Chapter 13 Exercises

Clarifying the Concepts

Question 13.1

What are the four assumptions for a within-groups ANOVA?

Question 13.2

What are order effects?

Question 13.3

Explain the source of variability called “subjects.”

Question 13.4

What is the advantage of the design of the within-groups ANOVA over that of the between-groups ANOVA?

Question 13.5

What is counterbalancing?

Question 13.6

Why is it appropriate to counterbalance when using a within-groups design?

Question 13.7

How do we calculate the sum of squares for subjects?

Question 13.8

How is the calculation of dfwithin different in a between-groups ANOVA from the calculation in a within-groups ANOVA?

Question 13.9

How could we turn a between-groups study into a within-groups study?

Question 13.10

What are some situations in which it might be impossible—or not make sense—to turn a between-groups study into a within-groups study?

Question 13.11

How is the calculation of effect size different for a one-way between-groups ANOVA versus a one-way within-groups ANOVA?

Question 13.12

What is a matched-groups research design?

Calculating the Statistics

Question 13.13

For the following data, assuming a within-groups design, determine:

Person
1 2 3 4
Level 1 of the independent variable   7 16   3   9
Level 2 of the independent variables 15 18 18 13
Level 3 of the independent variable 22 28 26 29
  1. dfbetween = Ngroups − 1

  2. dfsubjects = n − 1

  3. dfwithin = (dfbetween)(dfsubjects)

  4. dftotal = dfbetween + dfsubjects + dfwithin, or dftotal = Ntotal − 1

  5. SStotal = Σ(XGM)2

  6. SSbetween = Σ(MGM)2

  7. SSsubjects = Σ(MparticipantGM)2

  8. SSwithin = SStotalSSbetweenSSsubjects

  9. The rest of the ANOVA source table for these data

  10. The effect size

  11. The Tukey HSD statistic for the comparisons between level 1 and level 3

Question 13.14

For the following data, assuming a within-groups design, determine:

Person
1 2 3 4 5 6
Level 1 5 6 3 4 2 5
Level 2 6 8 4 7 3 7
Level 3 4 5 2 4 0 4
  1. dfbetween = Ngroups − 1

  2. dfsubjects = n − 1

  3. dfwithin = (dfbetween)(dfsubjects)

  4. dftotal = dfbetween + dfsubjects + dfwithin, or dftotal = Ntotal − 1

  5. SStotal = Σ(XGM)2

  6. SSbetween = Σ(MGM)2

  7. SSsubjects = Σ(MparticipantGM)2

  8. SSwithin = SStotalSSbetweenSSsubjects

  9. The rest of the ANOVA source table for these data

  10. The critical F value and your decision about the null hypothesis.

  11. If appropriate, the Tukey HSD statistic for all possible mean comparisons

  12. The critical q value; then, make a decision for each comparison in part (k)

  13. The effect size

Question 13.15

For the following incomplete source table below for a one-way within-groups ANOVA:

Source SS df MS F
Between 941.102 2
Subjects 3807.322
Within 20
Total 5674.502
  1. Complete the missing information.

  2. Calculate R2.

Question 13.16

Assume that a researcher had 14 individuals participate in all three conditions of her experiment. Use this information to complete the source table below.

Source SS df MS F
Between 60
Subjects
Within 50
Total 136

Applying the Concepts

Question 13.17

Fear of dogs and one-way within-groups ANOVA: Imagine a researcher wanted to assess people’s fear of dogs as a function of the size of the dog. He assessed fear among people who indicated they were afraid of dogs, using a 30-point scale from 0 (no fear) to 30 (extreme fear). The researcher exposed each participant to three different dogs, a small dog weighing 20 pounds, a medium-sized dog weighing 55 pounds, and a large dog weighing 110 pounds, and assessed the fear level after each exposure. Here are some hypothetical data; note that these are the data from Exercise 13.13, on which you have already calculated numerous statistics:

Person
1 2 3 4
Small dog   7 16   3   9
Medium dog 15 18 18 13
Large dog 22 28 26 29
  1. State the null and research hypotheses.

  2. Determine whether the assumptions of random selection and order effects were met.

  3. In Exercise 13.13, you calculated the effect size for these data. What does this statistic tell us about the effect of size of dog on fear levels?

  4. In Exercise 13.13, you calculated a Tukey HSD test for these data. What can you conclude about the effect of size of dog on fear levels based on this statistic?

