Statistics: ANOVA: Comparing Multiple Groups
Using one-way ANOVA to compare three or more means
Statistics: ANOVA: Comparing Multiple Groups
Using one-way ANOVA to compare three or more means
Statistics - Grade 9-12
- 1
A teacher compares average quiz scores for students who used flashcards, practice tests, or group study. Identify the explanatory variable, the response variable, and the number of groups.
The explanatory variable is the category used to form the groups.
The explanatory variable is study method. The response variable is quiz score. There are three groups: flashcards, practice tests, and group study. - 2
Write the null and alternative hypotheses for a one-way ANOVA comparing mean battery life for four phone brands.
The null hypothesis is that all four phone brands have the same mean battery life. The alternative hypothesis is that at least one phone brand has a different mean battery life. - 3
Explain why ANOVA is usually preferred over doing many separate two-sample t-tests when comparing four group means.
Think about what happens when you repeat a test many times.
ANOVA is preferred because doing many separate t-tests increases the chance of making a Type I error. ANOVA tests all group means at once while keeping the overall significance level under better control. - 4
A one-way ANOVA compares 5 treatment groups with a total of 40 observations. Find the degrees of freedom between groups, within groups, and total.
The degrees of freedom between groups is 5 - 1 = 4. The degrees of freedom within groups is 40 - 5 = 35. The total degrees of freedom is 40 - 1 = 39. - 5
In an ANOVA table, the mean square between groups is 120 and the mean square within groups is 30. Calculate the F statistic.
Use F = MS between divided by MS within.
The F statistic is 120 divided by 30, which equals 4. The ANOVA test statistic is F = 4. - 6
The dot plots show three groups. Group A, Group B, and Group C have clearly different centers, but each group has little spread. Describe whether the ANOVA F statistic would likely be small or large, and explain why.
A larger F happens when between-group variation is large compared with within-group variation.
The F statistic would likely be large because the group centers are far apart compared with the small amount of variation inside each group. - 7
A one-way ANOVA gives p = 0.018 when testing three exercise plans at alpha = 0.05. State the decision and conclusion in context.
Because 0.018 is less than 0.05, we reject the null hypothesis. There is enough evidence to conclude that at least one exercise plan has a different mean result. - 8
A one-way ANOVA gives p = 0.22 when comparing mean plant heights under four fertilizers at alpha = 0.05. State the decision and conclusion in context.
A large p-value means the observed differences could reasonably happen by chance if the group means are equal.
Because 0.22 is greater than 0.05, we fail to reject the null hypothesis. There is not enough evidence to conclude that the fertilizers have different mean plant heights. - 9
List three main assumptions or conditions for using a one-way ANOVA.
The observations should be independent, the data in each group should be approximately normal or the sample sizes should be large enough, and the groups should have roughly equal variances. - 10
The boxplots for four classes show similar centers, but Class 4 has a much larger spread than the other classes. Which ANOVA condition might be a concern?
Compare the widths of the boxes and whiskers.
The equal variances condition might be a concern because Class 4 has much more spread than the other groups. - 11
A one-way ANOVA has 3 groups and 18 total observations. The sum of squares between groups is 48 and the sum of squares within groups is 72. Find df between, df within, MS between, MS within, and F.
The df between is 3 - 1 = 2 and the df within is 18 - 3 = 15. The MS between is 48 divided by 2, which is 24. The MS within is 72 divided by 15, which is 4.8. The F statistic is 24 divided by 4.8, which is 5. - 12
An ANOVA test finds a significant difference among five group means. Does this result identify exactly which group means are different from each other? Explain.
ANOVA is an overall test.
No. A significant ANOVA result shows that at least one mean is different, but it does not identify which means differ. Follow-up comparisons or a post hoc test are needed. - 13
A researcher compares average sleep hours for students in three grade levels: 9th, 10th, and 11th grade. The graph shows the three sample means are close together. Predict whether the F statistic is likely to be large or small if the spreads are also large.
The F statistic is likely to be small because the group means are close together and the variation within the groups is large. - 14
Decide whether each situation calls for a one-way ANOVA: comparing mean test scores for students taught by three different teaching methods, or comparing mean test scores by teaching method and gender at the same time.
One-way ANOVA uses one categorical explanatory variable.
Comparing mean test scores for three teaching methods calls for a one-way ANOVA because there is one explanatory variable. Comparing by teaching method and gender at the same time would not be a one-way ANOVA because it uses two explanatory variables. - 15
In an ANOVA study, the sum of squares between groups is 40 and the total sum of squares is 100. Calculate eta squared and interpret it.
Eta squared is 40 divided by 100, which equals 0.40. This means about 40% of the total variation in the response is explained by differences among the groups.