Answers to the odd-numbered exercises appear in Appendix B.

Review Your Knowledge

12.01 ____-way ANOVA examines the impact of two explanatory variables at the same time.

12.02 A 2 × 3 ANOVA could also be called a ____ × ____ ANOVA.

12.03 “Between subjects” means ____ samples and “____ subjects” means dependent samples.

12.04 Two-way ANOVA, three-way ANOVA, and four-way ANOVA are all examples of ____ ANOVA.

12.05 There is an advantage in performing one ____-way ANOVA over two ____-way ANOVAs.

12.06 If every level of one explanatory variable occurs with every level of the other explanatory variable, then the two explanatory variables are said to be ____.

12.07 A two-way ANOVA has two ____ effects and one ____ effect.

12.08 An interaction occurs when the effect of one explanatory variable on the dependent variable ____ on the level of the other explanatory variable.

12.09 If the lines in a graph are ____, then an interaction exists.

12.10 If each cell has the same sample size, the cell means in a row can be averaged to find the ____.

12.11 Row means and column means give information about ____ effects.

12.12 If the interaction effect is statistically significant, ____ interpret the main effects.

12.13 In a between-subjects, two-way ANOVA, the groups are ____ samples.

12.14 The random samples assumption says that the sample is a ____ from the ____ to which one wishes to generalize the results.

12.15 If cases in a cell influence each other’s scores on the dependent variable, then the ____ assumption is violated.

12.16 The ____ assumption is tested by looking at cell standard deviations.

12.17 In a two-way ANOVA, there are ____ sets of hypotheses.

12.18 The row’s null hypothesis states that the ____ for all levels of the row variable are equal.

12.19 The alternative hypothesis for the columns says that ____ one of the population column means differs from ____ one of the other population column means.

12.20 The alternative hypothesis for the interaction effect says that there is ____ for at least one cell.

12.21 Decision rules exist for the columns main effect, the rows main effect, and the ____.

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12.22 The decision rule for the interaction effect involves comparing F_Interaction to ____.

12.23 The number of degrees of freedom for the row effect depends on ____.

12.24 The denominator degrees of freedom for all the F ratios calculated for a between-subjects, two-way ANOVA is degrees of freedom ____.

12.25 Finding F_cv in Appendix Table 4 depends on knowing the numerator ____ and denominator ____ for the F ratio.

12.26 If F_Interaction < F_{cv Interaction}, then the null hypothesis is ____.

12.27 Total variance in between-subjects, two-way ANOVA is divided into two factors: ____ and ____.

12.28 In between-subjects, two-way ANOVA, between-group variability is divided into variability due to ____, ____, and ____.

12.29 SS is the abbreviation for a sum of ____.

12.30 To calculate a mean square, one divides a ____ by its ____.

12.31 MS_Interaction divided by MS_Within calculates ____.

12.32 When an ANOVA summary table for a between-subjects, two-way ANOVA is completed, there are ____ F ratios.

12.33 If a null hypothesis is rejected, the result is called ____.

12.34 If results, in APA format, are written as p > .05, the null hypothesis was ____.

12.35 p < .05 means that alpha was set at ____.

12.36 Only interpret a main effect if ____.

12.37 ____ is the effect size used for between-subjects, two-way ANOVA.

12.38 Eta squared is calculated for both ____ effects and for the ____.

12.39 An effect of ≈ ____% is considered a medium effect.

12.40 Eta squared can alert one to the risk of ____ error if the effect was not statistically significant.

12.41 Post-hoc tests should only be used when an effect is ____.

12.42 If the row main effect is statistically significant and there are only two rows, it is/is not necessary to do a post-hoc test for the row effect.

12.43 If two cells differ by exactly the HSD amount, then the difference is ____.

12.44 The HSD value for the interaction effect is used to compare ____ means.

Apply Your Knowledge

Select the right test for the scenario from among a single-sample z test; single-sample t test; independent-samples t test; paired-samples t test; between-subjects, one-way ANOVA; one-way, repeated-measures ANOVA; and between-subjects, two-way ANOVA.

12.45 A dentist randomly assigns people to use one of three different forms of dental hygiene: (1) dental floss, (2) wooden toothpicks, or (3) anti-plaque rinse. After six months he measures, in grams, how much plaque he scrapes off the teeth. What statistical test should he use to see if form of dental hygiene has an impact on the amount of plaque?

