EPS 625 INTERMEDIATE STATISTICS ONE-WAY You are interested in whether male college students with black, blond, brunette, or red hair differ with respect to their social extrovertedness. For this study, 60 male college students from a local college (15 for each hair color) have been randomly selected to participate in the study using a stratified random sampling approach. The students were given a measure of social extroversion with a range of 0 (low level of social extroversion) to 10 (high level of social extroversion). Conduct an ANOVA to investigate the relationship between hair color and social extroversion. For this example, use an a priori alpha level of significance of =.05 for each statistical analysis (except, use an α =.001 for the Shapiro-Wilks test) to answer the following questions. 1. What would the null hypothesis be for this study? Show/write the appropriate symbols or expression in words. H 0 : µ black = µ blond = µ brunette = µ redhead or H 0 : µ 1 = µ 2 = µ 3 = µ The group means (average social extrovertedness scores) for the four (hair color) groups are equal. 2. What would the research/alternative hypothesis be for this study? Show/write the appropriate symbols or expression in words. H a : µ i µ k for some i, k or H a : µ i µ j for some i, j At least one of the (hair color) group means significantly differs from the other (hair color) group means. -or- At least two of the (hair color) group means are significantly different from each other. 3. Prior to examining whether group means differ, you need to test the assumptions underlying the one-way ANOVA. a. Was the assumption of independence met for these data? Indicate how you made this determination. YES, the assumption of independence was met. This is indicated by the four groups of male college students being independent of each other. A stratified random sampling technique was used, which focused on obtaining a random selection of male college students for each of the four hair colors. b. Was the assumption of normality met for these data? Indicate how you made this determination. YES, the assumption of normality was met for this set of data. This is indicated by the fact that none of the standardized skewness values exceeded +3.29, nor were any of the probability values less than (or equal to) the.001 alpha level set for the Shapiro-Wilks test.
Descriptives Social Extroversion Hair Color Black 95% Confidence Interval for Lower Bound Upper Bound Statistic Std. Error 7.33.532 6.19 8.7 Blond 5% Trimmed Median Variance Std. Deviation Minimum Maximum Range Interquartile Range Skewness Kurtosis 95% Confidence Interval for Lower Bound Upper Bound 7.37 8.00.238 2.059 10 6 -.122.580-1.05 1.121 6.27.613.95 7.58 Brunette 5% Trimmed Median Variance Std. Deviation Minimum Maximum Range Interquartile Range Skewness Kurtosis 95% Confidence Interval for Lower Bound Upper Bound 6.2 6.00 5.638 2.37 3 10 7.225.580-1.172 1.121 3.67.361 2.89. Redhead 5% Trimmed Median Variance Std. Deviation Minimum Maximum Range Interquartile Range Skewness Kurtosis 95% Confidence Interval for Lower Bound Upper Bound 3.63.00 1.952 1.397 2 6 3.329.580 -.970 1.121 5.13.568 3.92 6.35 5% Trimmed Median Variance Std. Deviation Minimum Maximum Range Interquartile Range Skewness Kurtosis 5.15 5.00.838 2.200 2 8 6 -.055.580-1.9 1.121 PAGE 2