Question 13.18

Chewing-gum commercials and one-way within-groups ANOVA: Commercials for chewing gum make claims about how long the flavor will last. In fact, some commercials claim that the flavor lasts too long, affecting sales and profit. Let’s put these claims to a test. Imagine a student decides to compare four different gums using five participants. Each randomly selected participant was asked to chew a different piece of gum each day for 4 days, such that at the end of the 4 days, each participant had chewed all four types of gum. The order of the gums was randomly determined for each participant. After 2 hours of chewing, participants recorded the intensity of flavor from 1 (not intense) to 9 (very intense). Here are some hypothetical data:

Person
1 2 3 4 5
Gum 1 4 6 3 4 4
Gum 2 8 6 9 9 8
Gum 3 5 6 7 4 5
Gum 4 2 2 3 2 1

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  1. Conduct all six steps of the hypothesis test.

  2. Are any additional tests warranted? Explain your answer.

Question 13.19

Pessimism and one-way within-groups ANOVA: Researchers Busseri, Choma, and Sadava (2009) asked a sample of individuals who scored as pessimists on a measure of life orientation about past, present, and projected future satisfaction with their lives. Higher scores on the life-satisfaction measure indicate higher satisfaction. The data below reproduce the pattern of means that the researchers observed in self-reported life satisfaction of the sample of pessimists for the three time points. Do pessimists predict a gloomy future for themselves?

Person
1 2 3 4 5
Past 18 17.5 19 16 20
Present 18.5 19.5 20 17 18
Future 22 24 20 23.5 21
  1. Perform steps 5 and 6 of hypothesis testing. Be sure to complete the source table when calculating the F ratio for step 5.

  2. If appropriate, calculate the Tukey HSD for all possible mean comparisons. Find the critical value of q and make a decision regarding the null hypothesis for each of the mean comparisons.

  3. Calculate the R2 measure of effect size for this ANOVA.

Question 13.20

Pessimism and one-way within-groups ANOVA: Exercise 13.19 describes a study conducted by Busseri and colleagues (2009) using a group of pessimists. These researchers asked the same question of a group of optimists: Optimists rated their past, present, and projected future satisfaction with their lives. Higher scores on the life-satisfaction measure indicate higher satisfaction. The data below reproduce the pattern of means that the researchers observed in self-reported life satisfaction of the sample of optimists for the three time points. Do optimists see a rosy future ahead?

Person
1 2 3 4 5
Past 22 23 25 24 26
Present 25 26 27 28 29
Future 24 27 26 28 29
  1. Perform steps 5 and 6 of hypothesis testing. Be sure to complete the source table when calculating the F ratio for step 5.

  2. If appropriate, calculate the Tukey HSD for all possible mean comparisons. Find the critical value of q and make a decision regarding the null hypothesis for each of the mean comparisons.

  3. Calculate the R2 measure of effect size for this ANOVA.

Question 13.21

Wagging tails and one-way within-groups ANOVA: How does a dog’s tail wag in response to seeing different people and other pets? Quaranta, Siniscalchi, and Vallortigara (2007) investigated the amplitude and direction of a dog’s tail wagging in response to seeing its owner, an unfamiliar cat, and an unfamiliar dog. The fictional data below are measures of amplitude. These data reproduce the pattern of results in the study, averaging leftward tail wags and rightward tail wags. Use these data to construct the source table for a one-way within-groups ANOVA.

Dog Participant Owner Cat Other Dog
1 69 28 45
2 72 32 43
3 65 30 47
4 75 29 45
5 70 31 44

Question 13.22

Memory, post hoc tests, and effect size: Luo, Hendriks, and Craik (2007) were interested in whether people might better remember lists of words if the lists were paired with either pictures or sound effects. They asked participants to memorize lists of words under three different learning conditions. In the first condition, participants just saw a list of nouns that they were to remember (word-alone condition). In the second condition, the words were also accompanied by a picture of the object (picture condition). In the third condition, the words were accompanied by a sound effect matching the object (sound effect condition). The researchers measured the proportion of words participants got correct in a later recognition test. Fictional data from four participants produce results similar to those of the original study. The average proportion of words recognized was M = 0.54 in the word-alone condition, M = 0.69 in the picture condition, and M = 0.838 in the sound effect condition. The source table below depicts the results of the ANOVA on the data from the four fictional participants.