12.46 The American Psychological Association wanted to determine if there was a difference in post-graduation success between BA and BS degrees in psychology. It put together a random sample of students with each degree and found out what the annual salary was for their first job after graduation. What statistical test should be used to analyze these data for the effect of type of degree?

12.47 Male and female athletes exercised for an hour on a treadmill. During this time, half of each sex was assigned to drink water and half was assigned to drink a sport beverage. At the end of the hour, lactic acid levels were measured and mean levels were calculated. What statistical test should be used to see how sex and type of beverage affect lactic acid production?

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12.48 To study accommodation to the loss of an eye, a sensory psychologist obtained 10 volunteers who agreed to wear an eye patch over one eye for 4 weeks. To test visual ability, the psychologist used a batting test: a pitching machine lobbed 50 pitches to each participant and the psychologist counted how many were hit. This test was conducted (a) before the eye patch was worn, (b) immediately after the eye patch was first put on, (c) 1 week into the study, (d) 2 weeks into the study, (e) 3 weeks into the study, (f) on the last day wearing the eye patch, and (g) 1 week later. What test should be used to see if batting ability changed over the course of the study?

Calculating row and column means

12.49 There are 10 participants in each cell. Each cell is an independent sample. The design is fully crossed. The value in each cell is the cell mean. Calculate (a) row means and (b) column means for this matrix.

12.50 There are 17 participants in each cell. Each cell is an independent sample. The design is fully crossed. The value in each cell is the cell mean. Calculate (a) row means and (b) column means for this matrix.

Speculating about main effects and interactions

12.51 Given the cell means, row means, and column means below, (a) graph the results to examine if there is an interaction; (b) speculate about which effects would be statistically significant; (c) indicate which effects should be interpreted.

12.52 Given the cell means, row means, and column means below, (a) graph the results to determine if you think there is an interaction; (b) speculate about which effects would be statistically significant; (c) indicate which effects should be interpreted.

Checking the assumptions

12.53 Eighty volunteers who read about a study in a newspaper were randomly assigned to be in the experimental or control group. Within each group, participants were randomly assigned to one of four different conditions. Each participant then participated in the study individually. After this, each participant’s status on the interval-level psychological variable was measured. Below are the means (and standard deviations). (a) Evaluate all four assumptions and (b) decide if it is OK to proceed with the test.

12.54 A researcher was interested in the impact of different types of day care on children’s development. She obtained samples of children who (a) had been watched by a babysitter at home; (b) had attended small, home-based day care; or (c) had gone to a large day-care center. She also classified the children as having received paid care for (1) less than 40 hours a week or (2) 40 or more hours per week. There were eight children in each cell and no siblings were included. She measured each child’s adjustment level, on an interval scale, in the first grade. Below are the means (and standard deviations). (a) Evaluate all four assumptions and (b) decide if it is OK to proceed with the test.

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Listing hypotheses

12.55 List all the hypotheses for the ANOVA described in Exercise 12.53 for (a) rows, (b) columns, and (c) interaction.

12.56 List all the hypotheses for the ANOVA described in Exercise 12.54 for (a) rows, (b) columns, and (c) interaction.

Calculating degrees of freedom

12.57 If R = 3, C = 4, and n = 8, calculate the degrees of freedom for the following effects: (a) rows, (b) columns, (c) interaction, (d) within groups, (e) between groups, and (f) total.

12.58 If R = 2, C = 3, and n = 11, calculate the degrees of freedom for the following effects: (a) rows, (b) columns, (c) interaction, (d) within groups, (e) between groups, and (f) total.

Finding F_cv

12.59 If α = .05, df_Rows = 2, df_Columns = 2, df_Interaction = 4, and df_Within = 36, find (a) F_{cv Rows}, (b) F_{cv Columns}, and (c) F_{cv Interaction}.

12.60 If α = .05, df_Rows = 3, df_Columns = 1, df_Interaction = 3, and df_Within = 40, find (a) F_{cv Rows}, (b) F_{cv Columns}, and (c) F_{cv Interaction}.

