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8. For the preceding problem you should find that there are significant differences among the three treatments. The primary reason for the significance is that the mean for treatment I is substantially smaller than the means for the other two treatments. To create the following data, we started with the values from problem 7 and added 3 points to each score in treatment I. Recall that adding a constant causes the mean to change but has no influence on the variability of the sample. In the resulting data, the mean differences are much smaller than those in problem 7. I II III_____________ n = 6 n = 6 n = 6 M = 4 M = 5 M = 6 N = 18 T = 24 T = 30 T = 36 G = 90 SS = 30 SS = 35 SS = 40 ΣX²_=567________ a. Before you begin any calculations, predict how the change in the data should influence the outcome of the analysis. That is, how will the F-ratio and the value of η² for these data compare with the values obtained in problem 7? b. Use an ANOVA with α = .05 to determine whether there are any significant differences among the three treatment means. (Does your answer agree with your prediction in part a?) c. Calculate 2 to measure the effect size for this study. (Does your answer agree with your prediction in part a?) 10. For the preceding problem you should find that there are significant differences among the three treatments. One reason for the significance is that the sample variances are relatively small. To create the following data, we started with the values from problem 9 and increased the variability (the SS values) within each sample. I II III n =5 n = 5 n = 5 M = 2 M = 5 M = 8 N = 15 T = 10 T = 25 T = 40 G = 75 SS = 64 SS =80 SS =96 ΣX² =705 a. Calculate the sample variance for each of the three samples. Describe how these sample variances compare with those from problem 9. b. Predict how the increase in sample variance should influence the outcome of the analysis. That is, how will the F-ratio for these data compare with the value obtained in problem 9? c. Use an ANOVA with α = .05 to determine whether there are any significant differences among the three treatment means. (Does your answer agree with your prediction in part b?) 12. A researcher reports an F-ratio with df = 3, 36 from an independent-measures research study. a. How many treatment conditions were compared in the study? b. What was the total number of participants in the study? 14. There is some evidence that high school students justify cheating in class on the basis of poor teacher skills or low levels of teacher caring (Murdock, Miller, and Kohlhardt, 2004). Students appear to rationalize their illicit behavior based on perceptions of how their teachers view cheating. Poor teachers are thought not to know or care whether students cheat, so cheating in their classes is okay. Good teachers, on the other hand, do care and are alert to cheating, so students tend not to cheat in their classes. Following are hypothetical data similar to the actual research results. The scores represent judgments of the acceptability of cheating for the students in each sample. Poor Average Good Teacher Teacher Teacher n = 6 n = 8 n = 10 N = 24 M = 6 M = 2 M =2 G = 72 SS = 30 SS = 33 SS = 42 ΣX² = 393 a. Use an ANOVA with α = .05 to determine whether there are significant differences in student judgments depending on how they see their teachers. b. Calculate 2 to measure the effect size for this study. c. Write a sentence demonstrating how a research report would present the results of the hypothesis test and the measure of effect size. 16. A pharmaceutical company has developed a drug that is expected to reduce hunger. To test the drug, two samples of rats are selected with n = 20 in each sample. The rats in the first sample receive the drug every day and those in the second sample are given a placebo. The dependent variable is the amount of food eaten by each rat over a 1-month period. An ANOVA is used to evaluate the difference between the two sample means and the results are reported in the following summary table. Fill in all missing values in the table. (Hint: Start with the df column.) Source SS df MS Between treatments ____ ____ 20 F = 4.00 Within treatments _____ _____ ____ Total _____ ______ 18. The following data were obtained from an independent-measures research study comparing three treatment conditions. Use an ANOVA with _ = .05 to determine whether there are any significant mean differences among the treatments. ________Treatment_________________ I II III____________ 2 5 7 N = 14 5 2 3 G = 42 0 1 6 ΣX² = 182 1 2 4 2 2___________________________________ T = 12 T =10 T = 20 SS = 14 SS = 9 SS = 10___________ 20. The following data represent the results from an independent-measures study comparing two