.122 Black: =. 2103.580.225 Blond: =. 3879.580.329.055 Brunette: =. 5672 Redhead: =. 098.580.580 Social Extroversion Hair Color Black Blond Brunette Redhead *. This is a lower bound of the true significance. a. Lilliefors Significance Correction Tests of Normality Kolmogorov-Smirnov a Shapiro-Wilk Statistic df Sig. Statistic df Sig..160 15.200*.915 15.160.170 15.200*.93 15.315.150 15.200*.900 15.097.167 15.200*.906 15.117 Black: p (.160) > (.001) Blond: p (.315) > (.001) Brunette: p (.097) > (.001) Redhead: p (.117) > (.001) c. Was the assumption of homogeneity of variance met for these data? Indicate how you made this determination. YES, the assumption of homogeneity is met, because p (.08) > (.05). This is indicated by the Levene s Test of Homogeneity of Variances, F(3, 56) = 2.3, p =.08. With an alpha level of.05, p (.08) > α (.05), which indicates nonsignificance, the null hypothesis (no variance difference) is retained as such, indicating that the assumption of homogeneity of variance is met. Test of Homogeneity of Variances Extroversion Social Extroversion Levene Statistic df1 df2 Sig. 2.335 3 56.08 PAGE 3
. The next question that needs to be answered is whether all of the groups are the same on their social extroversion means. This is answered by conducting a One-way Analysis of Variance using hair color as the independent variable and students scores on a measure of social extroversion as the dependent variable. If applicable, use the Welch statistic. What is your conclusion (at this point) from this analysis? Indicate how you came to your conclusion. Since the obtained F ratio (8.89) was significant at the.05 alpha level (p shown as.000, that is p <.001) reported as F(3, 56) = 8.89, p <.001 (or, p <.05), we can conclude that at least two of the four hair color groups differ significantly on their average social extrovertedness scores. That is, p (.000) < α (.05). The Welch statistic was not required since the equal variance assumption was met. However, beyond that, post hoc follow-up procedures will need to be conducted to test the difference between all unique pairwise comparisons. ANOVA Extroversion Social Extroversion Sum of Squares df Square F Sig. Between Groups 111.067 3 37.022 8.885.000 Within Groups 233.333 56.167 Total 3.00 59 5. Calculate the measure of association and interpret its meaning if applicable, or indicate why this measure would not be needed. 2 ω = SSB ( K 1) MS SS + MS T W W 111.067 ( 1).167 = 3.00 +.167 111.067 (3).167 111.067 12.501 = = = 38.567 38.567 98.566 38.567 =.282775 2 ω =.2828 Therefore, we conclude that approximately 28% (ω 2 =.28) of the total variance in the student s average social extrovertedness scores is accounted for by the four levels of hair color. -or- This indicates that the independent variable (four levels of hair color) accounts for approximately 28% (ω 2 =.28) of the total variance in the dependent variable (average social extrovertedness scores) for this sample. PAGE
6. Write the statistical strand for this one-way ANOVA analysis. F(3, 56) = 8.89, p <.001, ω 2 =.28 -or- F(3, 56) = 8.89, p <.05, ω 2 =.28 7. Assuming that you found a significant F, how do the pairs of groups differ? Indicate which post hoc procedure you used and why. Indicate your findings from the post hoc analysis. That is, how did you determine the pair to be significant or not (this must go beyond the * as an indication)? Be sure to discuss all unique pairwise comparisons. Tukey s post hoc procedure is used since the homogeneity of variance assumption was met Using an a priori alpha level of.05 Black vs. Blond ( difference = 1.067) is not significant, p (.86) > α (.05) Black vs. Brunette ( difference = 3.667) is significant, p (.000) < α (.05) Black vs. Redhead ( difference = 2.200) is significant, p (.023) < α (.05) Blond vs. Brunette ( difference = 2.600) is significant, p (.005) < α (.05) Blond vs. Redhead ( difference = 1.133) is not significant, p (.32) > α (.05) Brunette vs. Redhead ( difference = 1.67) is not significant, p (.212) > α (.05) Dependent Variable: Extroversion Social Extroversion Tukey HSD Multiple Comparisons (I) Hair Hair Color 1 Black 2 Blond 3 Brunette Redhead (J) Hair Hair Color 2 Blond 3 Brunette Redhead 1 Black 3 Brunette Redhead 1 Black 2 Blond Redhead 1 Black 2 Blond 3 Brunette *. The mean difference is significant at the.05 level. Difference 95% Confidence Interval (I-J) Std. Error Sig. Lower Bound Upper Bound 1.067.75.86 -.91 3.0 3.667*.75.000 1.69 5.6 2.200*.75.023.23.17-1.067.75.86-3.0.91 2.600*.75.005.63.57 1.133.75.32 -.8 3.11-3.667*.75.000-5.6-1.69-2.600*.75.005 -.57 -.63-1.67.75.212-3..51-2.200*.75.023 -.17 -.23-1.133.75.32-3.11.8 1.67.75.212 -.51 3. PAGE 5