Writing the decision rule

12.61 If F_{cv Interaction} is 2.668, what is the decision rule for the interaction effect?

12.62 If F_{cv Rows} is 3.295, what is the decision rule for the rows effect?

Calculating mean squares

12.63 If SS_Within = 168.48 and df_Within = 40, calculate MS_Within.

12.64 If SS_Interaction = 0.60 and df_Interaction = 4, calculate MS_Interaction.

Calculating F ratios

12.65 If MS_Rows = 17.30 and MS_Within = 3.33, what is F_Rows?

12.66 If MS_Columns = 66.54 and MS_Within = 78.88, what is F_Columns?

Completing a summary table

12.67 Given R = 2, C = 2, n = 5, SS_Between = 61.00, SS_Rows = 15.00, SS_Columns = 12.00, SS_Interaction = 34.00, SS_Within = 120.00, and SS_Total = 181.00, complete the ANOVA summary table for a between-subjects, two-way ANOVA.

12.68 Given R = 2, C = 4, n = 12, SS_Between = 225.00, SS_Rows = 78.00, SS_Columns = 23.00, SS_Interaction = 124.00, SS_Within = 350.00, and SS_Total = 575.00, complete the ANOVA summary table for a between-subjects, two-way ANOVA.

Applying the decision rule

12.69 If F_cv = 3.443 and F = 17.55, was the null hypothesis rejected?

12.70 If F_cv = 3.191 and F = 2.86, was the null hypothesis rejected?

Writing results in APA format

12.71 If F_Interaction = 3.70, df_Interaction = 2, df_Within = 66, and F_{cv Interaction} = 3.138, write the results in APA format. Use α = .05.

12.72 If F_Rows = 2.87, df_Rows = 2, df_Within = 171, and F_{cv Rows} = 3.053, write the results in APA format. Use α = .05.

12.73 If F_Columns = 2.45, df_Columns = 3, and df_Within = 168, write the results in APA format. Use α = .05.

12.74 If F_Columns = 4.00, df_Columns = 1, and df_Within = 36, write the results in APA format. Use α = .05.

12.75 Given the ANOVA summary table below, write the results in APA format for (a) the rows effect, (b) the columns effect, and (c) the interaction effect. Use α = .05.

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12.76 Given the ANOVA summary table below, write the results in APA format for (a) the rows effect, (b) the columns effect, and (c) the interaction effect. Use α = .05.

Deciding which effects to interpret

12.77 If the null hypothesis is rejected for all three effects (rows, columns, and interaction), which effects should be interpreted?

12.78 If the null hypothesis is rejected for the rows effect and the columns effect but not for the interaction effect, which effects should be interpreted?

12.79 If the null hypothesis is rejected for the interaction effect but not for the rows effect or the columns effect which effects should be interpreted?

12.80 If the null hypothesis is rejected for the rows effect and the interaction effect but not for the columns effect, which effects should be interpreted?

Calculating eta squared

12.81 If SS_Rows = 3.78 and SS_Total = 47.83, (a) calculate η²_Rows and (b) classify the effect as small, medium, or large.

12.82 If SS_Columns = 17.91 and SS_Total = 783.54, (a) calculate η²_Columns and (b) classify the effect as small, medium, or large.

12.83 Given the ANOVA summary table below, calculate η² for (a) the rows effect, (b) the columns effect, and (c) the interaction effect.

12.84 Given the ANOVA summary table below, calculate η² for (a) the rows effect, (b) the columns effect, and (c) the interaction effect.

Finding q

12.85 Given R = 3, C = 2, and n = 8, find (a) q_Rows, (b) q_Columns, and (c) q_Cells.

12.86 Given R = 3, C = 3, and n = 11, find (a) q_Rows, (b) q_Columns, and (c) q_Cells.

Calculating HSD

12.87 If q_Rows = 3.49, MS_Within = 4.56, and n_Rows = 12, what is HSD_Rows?

12.88 If q_Cells = 2.96, MS_Within = 12.75, and n_Cells = 4, what is HSD_Cells?

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12.89 If α = .05, for which effects should an HSD value be calculated?

12.90 If α = .05, for which effects should an HSD value be calculated?

Interpreting HSD

12.91 If M_{Row 1} = 117.66, M_{Row 2} = 113.63, M_{Row 3} = 128.91, and HSD_Rows = 5.89, determine (a) which rows have a statistically significant difference and (b) the direction of the differences.

12.92 If M_{Cell 1} = 55.54, M_{Cell 2} = 48.34, M_{Cell 3} = 36.44, M_{Cell 4} = 59.40, and HSD_Cells = 7.83, determine (a) which cells have a statistically significant difference and (b) the direction of the differences.