treatment conditions. a. Use an independent-measures t test with = = .05 to determine whether there is a significant mean difference between the two treatments. b. Use an ANOVA with α = .05 to determine whether there is a significant mean difference between the two treatments. Treatment____ I II______________ 8 2 N = 10 7 3 G = 50 6 3 ΣX² = 306 5 5 ___9 2_________________ M = 7 M = 3 T = 35 T = 15 SS = 10 SS = 6_______________________ 22. There is some research indicating that college students who use Facebook while studying tend to have lower grades than non-users (Kirschner & Karpinski, 2010). A representative study surveys students to determine the amount of Facebook use during the time they are studying or doing homework. Based on the amount of time spent on Facebook, students are classified into three groups and their grade point averages are recorded. The following data show the typical pattern of results. Facebook Use While Studying_______ Non-User Rarely Use Regularly Use 3.70 3.51 3.02 3.45 3.42 2.84 2.98 3.81 3.42 3.94 3.15 3.10 3.82 3.64 2.74 3.68 3.20 3.22 3.90 2.95 2.58 4.00 3.55 3.07 3.75 3.92 3.31 3.88 3.45 2.80 A, Use an ANOVA with α = .05 to determine whether there are significant mean differences among the three groups. b. Compute η² to measure the size of the effect. c. Write a sentence demonstrating how the result from the hypothesis test and the measure of effect size would appear in a research report. Chapter 13 2. The repeated-measures ANOVA can be viewed as a two-stage process. What is the purpose of the second stage? 4. A researcher conducts a repeated-measures experiment using a sample of n = 8 subjects to evaluate the differences among four treatment conditions. If the results are examined with an ANOVA, what are the df values for the F-ratio? 6. A published report of a repeated-measures research study includes the following description of the statistical analysis. “The results show significant differences among the treatment conditions, F(2, 20) = 6.10, p <.01.” a, How many treatment conditions were compared in the study? b. How many individuals participated in the study? 8. The following data were obtained from a repeated measures study comparing two treatment conditions. Use a repeated-measures ANOVA with _ .05 to determine whether there are significant mean differences between the two treatments. Treatments___________________ Person Person I II Totals A 3 5 P = 8 B 5 9 P = 14 N = 16 C 1 5 P = 6 G = 80 D 1 7 P = 8 ΣX² = 500 E 5 9 P = 14 F 3 7 P = 10 G 2 6 P = 8 H 4 8 P = 12______________ M = 3 M = 7 T = 24 T = 56 SS = 18 SS = 18 Data from problem 9 Treatments Person Person I II III Totals A 1 1 4 P = 6 B 3 4 8 P = 15 N = 15 C 0 2 7 P = 9 G = 45 D 0 0 6 P = 6 ΣX² = 231 E 1 3 5 P = 9_____________________ M = 1 M = 2 M = 6 T = 5 T = 10 T = 30 SS = 6 SS = 10 SS = 10 10. For the data in problem 9, you will find above a. Compute SStotal and SSbetween treatments. b. Eliminate the mean differences between treatments by adding 2 points to each score in treatment I, adding 1 point to each score in treatment II, and subtracting 3 points from each score in treatment III. (All three treatments should end up with M = 3 and T = 15.) c. Calculate SStotal for the modified scores. (Caution: You first must find the new value for ΣX².) d. Because the treatment effects were eliminated in part b, you should find that SStotal for the modified scores is smaller than SStotal for the original scores. The difference between the two SS values should be exactly equal to the value of SSbetween treatments for the original scores. 11. The following data were obtained from a repeated measures study comparing three treatment conditions. Treatment__________________________ Subject I II III P__________________ A 6 8 10 24 G = 48 B 5 5 5 15 ΣX² = 294 C 1 2 3 6 D 0 1 2 3______________________ T = 12 T = 16 T = 20 SS = 26 SS = 30 SS = 38________________________ Use a repeated-measures ANOVA with α = .05 to determine whether these data are sufficient to demonstrate significant differences between the treatments. 12. In Problem 11 the data show large and consistent differences between subjects. For example, subject A has the largest score in every treatment and subject D always has the smallest score. In the second stage of the ANOVA, the large individual differences are subtracted out of the denominator of the F-ratio, which results in a larger value for F. The following data were created by using the same numbers that appeared in Problem 11. However, we eliminated the consistent individual differences by scrambling the scores within each treatment. Treatment____________________________ Subject I II III P__________________ A 6 2 3 11 G = 48 B 5 1 5 11 ΣX² = 294 C 0 5 10 15 D 1 8 2 11____________________ T = 12 T = 16 T = 20 SS = 26 SS = 30 SS = 38 a. Use a repeated-measures ANOVA with α = .05 to determine whether these data are sufficient to demonstrate significant differences between the treatments. b. Explain how the results of this analysis compare with the results from Problem 11. 