CRJ_575_Mod_6_DiscussionFinalToPost

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Colorado State University, Global Campus *

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575

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Statistics

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Jan 9, 2024

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Hypothesis and null hypothesis: Hypothesis: There is a significant difference in the mean GPA scores among the different racial groups. In statistical terms, the research hypothesis can be viewed as follows: If 1 μ 1, 2 μ 2, and 3 μ 3 represent the average GPAs of Black, White, and Hispanic students respectively, the research hypothesis can be stated as at least one of the population means (1 μ 1, 2 μ 2, or 3 μ 3) is different from the others (Haans, 2019) suggesting that at least one of these racial groups have a different average GPA compared to the others. Null Hypothesis (HO ): There are no significant differences in mean GPA scores among the different racial groups. The null hypothesis μ 1= μ 2= μ 3 represents the average GPAs for Black, White, and Hispanic students. Both hypotheses and analysis performed are based on data acquired from the data file mtf11sdss.sav (Colorado State University Global, n.d.) Summary of ANOVA results: Between groups: SS = 6.0001, df = 2, MS = 3.001, F = 7.318, p > .001 Within groups: SS = 451.018, df = 1100, MS = .410 Total: SS = 457.020, df = 1102 Interpretation: The one-way ANOVA tests whether there are any statistically significant differences between the means of the GPA scores across the three racial groups (Chen et al., 2018). The F-statistic is 7.318 with 2 and 1100 degrees of freedom between and within groups. The p-value (<.001) is less than the chosen significance level of 0.05, indicating that there is a statistically significant difference in mean GPA scores among different racial groups. Effect sizes : Eta-squared (η²) = 0.013 represents the proportion of the total variance in GPA scores that is associated with the variance between the racial groups (Abebe, 2019). Roughly speaking, 1.3% of the variance in GPA scores can be attributed to the differences between racial groups. Omega-squared (ω²) = 0.011 (Fixed-effect model) , which is similar to Eta-squared, represents the proportion of the total variance in GPA scores accounted for by the variance between racial groups. As a result of the ANOVA, there is a statistically significant difference in mean GPA scores among racial groups. Descriptive Statistics: Black (1): Mean GPA = 3.0572, SD = 0.63440 White (2): Mean GPA = 3.2323, SD = 0.63649 Hispanic (3): Mean GPA = 3.0905, SD = 0.66169 These are values that show the means and standard deviations of GPA scores for each racial group.

Post Hoc Tests (Tukey’s HSD): Significant mean differences were found between Black and White (p =.0004) and between White and Hispanic (p = .021) groups. There is no significant difference between Black and Hispanic groups (p = .880). Tukey’s HSD test helped to identify which specific groups are different from each other regarding GPA scores. Homogenous Subsets: There are three subsets based on Tukey’s HSD with indicate groups with similar mean GPA scores: Subset 1 (Black (3.0572) and Hispanic (3.0905) and Subset 2: White (3.2323) Both the ANOVA and Tukey’s HSD test revealed that there are significant differences in mean GPA scores between Black and White students, as well as between White and Hispanic students. There did not show significant difference in mean GPA scores between Black and Hispanic students. The homogenous subsets further support the findings by grouping students with similar GPA scores together. Overall summary: The analysis of high school GPA scores across racial groups (Black, White, and Hispanic) yielded a statistically significant overall difference which was determined by the one-way ANOVA (F(2,100) = 7.318, p<.001). Using Tukey’s Honestly Significant Difference (HSD) post hoc test revealed significant mean differences between Black and White groups (p = .004) and between White and Hispanic groups (p = .021). There was no significant difference in mean GPA scores between Black and Hispanic groups (p = .880). Conclusion : Based on the results generated during my analysis and taking into account the hypotheses, one could conclude that there is enough evidence to reject the null hypothesis as there is a statistically significant difference in mean high school GPA scores among racial groups. However, if the p-value was greater than the chosen significance level (usually 0.05), there would not be enough evidence to reject the null hypothesis as there would be no significant difference in mean high school GPA among racial groups. My analysis shows a p-value less than 0.001 allowing me to reject the null hypothesis. Table 1 ANOVA One-way with effect sizes and descriptives Oneway ANOVA High School GPA (recoded from "grade") Sum of Squares df Mean Square F Sig. Between 6.001 2 3.001 7.318 <.001

Groups Within Groups 451.018 1100 .410 Total 457.020 1102 ANOVA Effect Sizes a Point Estimate 95% Confidence Interval Lower Upper High School GPA (recoded from "grade") Eta-squared .013 .002 .028 Epsilon-squared .011 .001 .027 Omega-squared Fixed- effect .011 .001 .027 Omega-squared Random-effect .006 .000 .014 a. Eta-squared and Epsilon-squared are estimated based on the fixed-effect model. Oneway Descriptives High School GPA (recoded from "grade") N Mean Std. Deviation Std. Error 95% Confidence Interval for Mean Lower Bound Upper Bound BLACK:(1) 166 3.0572 .63440 .04924 2.9600 3.1544 WHITE:(2) 758 3.2323 .63649 .02312 3.1869 3.2777 HISPANIC: (3) 179 3.0905 .66169 .04946 2.9929 3.1881 Total 1103 3.1830 .64399 .01939 3.1449 3.2210 Descriptives High School GPA (recoded from "grade") Minimum Maximum