Interpreting results

12.93 A consumer psychologist classified shoppers at a grocery store as (a) being males or females, and (b) shopping with or without children. These two variables were crossed and he took a random sample of five shoppers from each of the four cells. Then he gave them the Enjoyment of Shopping Experience Scale (ESES) to be completed for the current shopping experience. The ESES is an interval-level measure, with a mean of 50 indicating neutral feelings about shopping. Scores can range from 20 to 80, with scores above 50 an indication of enjoying shopping; scores below 50 indicate that shopping is not enjoyable. Here are the means:

No nonrobust assumptions were violated and a between-subjects, two-way ANOVA was completed. Here is the ANOVA summary table:

The researcher calculated η²_Rows = 0.50%, η²_Columns = 0.10%, η²_Interaction = 71.57%, and HSD_Cells = 8.27. Complete a four-point interpretation for the results. (Hint: Graphing the results will help.)

12.94 A cognitive psychologist decided to investigate whether two beliefs about healthy living had any real impact on performance in college. She took 40, first-semester college student volunteers and randomly assigned half to eat breakfast every day and the other half to skip breakfast every day. This was crossed with another variable. Half the students were randomly assigned to sleep at least 8 hours a night and half were randomly assigned to sleep less than 8 hours a night. At the end of the semester, she recorded each participant’s GPA. The results are shown in this table:

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No nonrobust assumptions for a between-subjects, two-way ANOVA were violated and the ANOVA was completed. Here is the summary table:

The eta squared values for rows, columns, and interaction are, respectively, 0.13%, 20.34%, and 0.91%. HSD_Cells is 0.50. Interpret the results. (Hint: Graphing the results will help.)

12.95 A developmental psychologist investigated the influence of two crossed variables—exposure to televised violence and type of parental discipline—on teens’ acceptance of violence. He obtained a random sample of seniors at the local high school and classified them on two dimensions: (1) whether or not their parents had restricted their television viewing and (2) whether their parents used positive reinforcement or punishment. He took a random sample of six teens from each of the four samples (no siblings were included) and his dependent variable was each teen’s score on the interval-level Approval of Violence Scale (AVS). Scores on the AVS range from 0 to 20, with higher scores indicating a greater acceptance of violence as a way to solve disagreements. The cell means (and standard deviations) are shown below. SS_Rows was calculated to be 210.04, SS_Columns as 222.04, SS_Interaction as 15.04, and SS_Within as 204.83.

12.96 An industrial organizational psychologist investigated the crossed effects of two variables—being a member of a team sport in high school and extroversion level—on how well professors are liked by their colleagues. The dependent variable was the interval-level Colleague Collegiality Scale (CCS). Scores on the CCS range from 0 to 24; higher scores indicate greater liking by colleagues. From a national and representative sample of college professors, the psychologist randomly selected professors until there were eight in each cell. The table below shows the cell means (and standard deviations). SS_Rows was 36.13, SS_Columns = 1.13, SS_Interaction = 36.13, and SS_Within = 217.50.

Expand Your Knowledge

12.97 Each cell in the table contains seven cases. (a) Given the cell, row, and column means below, calculate the missing cell means. If it can’t be done, say so. (b) Calculate the mean for all 28 cases. If it can’t be done, say so.

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12.98 Each cell in this table has the same number of cases, but that number is unknown. (a) Given the cell, row, and column means below, calculate the missing cell means. If it can’t be done, say so. (b) Calculate the mean for all the cases. If it can’t be done, say so.

12.99 Each cell in the table has five cases. (a) Given the cell, row, and column means below, calculate the missing cell means. If it can’t be done, say so. (b) Calculate the mean for all 30 cases. If it can’t be done, say so.

12.100 The means of five independent samples are being compared. Which ANOVA is being used?

12.101 The means of six independent samples are being compared. Which ANOVA is being used?

12.102 Given R = 3, C = 3, n = 12, and the ANOVA summary table below, calculate HSD values for the effects as necessary and appropriate.

12.103 If SS_Rows = 17.50, SS_Columns = 13.75, SS_Interaction = 22.25, and SS_Within = 44.75, calculate (a) SS_Between and (b) SS_Total.

12.104 If SS_Rows = 123.60, SS_Columns = 80.30, SS_Interaction = 20.80, and SS_Within = 168.20, calculate (a) SS_Between and (b) SS_Total.