Source SS df MS F
Between 0.177   2 0.089 8.900
Subjects 0.002   3 0.001 0.100
Within 0.059   6 0.010
Total 0.238 11
  1. Is it appropriate to perform post hoc comparisons on the data? Why or why not?

  2. Use the information provided in the ANOVA table to calculate R2. Interpret the effect size using Cohen’s conventions. State what this R2 means in terms of the independent and dependent variables used in this study.

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Question 13.23

Wagging tails, hypothesis-test decision making, and post hoc tests: Assume that we recruited a different sample of five dogs and attempted to replicate the Quaranta and colleagues (2007) study described in Exercise 13.21. The source table for our fictional replication appears below. Find the critical F value and make a decision regarding the null hypothesis. Based on this decision, is it appropriate to conduct post hoc comparisons? Why or why not?

Source SS df MS F
Between   58.133   2 29.067 0.066
Subjects 642.267   4 160.567 0.364
Within 532.533   8 441.567
Total 4232.933 14

Question 13.24

Pilots’ mental efforts and a one-way within-groups ANOVA: Researchers examined the amount of mental effort that participants felt they were expending on a cognitively complex task, piloting an unmanned air vehicle (UAV) (Ayaz, Shewokis, Bunce, Izzetoglu, Willems, & Onaral, 2012). The researchers used the Task Load Index (TLX), a measure that assesses participants’ perception of their mental effort following a series of approach and landing tasks in simulated UAV tasks. They wondered whether expertise would have an effect on perceptions of mental effort. In the results section, the researchers reported the results of their analyses, a series of one-way repeated-measures ANOVA. “The results indicated a significant main effect of practice level (beginner/intermediate/advanced conditions) for mental demand (F (2,8) = 17.87, p < 0.01, η2 = 0.817), effort (F (2,8) = 16.32, p < 0.01, η2 = 0.803), and frustration (F (2,8) = 8.60, p < 0.01, η2 = 0.682).” They went on to explain that mental demand, effort, and frustration all tended to decrease with expertise.

  1. What is the independent variable in this study?

  2. What are the dependent variables in this study?

  3. Explain why the researchers were able to use a one-way within-groups ANOVA in this situation.

  4. η2 is roughly equivalent to R2. How large are each of these effects, based on Cohen’s conventions?

  5. The researchers drew a specific conclusion beyond that there was some difference, on average, in the dependent variables, depending on the particular levels of the independent variable. What additional test were they likely to have conducted? Explain your answer.

Putting It All Together

Question 13.25

Eye glare, football, and one-way within-groups ANOVA: Does the black grease beneath football players’ eyes really reduce glare or does it just make them look intimidating? In a variation of a study actually conducted at Yale University, 46 participants placed one of three substances below their eyes: black grease, black antiglare stickers, or petroleum jelly. The researchers assessed eye glare using a contrast chart for each participant that gives a value on a scale measure. Every participant was assessed with each of the three substances, one at a time. Black grease led to a reduction in glare compared with the two other conditions, antiglare stickers or petroleum jelly (DeBroff & Pahk, 2003).

Person Black Grease Antiglare Stickers Petroleum Jelly
1 19.8 17.1 15.9
2 18.2 17.2 16.3
3 19.2 18.0 16.2
4 18.7 17.9 17.0
  1. What is the independent variable? What are its levels?

  2. What is the dependent variable?

  3. What kind of ANOVA is this?

  4. What is the first assumption for ANOVA? Is it likely that the researchers met this assumption? Explain your answer.

  5. What is the second assumption for ANOVA? How could the researchers check to see if they had met this assumption? Be specific.

  6. What is the third assumption for ANOVA? How could the researchers check to see if they had met this assumption? Be specific.

  7. What is the fourth assumption, specific to the within-groups ANOVA? What would the researchers need to do to ensure that they meet this assumption?

  8. Perform steps 5 and 6 of hypothesis testing. Be sure to complete the source table when calculating the F ratio for step 5.

  9. If appropriate, calculate the Tukey HSD for all possible mean comparisons. Find the critical value of q and make a decision regarding the null hypothesis for each of the mean comparisons.

  10. Calculate the R2 measure of effect size for this ANOVA.

  11. How could this study be conducted using a between-groups design?

  12. How could this study be conducted using a matched-groups design?

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