Analysis of Variance - Assignment Example

PART A – MULTIPLE CHOICE Question Analysis of variance allows researchers to compare several different treatment conditions without conducting several different hypothesis tests. Answer True False  Question 2   In an analysis of variance, df between is determined by counting the number of treatment conditions and subtracting one. Answer True False  Question 3   When there are more than two treatments in an ANOVA, rejecting the null hypothesis means that all of the treatment means are significantly different from each other. Answer True False  Question 4   Analysis of variance is used to determine whether the variance in any one treatment is significantly different than the variance in any other treatment. Answer True False  Question 5   When the null hypothesis is true, the F-ratio for analysis of variance is expected, on average, to be zero. Answer True False   Question 6   In general, a large value for an F-ratio indicates that the null hypothesis is wrong. Answer True False  Question 7   The null hypothesis for an ANOVA states that ____. Answer a. there are no differences between any of the population means b. at least one of the population means is different from the others c. all of the population means are different from each other d. None of the other three choices is correct.  Question 8   An analysis of variance produces SS within = 40 and MS within = 10. In this analysis, how many treatment conditions are being compared? Answer a. 4 b. 5 c. 30 d. 50  Question 9   In an independent-measures experiment with three treatment conditions, all three treatments have the same mean, M 1 = M 2 = M 3. For these data SS between equals ____. Answer a. 0 b. 1.00 c. 3(5.50) d. cannot be determined from the information given   Question 10   In the notation for ANOVA, the letter n refers to ____ and the letter N refers to ____. Answer a. the number of scores in each treatment, the number of scores in the entire study b. the number of scores in the entire study, the number of scores in each treatment c. the sum of the scores in each treatment, the sum of the scores for the entire study d. the sum of the scores for the entire study, the sum of the scores in each treatment  Question 11   In general, the largest F-ratio will be obtained when the differences between sample means are ____ and the magnitudes of the sample variances are ____. Answer a. small, small b. large, large c. large, small d. large, large  Question 12   In analysis of variance, the term "factor" refers to ____. Answer a. a dependent variable b. an independent (or quasi-independent) variable c. a treatment mean d. a treatment total  Question 13   Which combination of factors will produce a large value for η 2 (eta squared)? Answer a. large mean differences and large sample variances b. large mean differences and small sample variances c. small mean differences and large sample variances d. small mean differences and small sample variances  Question 14   In analysis of variance, the F-ratio is a ratio of _____. Answer a. two (or more) sample means b. two variances c. sample means divided by sample variances d. None of the other three choices is correct.  Question 15   An analysis of variance is used to evaluate the mean differences for a research study comparing four treatment conditions with a separate sample of n = 5 in each treatment. The analysis produces SS within treatments = 32, SS between treatments = 40, and SS total = 72. For this analysis, what is MS within treatments? Answer a. 32/5 b. 32/4 c. 32/16 d. 32/20  Question 16   A researcher uses analysis of variance to test for mean differences among four treatments with a sample of n = 6 for each treatment. The F-ratio for this analysis would have what df values? Answer a. df = 3, 5 b. df = 3, 15 c. df = 3, 20 d. df = 4, 24  Question 17   In analysis of variance, large values for the sample variances will produce a large value for _____. Answer a. SSbetween treatments b. SSwithin treatments c. SStotal d. Large sample variances will cause all three SS values to be large.  Question 18   In an ANOVA, which of the following is most likely to produce a large value for the F-ratio? Answer a. large mean differences and small sample variances b. large mean differences and large sample variances c. small mean differences and small sample variances d. small mean differences and large sample variances  Question 19   A research report concludes that there are significant differences among treatments, with " F(2,22) = 8.62, p < .01." How many treatment conditions were compared in this study? Answer a. 2 b. 3 c. 29 d. 30  Question 20   Post hoc tests are necessary after an ANOVA whenever _____. Answer a. H0 is rejected b. there are more than two treatments c. H0 is rejected and there are more than two treatments d. You always should do post hoc tests after an ANOVA.  Question 21   An analysis of variance produces SS between = 40 and SS within = 60. Based on this information, the percentage of variance accounted for, η 2, is equal to ______. Answer a. 40/60 = 67% b. 40/100 = 40% c. 60/40 = 150% d. 60/100 = 60% PART B – SHORT ANSWER QUESTIONS 1. The following data summarize the results of two experiments. Each experiment compares three treatment conditions, and each experiment uses separate sample of n = 10 for each treatment. Experiment A Treatment   Experiment B Treatment 1 2 3   1 2 3 M = 1 M = 3 M = 5   M = 1 M = 10 M = 20 s = 15 s = 12 s = 18   s = 3 s = 5 s = 4 Just looking at the data—without doing an calculations—answer each of the following questions. a. Which experiment will produce the larger MSbetween? Experiment B b. Which experiment will produce the larger MSwithin? Experiment A c. Which experiment will produce the larger F-ratio? Experiment B 2. Use an analysis of variance with α = .05 to determine whether there are any significant differences among the three treatments. Treatments   I II III   4 4 9 N = 12 2 0 3  σ = 48 6 3 6 ΣX2 = 260   4 1 6   M = 4 M = 2 M = 6   SS = 8 SS = 10 SS = 18 STEP 1 The null and alternate hypotheses are The selected level of significance, α is .05. STEP 2 The total degree of freedom dftotal = N – 1 = 12 – 1 = 11 The between treatments degree of freedom dfbetween = k – 1 = 3 – 1 = 2 The within treatments degree of freedom dfwithin = N – k = 12 – 3 = 9 Thus, the F-ratio for these data has DF = 2, 9. The critical value of F-ratio with DF = 2, 9 is 4.256. Therefore, decision rule will be Reject H0 if F > 4.256. Otherwise, do not reject H0. STEP 3 ANOVA Source of Variation SS df MS F P-value F crit Between Groups 32 2 16 4 0.057157 4.256 Within Groups 36 9 4 Total 68 11 STEP 4 The F value obtained , F = 4, is not in the critical region. Therefore, we reject H0 and conclude that there are non-significant differences among the three treatments. 