14. The following data are from an experiment comparing three different treatment conditions: A B C______________ 0 1 2 N = 15 2 5 5 ΣX² = 354 1 2 6 5 4 9 2 8 8_________________ T = 10 T = 20 T = 30 SS = 14 SS = 30 SS = 30 a. If the experiment uses an independent-measures design, can the researcher conclude that the treatments are significantly different? Test at the .05 level of significance. b. If the experiment is done with a repeated-measures design, should the researcher conclude that the treatments are significantly different? Set alpha at .05 again. c. Explain why the analyses in parts a and b lead to different conclusions. 16. The following summary table presents the results from a repeated-measures ANOVA comparing three treatment conditions with a sample of n = 11 subjects. Fill in the missing values in the table. (Hint: Start with the df values.) Source SS df MS_________ Between treatments ___ ___ ___ F = 5.00 Within treatments 80 ___ Between subjects ___ ____ ____ Error 60 ____ ____ Total ___ ____ 18. A recent study indicates that simply giving college students a pedometer can result in increased walking (Jackson & Howton, 2008). Students were given pedometers for a 12-week period, and asked to record the average number of steps per day during weeks 1, 6, and 12. The following data are similar to the results obtained in the study. Number of steps (x1000)_____________ Week______________________ Participant 1 6 12 P_______________ A 6 8 10 24 B 4 5 6 15 C 5 5 5 15 G = 72 D 1 2 3 6 ΣX² = 400 E 0 1 2 3 ____F 2 3 4 9___________________ T = 18 T = 24 T = 30 SS = 28 SS = 32 SS = 40 a. Use a repeated-measures ANOVA with α = .05 to determine whether the mean number of steps changes significantly from one week to another. b. Compute η ² to measure the size of the treatment effect. c. Write a sentence demonstrating how a research report would present the results of the hypothesis test and the measure of effect size. 20. For either independent-measures or repeated-measures designs comparing two treatments, the mean difference can be evaluated with either a t test or an ANOVA. The two tests are related by the equation F = t². For the following data, a. Use a repeated-measures t test with α = .05 to determine whether the mean difference between treatments is statistically significant. b. Use a repeated-measures ANOVA with α = .05 to determine whether the mean difference between treatments is statistically significant. (You should find that F = t².) Person Treatment 1 Treatment 2 Difference A 4 7 3 B 2 11 9 C 3 6 3 D 7 10 3 M _ 4 M _ 8.5 MD _ 4.5 T _ 16 T _ 34 SS _ 14 SS _ 17 SS _ 27 Chapter 14 2. The structure of a two-factor study can be presented as a matrix with the levels of one factor determining the rows and the levels of the second factor determining the columns. With this structure in mind, describe the mean differences that are evaluated by each of the three hypothesis tests that make up a two-factor ANOVA. 4. For the data in the following matrix: No Treatment Treatment Male M = 5 M = 3 Overall M = 4 Female M = 9 M = 13 Overall M = 11 overall M = 7 overall M = 8 a, Which two means are compared to describe the treatment main effect? b. Which two means are compared to describe the gender main effect? c. Is there an interaction between gender and treatment? Explain your answer. 6. The following matrix presents the results of a two factor study with n = 10 scores in each of the six treatment conditions. Note that one of the treatment means is missing. Factor B_____________ B1 B2 B3 A1 M = 10 M = 20 M = 40 Factor A A2 M = 20 M = 30________________ a. What value for the missing mean would result in no main effect for factor A? b. What value for the missing mean would result in no interaction? 8. A researcher conducts an independent-measures, two-factor study using a separate sample of n = 15 participants in each treatment condition. The results are evaluated using an ANOVA and the researcher reports an F-ratio with df = 1, 84 for factor A, and an F-ratio with df = 2, 84 for factor B. a. How many levels of factor A were used in the study? b. How many levels of factor B were used in the study? c. What are the df values for the F-ratio evaluating the interaction? 