_{within}}\]
\[MS_{within} = \frac{2794.13}{28}\]
\[MS_{within} = {99.79}\]
Calculate the F-Ratio
\[F_{obt} = \frac{MS_{between}}{MS_{within}}\]
\[F_{obt} = \frac{40.84}{99.79}\]
\[F_{obt} = {0.41}\]
Calculate \(\eta^2\)
\[\eta^2 = \frac{SS{between}}{SS_{total}}\]
\[\eta^2 = \frac{40.84}{2834.97}\]
\[\eta^2 = {0.01}\]
The results:
fail to reject the null hypothesis, results not significant,
\(F({1}, {28}) = {0.41}, p > {0.01}, \eta^2 = {0.01}\)
7.5 Problem 4
Given the following between-subjects data, use a one-way ANOVA with \(\alpha = {0.01}\)
| A | B |
|---|---|
| 55 | 77 |
| 50 | 55 |
| 39 | 72 |
| 48 | 66 |
| 46 | 61 |
| 56 | 69 |
| 36 | 70 |
| 45 | 59 |
Summary statistics for these data:
\(G = {904} \quad \Sigma{X^2} = {53280} \quad k = {2} \quad N = {16}\)
\(T_{A} = {375} \quad SS_{A} = {344.88}\)
\(T_{B} = {529} \quad SS_{B} = {376.88}\)
State the Hypotheses
\(H_0: \mu_A = \mu_B\)
\(H_1:\) At least one mean is different
The decision criteria:
\(F_{crit} = {8.86}, \alpha = {0.01}\)
Calculate the Degrees of Freedom
\[df_{total} = N - 1\]
\[df_{total} = {16} - 1\]
\[df_{total} = {15}\]
\[df_{between} = k - 1\]
\[df_{between} = {2} - 1\]
\[df_{between} = {1}\]
\[df_{within} = N - K\]
\[df_{within} = {16} - {2}\]
\[df_{within} = {14}\]
Calculate the Sum of Squares
\[SS_{total} = \Sigma X^2 - \frac{G^2}{N}\]
\[SS_{total} = {53280} - \frac{904^2}{16}\]
\[SS_{total} = {53280} - \frac{817216}{16}\]
\[SS_{total} = {53280} - {51076.0}\]
\[SS_{total} = {2204.0}\]
\[SS_{within} = \Sigma SS_{inside\_each\_condition}\]
\[SS_{within} = {344.88 + 376.88}\]
\[SS_{within} = {721.76}\]
\[SS_{between} = SS_{total} - SS_{within}\]
\[SS_{between} = {2204.0} - {721.76}\]
\[SS_{between} = {1482.24}\]
note: the other way to calculate \(SS_{betwen}\) is:
\[SS_{between} = \Sigma{\frac{T^2}{n}} - \frac{G^2}{N}\]
Calculate the Mean Squares
\[MS_{between} = \frac{SS_{between}}{df_{between}}\]
\[MS_{between} = \frac{1482.24}{1}\]
\[MS_{between} = {1482.24}\]
\[MS_{within} = \frac{SS_{within}}{df_{within}}\]
\[MS_{within} = \frac{721.76}{14}\]
\[MS_{within} = {51.55}\]
Calculate the F-Ratio
\[F_{obt} = \frac{MS_{between}}{MS_{within}}\]
\[F_{obt} = \frac{1482.24}{51.55}\]
\[F_{obt} = {28.75}\]
Calculate \(\eta^2\)
\[\eta^2 = \frac{SS{between}}{SS_{total}}\]
\[\eta^2 = \frac{1482.24}{2204.0}\]
\[\eta^2 = {0.67}\]
The results:
reject the null hypothesis, results are significant,
\(F({1}, {14}) = {28.75}, p < {0.01}, \eta^2 = {0.67}\)
7.6 Problem 5
Given the following between-subjects data, use a one-way ANOVA with \(\alpha = {0.05}\)
| A | B | C | D |
|---|---|---|---|
| 66 | 65 | 71 | 68 |
| 65 | 65 | 73 | 79 |
| 70 | 59 | 55 | 70 |
| 74 | 55 | 72 | 79 |
| 68 | 62 | 73 | 64 |
| 66 | 70 | 69 | 70 |
| 75 | 63 | 66 | 71 |
| 68 | 60 | 71 | 73 |
| 75 | 54 | 75 | 72 |
| 77 | 60 | 59 | 77 |
| 79 | 59 | 69 | 70 |
| 69 | 60 | 67 | 60 |
| 69 | 54 | 72 | 69 |
Summary statistics for these data:
\(G = {3521} \quad \Sigma{X^2} = {240637} \quad k = {4} \quad N = {52}\)
\(T_{A} = {921} \quad SS_{A} = {253.69}\)
\(T_{B} = {786} \quad SS_{B} = {259.23}\)
\(T_{C} = {892} \quad SS_{C} = {401.08}\)
\(T_{D} = {922} \quad SS_{D} = {354.92}\)
State the Hypotheses
\(H_0: \mu_A = \mu_B = \mu_C = \mu_D\)
\(H_1:\) At least one mean is different
The decision criteria:
\(F_{crit} = {2.8}, \alpha = {0.05}\)
Calculate the Degrees of Freedom
\[df_{total} = N - 1\]
\[df_{total} = {52} - 1\]
\[df_{total} = {51}\]
\[df_{between} = k - 1\]
\[df_{between} = {4} - 1\]
\[df_{between} = {3}\]
\[df_{within} = N - K\]
\[df_{within} = {52} - {4}\]
\[df_{within} = {48}\]
Calculate the Sum of Squares
\[SS_{total} = \Sigma X^2 - \frac{G^2}{N}\]
\[SS_{total} = {240637} - \frac{3521^2}{52}\]
\[SS_{total} = {240637} - \frac{12397441}{52}\]
\[SS_{total} = {240637} - {238412.33}\]
\[SS_{total} = {2224.67}\]
\[SS_{within} = \Sigma SS_{inside\_each\_condition}\]
\[SS_{within} = {253.69 + 259.23 + 401.08 + 354.92}\]
\[SS_{within} = {1268.92}\]
\[SS_{between} = SS_{total} - SS_{within}\]
\[SS_{between} = {2224.67} - {1268.92}\]
\[SS_{between} = {955.75}\]
note: the other way to calculate \(SS_{betwen}\) is:
\[SS_{between} = \Sigma{\frac{T^2}{n}} - \frac{G^2}{N}\]
Calculate the Mean Squares
\[MS_{between} = \frac{SS_{between}}{df_{between}}\]
\[MS_{between} = \frac{955.75}{3}\]
\[MS_{between} = {318.58}\]
\[MS_{within} = \frac{SS_{within}}{df_{within}}\]
\[MS_{within} = \frac{1268.92}{48}\]
\[MS_{within} = {26.44}\]
Calculate the F-Ratio
\[F_{obt} = \frac{MS_{between}}{MS_{within}}\]
\[F_{obt} = \frac{318.58}{26.44}\]
\[F_{obt} = {12.05}\]
Calculate \(\eta^2\)
\[\eta^2 = \frac{SS{between}}{SS_{total}}\]
\[\eta^2 = \frac{955.75}{2224.67}\]
\[\eta^2 = {0.43}\]
The results:
reject the null hypothesis, results are significant,
\(F({3}, {48}) = {12.05}, p < {0.05}, \eta^2 = {0.43}\)