3. The results from an independent-measures research study indicate that there are significant differences between treatments, F(3, 36) = 38, p < .05. a.  How many treatment conditions were compared in the study? dfbetween = k – 1 = 3  k = 4 4 treatment conditions were compared in the study. b.  How many individuals participated in the entire study? dfwithin = N – k = N – 4 = 36  N = 40 40 individuals participated in the entire study. 4. 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 class 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. Use an ANOVA with α = .05 to determine whether there are significant differences in the student judgments depending on how they see their teachers. Poor Teacher Average Teacher Good Teacher   n = 6 n = 8 n = 10 N = 24 M = 6 M = 2 M = 2  σ = 72 SS = 30 SS = 33 SS = 42 ΣX2 = 393 STEP 1 The null and alternate hypotheses are The selected level of significance, α is .05. STEP 2 The total degree of freedom dftotal = N – 1 = 24 – 1 = 23 The between treatments degree of freedom dfbetween = k – 1 = 3 – 1 = 2 The within treatments degree of freedom dfwithin = N – k = 24 – 3 = 21 Thus, the F-ratio for these data has DF = 2, 21. The critical value of F-ratio with DF = 2, 21 is 3.467. Therefore, decision rule will be Reject H0 if F > 3.467. Otherwise, do not reject H0. STEP 3 ANOVA Source of Variation SS df MS F P-value F crit Between Groups 72 2 36 7.2 0.004157 3.467 Within Groups 105 21 5 Total 177 23 STEP 4 The F value obtained , F = 7.2, is in the critical region. Therefore, we reject H0 and conclude that there are significant differences in the student judgments depending on how they see their teachers. 5. Describe the circumstances under which you should use ANOVA instead of t tests, and explain why t tests are inappropriate in these circumstances. We should use ANOVA instead of t tests when there are more than two treatment conditions to compare. The ANOVA can be used to compare two or more treatments. t tests are inappropriate in these circumstances because we need to perform many tests when treatments are more than two. 6. Describe the circumstances in which post hoc tests are used and explain why these tests are necessary. Post hoc tests are used after ANOVA when (1) We reject H0 and (2) There are three or more treatments (k ≥ 3). Rejecting H0 indicates that at least one difference exists among the treatments. The post hoc tests determine exactly which means are significantly different when there are three or more treatments (k ≥ 3). 7. A researcher used an analysis of variance to compare four treatment conditions, with a separate sample of n = 9 participants in each treatment. The results of the analysis are shown in the following summary. Fill in all missing values in the table. (Hint: Start with the df values.) Source SS df MS   Between Treatments 45 3 15 F =  5 Within Treatments 96 32 3   Total 141 35     8. A psychologist would like to examine how the rate of presentation affects peoples ability to memorize a list of words. A list of 20 words is prepared. For one group of participants, the list is presented at the rate of one word every 0.5 second. The next group gets one word every second. The third group has one word every 2 seconds, and the fourth group has one word every 3 seconds. After the list is presented, the psychologist asks each person to recall the entire list. The dependent variable is the number of errors in recall. The data from this experiment are as follows: 0.5 second 1 second 2 seconds 3 seconds 4 0 3 0 6 2 1 2  σ = 32 2 2 2 1 ΣX2 = 104 4 0 2 1 M = 4 M = 1 M = 2 M = 1 SS = 8 SS = 4 SS = 2 SS = 2 a. Can the psychologist conclude that the rate of presentation has a significant effect on memory? Test at the .05 level. STEP 1 The null and alternate hypotheses are The selected level of significance, α is .05. STEP 2 The total degree of freedom dftotal = N – 1 = 16 – 1 = 15 The between treatments degree of freedom dfbetween = k – 1 = 4 – 1 = 3 The within treatments degree of freedom dfwithin = N – k = 16 – 4 = 12 Thus, the F-ratio for these data has DF = 3, 15. The critical value of F-ratio with DF = 3, 15 is 3.490. Therefore, decision rule will be Reject H0 if F > 3.490. Otherwise, do not reject H0. STEP 3 ANOVA Source of Variation SS df MS F P-value F crit Between Groups 24 3 8 6 0.009731 3.490 Within Groups 16 12 1.333 Total 40 15 STEP 4 The F value obtained , F = 6, is in the critical region. Therefore, we reject H0 and conclude that the rate of presentation has a significant effect on memory. Yes, the psychologist can conclude that the rate of presentation has a significant effect on memory. b. Use the Tukeys HSD test to determine which rates of presentation are statistically different and which are not. With k = 4 treatments, n = 4, and α = .05, the value of q for the test is 4.20. Tukey’s HSD is Thus, the mean difference between any two samples must be at least 2.425 to be significant. Using the value, we can make the following conclusions: (1) The first group (one word every 0.5 seconds) is significantly different from the second group (one word every 1 second). (2) The first group (one word every 0.5 seconds) is significantly different from the fourth group (one word every 3 second). 1 second 2 seconds 3 seconds 0.5 second 3 2 3 1 second 1 0 2 seconds 1 Read More