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1 Testing Hypothesis for a Mean DrPh, Walden University RSCH-8210R: Quantitative Reasoning and Analysis Dr. Nancy Rea July, 2020

2 Testing Hypothesis for a Mean Researchers are required to have the ability to test their hypothesis. Statistical data provides the researcher with the ability to test their hypothesis and determine if it will be rejected or retained (Frankfort-Nachmias et al., 2020). This week there are three case scenarios; each scenario has an alignment to an appropriate t-test. There are three types of t-test, which are a one-sample t-test, independent samples t-test, and paired samples t-test (Frankfort-Nachmias et al., 2020). Two types of datasets will be used to verify the datasets utilized. The mean of the Afrobarometer will be calculated (IBM SPSS vs25, 2017a). The mode will be calculated using the High School Longitudinal Survey (IBM SPSS vs25, 2017b). Verification of Datasets Utilized Mean Statistics Table 1 reveals, the mean age for the respondents is 37.39 Ta ble 1 Mean Age of Respondents N Valid 10232 Missing 85 Mean 37.39 Std. Deviation 14.863 Range 87 Minimum 18 Maximum 105 From “Afrobarometer: Dataset B,” IBM SPSS v25, 2017a. Table 2 reveals, the mode of parent highest level of education is 2.

3 Table 2 Parent Highest Level of Education Valid 3332 Missing 1368 Mean 2.99 Median 3.00 Mode 2 From “High School Longitudinal Survey: Dataset B,” IBM SPSS v25, 2017b. Scenarios Scenario One: One Sample t-Test As an international development researcher, you would like to know African Citizen’s perceptions about current levels of democracy. Your working hypothesis is that a series of reforms have increased African views of the level of democracy today. You do not have a good research design to compare attitudes before and after the reforms, but know that leaders and development experts would like to see a value of 6, on the scale of 1–10. Using the data from the 2015 Afrobarometer, determine whether perceptions about current levels of democracy statistically differ from a value of 6 (Laureate Education, 2016, para. 1) Interpretation for One Sample t-Test The one-sample t-test has a continuous dependent variable, and the independent variable is implied (Laureate Education, n.d.). The dependent variable is the level of democracy (Laureate Education, 2016). The results of the one-sample t-test displayed, in Table 3, the mean level of democracy among African citizens [Mean = 5.54, SD = 2.871]. Table 4 reveals the mean was statistically significant at the .05 level of significance (t = -15.619, df = 9394, p < .001) from the [Test value = 6]. The [ Mean difference = -.463, 95% CI (-.52, -.40)] as displayed in Table 4.

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4 The null hypothesis suggested that there was no significant difference in the mean democracy level of African citizens. The null hypothesis is rejected. The Cohen’s effect size value (d = - 0.16.0) suggests a low practical significance and small effect size. Cohens effect sizes refer as small (d = 0.2), medium (d = 0.5), and large (d = 0.8) (Lakens, 2013). Table 3 One-Sample Statistics Level of Democracy in Africa N Mean Std. Deviation Std. Error Mean Q46a. Level of democracy: today 9395 5.54 2.871 .030 From “Afrobarometer: Dataset B,” IBM SPSS v25, 2017. Table 4 One-Sample Test Level of Democracy in Africa Test Value = 6 t df Sig. (2- tailed) Mean Difference 95% Confidence Interval of the Difference Lower Upper Q46a. Level of democracy: today -15.619 9394 .000 -.463 -.52 -.40 From “Afrobarometer: Dataset B,” IBM SPSS v25, 2017. Scenario Two: Independent Samples t-Test As an international development researcher, you have already tested whether perceptions about the current levels of democracy differ from a value of 6. Given recent social change movements in North Africa, you now want to determine whether there is a statistical difference in these perceptions between North Africa and Southern Africa. Using the data from the 2015 Afrobarometer (Laureate Education, 2016, para. 2) Interpretation for Independent Samples t-Test