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BLACK:(1) 1.00 4.00 WHITE:(2) 1.00 4.00 HISPANIC: (3) 1.00 4.00 Total 1.00 4.00 ANOVA High School GPA (recoded from "grade") Sum of Squares df Mean Square F Sig. Between Groups 6.001 2 3.001 7.318 <.001 Within Groups 451.018 1100 .410 Total 457.020 1102 ANOVA Effect Sizes a Point Estimate 95% Confidence Interval Lower Upper High School GPA (recoded from "grade") Eta-squared .013 .002 .028 Epsilon-squared .011 .001 .027 Omega-squared Fixed- effect .011 .001 .027 Omega-squared Random-effect .006 .000 .014 a. Eta-squared and Epsilon-squared are estimated based on the fixed-effect model. Note . Adapted from Adapted from Data Sets and Codebooks , by Colorado State University Global, n.d., Canvas ( https://csuglobal.instructure.com/courses/81945/modules ) and [DataSet1] C:\Users\SnS_M\OneDrive\Desktop\CRJ_575_Analytical Methods\Datasets Codebooks\mtf11sdss.sav. Data analytics were performed using IBM SPSS statistics for Windows (Version 28.0) [Computer Software] by IBM (https://www.ibm.com/products/spss-statistics). Table 2 Post Hoc Tests

Post Hoc Tests Multiple Comparisons Dependent Variable: High School GPA (recoded from "grade") Tukey HSD (I) Respondent's race (trichotomized B/W/H) (J) Respondent's race (trichotomized B/W/H) Mean Difference (I- J) Std. Error Sig. BLACK:(1) WHITE:(2) -.17509 * .05487 .004 HISPANIC:(3) -.03327 .06900 .880 WHITE:(2) BLACK:(1) .17509 * .05487 .004 HISPANIC:(3) .14182 * .05321 .021 HISPANIC:(3) BLACK:(1) .03327 .06900 .880 WHITE:(2) -.14182 * .05321 .021 Multiple Comparisons Dependent Variable: High School GPA (recoded from "grade") Tukey HSD (I) Respondent's race (trichotomized B/W/H) (J) Respondent's race (trichotomized B/W/H) 95% Confidence Interval Lower Bound Upper Bound BLACK:(1) WHITE:(2) -.3039 -.0463 HISPANIC:(3) -.1952 .1287 WHITE:(2) BLACK:(1) .0463 .3039 HISPANIC:(3) .0169 .2667 HISPANIC:(3) BLACK:(1) -.1287 .1952 WHITE:(2) -.2667 -.0169 The mean difference is significant at the 0.05 level.

Homogeneous Subsets High School GPA (recoded from "grade") Tukey HSD a,b Respondent's race (trichotomized B/W/H) N Subset for alpha = 0.05 1 2 BLACK:(1) 166 3.0572 HISPANIC:(3) 179 3.0905 WHITE:(2) 758 3.2323 Sig. .842 1.000 Means for groups in homogeneous subsets are displayed. a. Uses Harmonic Mean Sample Size = 232.019. b. The group sizes are unequal. The harmonic mean of the group sizes is used. Type I error levels are not guaranteed. Note . Adapted from Data Sets and Codebooks , by Colorado State University Global, n.d., Canvas ( https://csuglobal.instructure.com/courses/81945/modules ) and [DataSet1] C:\Users\SnS_M\OneDrive\Desktop\CRJ_575_Analytical Methods\Datasets Codebooks\mtf11sdss.sav. Data analytics were performed using IBM SPSS statistics for Windows (Version 28.0) [Computer Software] by IBM (https://www.ibm.com/products/spss-statistics). References Abebe, T. H. (2019). The derivation and choice of appropriate test statistic (z, t, f and Chi-Square test) in research methodology. Mathematics Letters , 5 (3). https://doi.org/10.11648/j.ml.20190503.11Chen, T., Xu, M., Tu, J., Wang, H., & Niu, X. (2018). Relationship between omnibus and post-hoc tests: An investigation of performance of the f test in ANOVA. Shanghai Archives of Psychiatry , 30 (1). https://doi.org/10.11919/j.issn.1002-0829.21801

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Chen, T., Xu, M., Tu, J., Wang, H., & Niu, X. (2018). Relationship between omnibus and post- hoc tests: An investigation of performance of the f test in ANOVA. Shanghai Archives of Psychiatry , 30 (1). https://doi.org/10.11919/j.issn.1002-0829.21801 Colorado State University Global. (n.d.). Modules: Data Sets and Codebooks , Canvas. https://csuglobal.instructure.com/courses/81945/modules Haans, A. (2019). “contrast analysis”: A tutorial. Practical Assessment, Research, and Evaluation , 23 (9). https://scholarworks.umass.edu/pare/vol23/iss1/9 IBM. (n.d.). IBM SPSS Statistics for Windows (Version 28.0) [Computer Software]. https://www.ibm.com/products/spss-statistics

CRJ_575_Mod_6_DiscussionFinalToPost

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