12. Most sports injuries are immediate and obvious, like a broken leg. However, some can be more subtle, like the neurological damage that may occur when soccer players repeatedly head a soccer ball. To examine long-term effects of repeated heading, Downs and Abwender (2002) examined two different age groups of soccer players and swimmers. The dependent variable was performance on a conceptual thinking task. Following are hypothetical data, similar to the research results. a. Use a two-factor ANOVA with α = .05 to evaluate the main effects and interaction . b. Calculate the effects size (η ²) for the main effects and the interaction. c. Briefly describe the outcome of the study. Factor B: Age________________________ College Older n = 20 n = 20 Soccer M = 9 M = 4 T = 180 T = 80 Factor A: SS = 380 SS = 390 Sport n = 20 n = 20 Swimming M _ 9 M = 8 T _ 180 T = 160 SS = 350 SS = 400 ΣX² = 6360 14. The following table summarizes the results from a two-factor study with 2 levels of factor A and 3 levels of factor B using a separate sample of n = 8 participants in each treatment condition. Fill in the missing values. (Hint: Start with the df values.) Source SS df MS________________________ Between treatments 60 ____ Factor A ____ ____ 5 F = _____ Factor B _____ _____ ____ F= ______ A X B Interaction 25 _____ _____ F = ______ Within treatments _____ ______ 2.5 Total ______ ______ 16. The Preview section for this chapter described a two-factor study examining performance under two audience conditions (factor B) for high and low self-esteem participants (factor A). The following summary table presents possible results from the analysis of that study. Assuming that the study used a separate sample of n = 15 participants in each treatment condition (each cell), fill in the missing values in the table. (Hint: Start with the df values.) Source SS df MS____________ Between treatments 67 ____ Audience ____ ____ _____ F = ____ Self-esteem 29 ____ _____ F = ____ Interaction ____ _____ _____ F = 5.50 Within treatments ____ _____ 4 Total ____ _____ 18. The following data are from a two-factor study examining the effects of two treatment conditions on males and females. a. Use an ANOVA with α = .05 for all tests to evaluate the significance of the main effects and the interaction. b. Compute η ² to measure the size of the effect for each main effect and the interaction. Treatments__________________________ I ____________II_ 3 2 8 8 9 7 Male 4 7 Tmale = 48 7 M = 6 M = 6 T = 24 T = 24 N = 16 Factor A: SS = 26 SS = 22 G = 96 Gender 0 12 ΣX² = 806 0 6 2 9 6 13 Female M = 2 M = 10 Tfemale = 48 T = 8 T = 40 SS = 24 SS = 30 TI = 32 TЏ = 64 20. Mathematics word problems can be particularly difficult, especially for primary-grade children. A recent study investigated a combination of techniques for teaching students to master these problems (Fuchs, Fuchs, Craddock, Hollenbeck, Hamlett, & Schatschneider, 2008). The study investigated the effectiveness of small-group tutoring and the effectiveness of a classroom instruction technique known as “hot math.” The hot-math program teaches students to recognize types or categories of problems so that they can generalize skills from one problem to another. The following data are similar to the results obtained in the study. The dependent variable is a math test score for each student after 16 weeks in the study. No Tutoring With Tutoring 3 9 6 4 2 5 Traditional Instruction 2 8 4 4 7 6________ 7 8 7 12 2 9 Hot-Math Instruction 6 13 8 9 6 9_______ a. Use a two-factor ANOVA with α = .05 to evaluate the significance of the main effects and the interaction. b. Calculate the η ² values to measure the effect size for the two main effects. c. Describe the pattern of results. (Is tutoring significantly better than no tutoring? Is traditional classroom instruction significantly different from hot math? Does the effect of tutoring depend on the type of classroom instruction?) 22. In Chapter 11, we described a research study in which the color red appeared to increase men’s attraction to women (Elliot & Niesta, 2008). The same researchers have published other results showing that red also increases women’s attraction to men but does not appear to affect judgments of same sex individuals (Elliot, et al., 2010). Combining these results into one study produces a two-factor design in which men judge photographs of both women and men, which are shown on both red and white backgrounds. The dependent variable is a rating of attractiveness for the person shown in the photograph. The study uses a separate group of participants for each condition. The following table presents data similar to the results from previous research. a. Use a two-factor ANOVA with α = .05 to evaluate the main effects and the interaction. Person Shown in Photograph______________ Female Male____ White n = 10 n = 10 M = 4.5 M = 4.4 Background Color SS = 6 SS = 7____ For Photograph n = 10 n = 10 Red M = 7.5 M = 4.6 SS = 9 SS = 8 b. Describe the effect of background color on judgments of males and females.

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