5 The independent samples t-test has categorical independent variables, and the dependent variable is continuous (Laureate Education, n.d.). North African and Southern Africa are the independent variables, while the level of democracy is the dependent variable. Table 5 reveals the independent samples t-test indicated that scores were significantly higher for Southern Africa (M = 5.78, SD = 2.755, n = 3207) than for North Africa (M = 4.93, SD = 3.026, n = 1130), t(1829.904), df = -8.347, p < .001 as shown in Table 6. Levene’s test for equality of variances in Table 6 indicated unequal variances (F = 20.5, p = .083) and equal variances not assumed, so it is not fulfilled. Cohen’s size effect (d = 0.293744), which shows a small effect size. Table 7 and 8 shows the [Mean difference = -.854, SD = .102, 95% CI (-1.055, -.653). The null hypothesis is rejected. Table 5 Group Statistics North Africa and Southern Africa Perceptions Country by region N Mean Std. Deviation Std. Error Mean Q46a. Level of democracy: today North Africa 1130 4.93 3.026 .090 Southern Africa 3207 5.78 2.755 .049 From “Afrobarometer: Dataset B,” IBM SPSS v25, 2017.

6 Table 6 Independent Samples Test Africa and Southern Africa Perceptions Levene's Test for Equality of Variances t-test for Equality of Means F Sig. t df Q46a. Level of democracy: today Equal variances assumed 20.588 .000 -8.729 4335 Equal variances not assumed -8.347 1829.904 From “Afrobarometer: Dataset B,” IBM SPSS v25, 2017. Table 7 Independent Samples Test Africa and Southern Africa Perceptions t-test for Equality of Means Sig. (2- tailed) Mean Difference Std. Error Difference 95% Confidence Interval of the Difference Lower Q46a. Level of democracy: today Equal variances assumed .000 -.854 .098 -1.046 Equal variances not assumed .000 -.854 .102 -1.055 From “Afrobarometer: Dataset B,” IBM SPSS v25, 2017.

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7 Table 8 Independent Samples Test Africa and Southern Africa Perceptions t-test for Equality of Means 95% Confidence Interval of the Difference Upper Q46a. Level of democracy: today Equal variances assumed -.662 Equal variances not assumed -.653 From “Afrobarometer: Dataset B,” IBM SPSS v25, 2017. Scenario Three: Paired Sample t-Test As an educational researcher, would like to know whether high school student’s perceptions about mathematical utility changed between their freshman and senior year. Each respondent is asked the same series of questions about the utility of mathematics in their future during their freshman and senior year. These questions are combined to form one variable of mathematical utility; higher values indicate higher levels of mathematical utility. Using the High School Longitudinal Survey (Laureate Education, 2016, para. 3) Interpretation for Paired Sample t-Test A paired samples t-test has a dependent variable, and the independent variable is implied (Laureate Education, n.d.). The dependent variables are freshman year and senior year in high school. The independent variable is mathematical utility. Table 9 shows a paired sample t-test indicates the scores were lower but not significantly for mathematical utility of freshman (M = .0253, SD = .98557, n = 3242) than for the seniors (M = .0298, SD = .98933, n = 3242), (t = -.215, df = 3241, p > .830) as shown in Table 10 and 11. The p-value is not significant because it

8 is greater than alpha, which is set at .05. Therefore, the hypothesis is retained. Cohen’s effect size value (d = .9911) suggests a high practical significance. Table 9 Paired Samples Statistics Mathematical Utility Freshman and Senior Mean N Std. Deviation Std. Error Mean Pair 1 T1 Scale of student's mathematics utility .0253 3242 .98557 .01731 T2 Scale of student's mathematics utility .0298 3242 .98933 .01738 From “High School Longitudinal Survey: Dataset B,” IBM SPSS v25, 2017b. Table 10 Paired Samples Test Mathematical Utility Freshman and Senior Paired Differences t Mean Std. Deviation Std. Error Mean 95% Confidence Interval of the Difference Lower Upper Pair 1 T1 Scale of student's mathematics utility - T2 Scale of student's mathematics utility -.00446 1.17970 .02072 -.04508 .03616 -.215 From “High School Longitudinal Survey: Dataset B,” IBM SPSS v25, 2017b.