8. For all significant pairwise comparisons, calculate and report the effect size. Using ES X i X k = where MS W =.167 = 2. 01323 MS W 3.667 ES for Black vs. Brunette = ES = = 1. 7967 = 2.01 1.80 2.200 ES for Black vs. Redhead = ES = = 1. 0779 = 2.01 1.08 2.600 ES for Blond vs. Brunette = ES = = 1. 2739 = 2.01 1.27 9. Complete the three tables below that present your statistical results from these analyses. Use the values reported in the SPSS output you do not need to round (Note that in reporting your data, APA suggests being consistent in your reporting of values, e.g., all rounded to two or three decimals, with the exception of p values, which are typically reported with three decimal places.). Results A One-way Analysis of Variance (ANOVA) was used to examine the question of whether male college students with black, blond, brunette, or red hair differ with respect to their social extrovertedness. The independent variable represented the different hair colors with four groups being represented: 1) black; 2) blond; 3) brunette; and ) red. The dependent variable was the average score that students made on a measure of social extroversion with a range of 0 (low level of social extroversion) to 10 (high level of social extroversion). See Table 1 for the means and standard deviations for each of the four groups. PAGE 6
Table 1 s and Standard Deviations of Social Extroversion Scores by Hair Color Hair Color n SD Black 15 7.33 2.06 Blond 15 6.27 2.37 Brunette 15 3.67 1.0 Red 15 5.13 2.20 Total 60 5.60 2.2 The test for normality, examining standardized skewness and the Shapiro-Wilks test, indicated the data were statistically normal. Additionally, homogeneity of variance was not significant, Levene s F(3, 56) = 2.3, p =.08, indicating that this assumption underlying the application of ANOVA was met. An alpha level of.05 was used for all subsequent analyses. The one-way ANOVA of student s average score on the measure of social extroversion (see Table 2) revealed a statistically significant main effect, F(3, 56) = 8.89, p <.001, indicating that not all hair colors had the same average score on the measure of social extroversion. Omega squared (ω 2 =.28) indicated that approximately 28% of the total variation in average score on students measure of social extroversion is attributable to differences between the four colors of hair. Table 2 Analysis of Variance for Social Extroversion Scores by Hair Color Source SS df MS F p Between 111.07 3 37.02 8.89.000 Within 233.33 56.17 Total 3.0 59 PAGE 7
Post hoc comparisons, using the Tukey post hoc procedure, were conducted to determine which pairs of the four hair color means differed significantly. These results are given in Table 3 and indicate that students with black hair (M = 7.33, SD = 2.06) had a significantly higher average score on the measure of social extroversion than students with brunette hair (M = 3.67, SD = 1.0) as well as students with red hair (M = 5.13, SD = 2.20). The effect sizes for these two significant effects were 1.80 and 1.08, respectively. Additionally, students with blond hair (M = 6.27, SD = 2.37) had a significantly higher average score on the measure of social extroversion than students with brunette hair, with an effect size of 1.27. Table 3 Post Hoc Results for Social Extroversion Scores by Hair Color Hair Color Differences ( X i X j ) (Effect Sizes are indicated in parentheses) 1 2 3 1. Black 7.33 2. Blond 6.27 1.06 3. Brunette 3.67 3.66 *** (1.80) 2.60 ** (1.27). Red 5.13 2.20 * (1.08) 1.1 1.6 *p <.05, ** p <.01, *** p <.001 PAGE 8