ANOVA Practice Worksheet: One-Way and Two-Way

Understanding ANOVA: The Basics

Analysis of Variance (ANOVA) is a statistical test used to compare the means of two or more independent groups. It determines if there is a statistically significant difference between the means by analyzing the variance (spread) within and between the groups.

Factors (Independent Variables): The categorical variables being tested (e.g., 'Drug Type' or 'Gender').
Levels: The different groups within a factor (e.g., for the 'Drug Type' factor, the levels might be 'Drug A', 'Drug B', and 'Placebo').
F-Statistic: The test statistic calculated as the ratio of the variance between the groups to the variance within the groups. A larger F-statistic suggests a greater difference between the group means.
Null Hypothesis (H0): Always states that there is no significant difference between the group means. In other words, all group means are equal.
Assumption of Normality: The population distribution of the dependent variable in each group must be approximately normal.
Assumption of Homogeneity of Variance: The variance of the dependent variable must be approximately equal across all groups.

One-Way vs. Two-Way ANOVA

One-Way ANOVA is used when you have one categorical independent variable (one factor) and one quantitative dependent variable (e.g., testing the effect of three different teaching methods on a single test score).

Two-Way ANOVA is used when you have two categorical independent variables (two factors) and one quantitative dependent variable (e.g., testing the effect of both teaching method AND student gender on a single test score). It allows you to test for the main effect of each factor and, crucially, their interaction effect.

🧪 One-Way ANOVA Practice Questions

These questions involve a single independent variable with two or more levels.

Question 1: Null Hypothesis A psychologist wants to test if five different colors of background music (Classical, Jazz, Pop, Rock, None) affect students' memory scores. This is a One-Way ANOVA. State the null hypothesis in plain language.

Hint: The null hypothesis always assumes no difference between the population means of the groups being compared.

Question 2: Factor and Levels A quality control manager compares the defect rates of products made on three different shifts (Morning, Afternoon, Night). Identify the factor and the number of its levels.

Hint: The factor is the independent variable, and the levels are the specific groups or categories within it.

Question 3: F-Statistic Logic If a One-Way ANOVA yields an F-statistic very close to 1.0, what does this suggest about the difference between the group means?

Hint: The F-value is the ratio of variance between groups to variance within groups. A value near 1 means the variances are similar, so the group means likely do not differ significantly.

Question 4: Assumption Check (Normality) Before running a One-Way ANOVA on customer satisfaction scores from four different stores, you check the assumption of normality. Which specific scores need to be approximately normally distributed?

Hint: The dependent variable's distribution within each group should be approximately normal.

Question 5: Post-Hoc Test Necessity A One-Way ANOVA comparing four fertilizer types on plant growth shows a significant result (p-value less than 0.05). What is the necessary next step to determine which specific fertilizer types differ from each other?

Hint: ANOVA tells you if there is any overall difference, but you need a post-hoc pairwise comparison test to identify specific group differences.

Question 6: Decision Making The critical F-value at significance level 0.05 is 3.24. Your calculated F-statistic is 4.51. Should you reject or fail to reject the null hypothesis?

Hint: If the calculated F-statistic is greater than the critical F-value, reject the null hypothesis.

Question 7: Dependent Variable A study uses One-Way ANOVA to compare the average response time (in milliseconds) of participants who received a low, medium, or high dose of a drug. What is the dependent variable?

Hint: The dependent variable is the outcome being measured and compared across groups.

Question 8: Total Sum of Squares In the context of ANOVA, the Total Sum of Squares (SSTotal) is partitioned into which two main components in a One-Way analysis?

Hint: Variation explained by the factor and variation due to random error (within groups).

Question 9: Homogeneity of Variance Test Which statistical test is commonly used to check the assumption of homogeneity of variance before performing a One-Way ANOVA?

Hint: Levene's test is commonly used to assess equality of variances across groups.

Question 10: Equal Sample Sizes Why is having equal sample sizes (a balanced design) generally preferred in One-Way ANOVA, especially when the homogeneity of variance assumption might be violated?

Hint: Equal sample sizes make the test more robust against violations of assumptions.

Calculate One-Way ANOVA

📊 Two-Way ANOVA Practice Questions

These questions involve two independent variables (factors) and their potential interaction effect.

Question 11: Main Effects vs. Interaction A researcher studies the effect of 'Diet Type' (Factor A: Low-Carb, High-Carb) and 'Exercise Frequency' (Factor B: Daily, Weekly) on weight loss. Name the three separate effects that a Two-Way ANOVA will test for.

Hint: Two main effects (Diet and Exercise) and one interaction effect (Diet × Exercise).

Question 12: Interaction Interpretation In the study from Question 11, the interaction term (Diet × Exercise) is found to be statistically significant. What does this significance imply about the relationship between diet and exercise on weight loss?

Hint: A significant interaction means the effect of one factor depends on the level of the other factor.

Question 13: Null Hypothesis for Interaction State the null hypothesis for the interaction effect in a Two-Way ANOVA testing 'Gender' and 'Medication Dosage' on recovery time.

Hint: The null hypothesis assumes that the effects of Gender and Dosage are additive and independent (no interaction).

Question 14: Factors and Levels (2 by 3 Design) A Two-Way ANOVA is described as a 2 by 3 design. If the two factors are 'Location' and 'Time of Day', how many levels does each factor have, and how many total experimental groups are there?

Hint: Factor 1 has 2 levels; Factor 2 has 3 levels; total groups = 2 × 3 = 6.

Question 15: Graphical Interpretation If you create a line plot of the cell means for a Two-Way ANOVA and the lines are perfectly parallel, what does this visually suggest about the interaction effect?

Hint: Parallel lines suggest no interaction; non-parallel lines suggest an interaction.

Question 16: Two Main Effects Not Significant A Two-Way ANOVA shows that the main effect of Factor A is not significant, the main effect of Factor B is not significant, but the interaction effect (A × B) is significant. Is this possible, and what is the primary conclusion?

Hint: Yes, it is possible. The significant interaction suggests that the effect of one factor depends on the other, even if main effects are not significant.

Question 17: Assumptions (Independence) Which crucial assumption of Two-Way ANOVA would be violated if the same group of participants was measured for their performance under four different combinations of the two factors?

Hint: Independence assumption is violated because measurements are not independent.

Question 18: Sum of Squares Partitioning In a Two-Way ANOVA, the Total Sum of Squares (SSTotal) is partitioned into which four key components?

Hint: Main effect A, main effect B, interaction effect (A × B), and residual error.

Question 19: The Need for Replication Why must a Two-Way ANOVA have more than one observation per cell (replication) to be able to calculate an F-statistic for the interaction effect?

Hint: Replication allows estimation of variability within cells (error term), needed for testing interaction.

Question 20: Reporting Main Effects with Interaction If the interaction effect in a Two-Way ANOVA is statistically significant, should you still prioritize the interpretation of the two main effects? Explain briefly.

Hint: Significant interaction typically means main effects should be interpreted cautiously or with context.

Calculate Two-Way ANOVA