9 From “High School Longitudinal Survey: Dataset B, “IBM SPSS v25, 2017b. Table 11 Paired Samples Test Mathematical Utility Freshman and Senior df Sig. (2-tailed) Pair 1 T1 Scale of student's mathematics utility - T2 Scale of student's mathematics utility 3241 .830 Social Implications Social change and statistics work well together to contribute to the everchanging world. The comprehension of the level of democracy in Africa is imperative in moving the country forward. The implementation of democracy required a change in the South African political and determinants of health, including restructuring the educational and employment aspects to reduce inequities (Arendse & Smith, 2018). The current democracy levels in different parts of Africa will provide a robust knowledge base of the continued social changes required. Data analysis is used to provide evidence to answer questions and make decisions about advocating for social justice and human rights (Price, 2015). There is a need for more statisticians for analyzing data to assist in answering questions that bring about change and exploration (Simpson, 2015). Math utility in high school can lay a foundation for creating future statisticians. Conclusion The mean age of respondents from the Afrobarometer dataset was used for verification for the first and second scenarios for the one sample t=test and independent samples t-test (IBM SPSS vs25, 2017a). The High School Longitudinal Survey dataset was used to calculate the mode for verification for the third scenario for a paired test (IBM SPSS vs25, 2017b). The paper further reviewed scenarios that included the linkage to the appropriate t-test. Each t-test had

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10 charts to coincide with the scenario and description of the variables, the hypothesis rejecting or retainment, significance of t-test, and calculation of the Cohen’s d and Levene’s test with the independent samples t-test. Lastly, a social change regarding democracy in Africa and the implications of math utility in high schools to increase the feasibility of future statisticians. References

11 Arendse, A., & Smith, J. (2018). Economic Transformation and Emancipation through Active Citizenship Education . BCES Conference Books, 16 , 45–51. Frankfort-Nachmias, C., Leon-Guerrero, A., & Davis, G. (2020). Social statistics for a diverse society (9th ed.). IBM Corp. Released 2017a. Afrobarometer [Dataset B]. IBM SPSS Statistics for Windows, Version 25.0. Armonk, NY: IBM Corp IBM Corp. Released 2017b. HS Long Study Dataset [Dataset B]. IBM SPSS Statistics for Windows, Version 25.0. Armonk, NY: IBM Corp Lakens D. (2013). Calculating and reporting effect sizes to facilitate cumulative science: a practical primer for t-tests and ANOVAs. Frontiers in psychology, 4 , 863. https://doi.org/10.3389/fpsyg.2013.00863 Laureate Education. (n.d.). General statistics : T-tests [Academic Skills Center Resources]. Walden University. https://academicguides.waldenu.edu/academic-skills-center/skills/statistics/general-stats Laureate Education. (n.d.). Week 6 statistics drop-in sessions for RSCH 8210 [Tutor PowerPoint]. Walden University. https://academicguides.waldenu.edu/academic-skills- center/tutors/stats-drop-in-rsch-8210 Laureate Education. (2016). Week 6 t test scenarios [Course Materials]. Walden University Blackboard. https://class.content.laureate.net/f66d3004e930abfeb29d19c7ffa8a876.pdf Price, M. (2015). Statistician promotes social justice and human rights [Video file]. Retrieved from https://www.youtube.com/watch?v=orW01w8a4zY

12 Simpson S. H. (2015). Creating a Data Analysis Plan: What to Consider When Choosing Statistics for a Study. The Canadian journal of hospital pharmacy, 68(4), 311–317. https://doi.org/10.4212/cjhp.v68i4.1471 Wagner, III, W. E. (2020). Using IBM® SPSS® statistics for research methods and social science statistics (7th ed.). Thousand Oaks, CA: Sage Publications.

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