jimjim harperbusi 820 – quantitative research methodsdecember 12 B u s i n e s s F i n a n c e

Peer reply to JimJim HarperBUSI 820 – Quantitative Research MethodsDecember 12, 2021Discussion Forum #8 D8.9.1: One Sample t-testD8.9.1.a            The one-sample t-test is conducted whenever the objective of the analysis is to compare the mean of a sample with a hypothesized population mean to determine if the sample is significantly different. The assumptions for using the one-sample t-test require that the dependent variable be normally distributed, and the participants must be independent of each other.D8.9.1.b            An example of an alternative use of the one sample t-test would be: “Are math achievement scores significantly different from 12.5?”D8.9.2: Independent Samples t-testD8.9.2.a            For the math achievement test, the p-value for Levene’s Test (p = .466) is not significant, indicating that the variances must be assumed to be equal. For grades in high school, the p-value for Levene’s Test (p = .451) is not significant, indicating that the variances must be assumed to be equal. For visualization test, the p-value for Levene’s Test (p = .013) is statistically significant, indicating that the assumption that the variances are equal is violated.D8.9.2.b            For the math achievement test, fast track students were statistically significantly different from regular track students, t(73) = 2.697, p = .009. For grades in high school, fast track students were not statistically significantly different from regular track students, t(73) = -0.903, p = .369. For the visualization test, fast track students were statistically significantly different from regular track students, t(57.2) = 2.385, p = .020. In this case, the results for t and df were adjusted due to unequal variances.D8.9.2.c            As seen in D8.9.2.b, fast track students were statistically significantly different from regular track students on math achievement test and visualization test, while the t-test for grades in high school was not statistically significant.D8.9.2.d            Fast track students were statistically significantly different from regular track students on math achievement scores, t(73) = 2.697, p = .009. Fast track students did not differ from regular track students on grades in high school, t(73) = -90, p = .369), but fast track students did score higher on the visualization test, t(57.2) = 2.39, p = .020), which was statistically significantly different.D8.9.2.e            The 95% Confidence Interval of the Difference means that if we conducted an infinite number of studies using the same conditions, and computed a 95% confidence interval for each study, 95% of the intervals would contain the true population difference between means. For grades in high school, the 95% confidence interval is between -1.056 points and 0.397 points. Since the range from lower to upper contains the point of zero difference, this indicates that the t-test for the difference is not statistically significant. For the visualization test, the 95% confidence interval is between 0.348 points and 3.981 points. The lower and upper limits of the confidence interval on visualization test indicate that the difference between fast track and regular track could be as small as 0.348 points or as large as 3.981 points.D8.9.2.f            Effect sizes for t tests are not provided in the printout but can be easily calculated. For math achievement, the difference between the means (4.01) is divided by about 6.4, an estimate of the pooled standard deviation. In this case, d is approximately .60, which is a medium to large effect size. For grades in high school, the difference between the means (-0.329) is divided by about 1.57, an estimate of the pooled standard deviation. In this case, d is approximately .21, which is a small to medium effect size. For the visualization test, the difference between the means (2.16) is divided by about 3.725, an estimate of the pooled standard deviation. In this case, d is approximately .58, which is a medium to large effect size.D8.9.3: Mann-Whitney U testD8.9.3.a            The Mann-Whitney U test produces the average ranks for fast track and regular track students for each of the three dependent variables, with the group with the higher rank having the higher grades or test scores. For math achievement test, fast track students were statistically significantly different from regular track students, U = 455.5, p = .010, r = –.30. The effect size is small to medium. For grades in high school, fast track students were not statistically significantly different from regular track students, U = 621.5, p = .413, r = –.09. The effect size is small. For the visualization test, fast track students were statistically significantly different from regular track students, U = 505, p = .040, r = –.24. The effect size is small to medium.            The Mann–Whitney test is only slightly less powerful than the t test, so it is a good alternative if the assumptions of the t test are violated. For each variable, the p-value is slightly larger for the Mann-Whitney test than for the t-test.D8.9.3.b            The Mann-Whitney U test is appropriate when the assumption of normality is violated and is used in the case of a between-groups design having two levels for the independent variable. It is used as a non-parametric substitute for the independent samples t-test.D8.9.4: Paired Samples t-testD8.9.4.a            The correlation (r = .68) between mother’s and father’s education indicates that more educated men tend to marry more educated women and vice versa.D8.9.4.b            A paired samples t-test indicated that students’ fathers tend to have significantly more education than their mothers, t(72) = 2.396, p = .019, d = .28. The effect size of the difference is small to medium.D8.9.4.c            The correlation between mother’s and father’s education indicates that there is a significant relationship between the variables such that educated men tend to marry educated women and vice versa. The paired samples t-test determines if there is a difference in the education levels of the mothers and fathers.D8.9.4.d            A correlation coefficient equal to 0.9 would indicate a strong relationship between mother’s and father’s education, meaning that educated men marry educated women at almost every opportunity. If t = 0, the test would indicate that the difference between father’s education and mother’s education is not statistically different from zero, meaning on average their education levels are the same.D8.9.4.e            A correlation coefficient equal to zero would indicate no relationship between mother’s and father’s education, meaning that education has no influence. If t = 0.9, the test would indicate that the difference between father’s education and mother’s education is statistically significantly different from zero, meaning that on average fathers are more highly educated than mothers.D8.9.5: Wilcoxon Signed Ranks TestD8.9.5.a:            The Wilcoxon Signed Ranks test was used to evaluate the difference between mother’s and father’s education levels. Of the 73 students, 27 fathers had more education than the mother, 21 mothers had more education than the father, 25 were tied. The difference indicates that fathers have statistically significantly more education than mothers on average, Z = 2.09, p = .037, r = -.24, which is a small to medium effect size. There was not a statistically significant difference between the visualization test and visualization 2 test, Z = -.373, p = .709, r = -.04, which is a small effect size.            The Wilcoxon Signed Ranks test is only slightly less powerful than the paired t-test, so it is a good alternative if the assumptions of the t-test are violated. For the difference between mother’s and father’s education, the p-value is slightly larger for the Wilcoxon test (p = .037) relative to the paired t-test (p = .019), but both tests found the difference to be statistically significantly different.D8.9.5.b:            The Wilcoxon Signed Ranks test is used in cases where one or more paired variables is not normally distributed. It is used as a nonparametric substitute for the paired samples t-test.ReferencesMorgan, G. A., Barrett, K. C., Leech, N. L., & Gloeckner, G. W. (2020). Ibm spss for introductory statistics: Use and interpretation (6th ed.). Routledge. https://doi.org/10.4324/9780429287657