Reply from Jediah Logiodice
Week 07 Discussion 01: Comparing Groups Part 1
Jediah Logiodice
Liberty University, School of Business
BUSI820: Quantitative Research Methods
July 1, 2026
Author Note
Jediah Logiodice
I have no known conflict of interest to disclose.
Correspondence concerning this article should be addressed to Jediah Logiodice.
Email: [email protected]
D7.9.1 Information on the One-Sample t Test
The following questions are answered using the hsbdataNEW.sav file provided by Barrett et al. (2025) for IBM SPSS for introductory statistics: Use and interpretation.
D7.9.1.a Conditions for One-Sample t Test
This week’s reading discusses three types of t tests: the one-sample t test, the independent-samples t test, and the paired-samples t test (Morgan et al., 2020). The one-sample t test is used when a researcher has a hypothesized mean and wants to compare it with the sample mean. The test is designed to determine whether the difference between the sample mean and a hypothesized mean exceeds what would be explained by sampling error alone. According to Morgan et al. (2020), the one-sample t test has two primary assumptions: (a) the dependent variable “is normally distributed within the population” (p. 177), and (b) the case values being assessed are independent of each other.
D7.9.1.b Examples of One-Sample t Test
An example of a one-sample t test would be: Is the mean motivation score statistically significantly different from the midpoint of 2.5? This question could be used to evaluate whether participants are, on average, more or less motivated than the midpoint of the range (i.e., [1, 4]).
D7.9.2 Interpreting Outputs 9.2
The following section provides answers related to Output 9.2 using the Morgan et al. (2020) sixth edition.
D7.9.2.a Variances for Math Achievement, Grades in High School, and Visualization Tests
For Output 9.2 in Morgan et al. (2020), Levene’s test for equality of variances across academic track groups and the dependent variables of math achievement test, F(1, 73) = .54, p = .466, and high school grades, F(1, 73) = .57, p = .451, were both nonsignificant, meaning the assumption of equal variances held true. Levene’s test was significant for academic track and visualization test scores, F(1, 73) = 6.51, p = .013, meaning the assumption of equal variances was not met.
D7.9.2.b Listing the Appropriate t Test Statistics
The following lists the appropriate t, df, and p (sig.) for each t-test as they would be written inside of an article:
- Math achievement test: t(73) = 2.70, p = .009
- Visualization test scores: t(57.2) = 2.39, p = .020
- High school grades: t(73) = -0.90, p = .369
D7.9.2.c Identifying Statistically Significant t Tests
For the independent-samples t test on academic track, the differences between academic track and math achievement test scores were statistically significant (p = .009), the differences for visualization test scores were statistically significant (p = .020), while the differences for high school grades were not statistically significant (p = .369).
D7.9.2.d Academic Track Difference Between Visualization and Grades in High School
For the independent-samples t test, the visualization test scores for fast-track participants were statistically significantly different from those of regular-track participants, t(57.2) = 2.39, p = .020, with fast-track participants scoring on average higher (M = 6.43) than regular-track participants (M = 4.26), with a medium (Cohen, 1988) effect size, d = .57. In contrast, fast-track participants were similar (M = 5.50) to regular-track participants (M = 5.83) on high school grades; the differences were not statistically significant, t(73) = -0.90, p = .369, and represented a small (Cohen, 1988) effect size, d = -.21.
D7.9.2.e Interpreting the Confidence Intervals
The 95% confidence interval provides a range of likely values for the true mean difference, such that, if an interval were computed for many repeat samplings, 95% of the time the interval would contain the true mean difference (Morgan et al., 2020). For visualization test scores, the confidence interval [0.35, 3.98] excludes 0, indicating a statistically significant difference (consistent with p = .020), and provides 95% confidence that fast-track participants would score 0.35 to 3.98 points higher. For the high school grades, the confidence interval [-1.06, 0.40] includes 0, indicating that the true mean difference could be 0 (i.e., no difference), which aligns with the non-significant findings (p = .369).
D7.9.2.f Describing the Effect Size
The differences between fast-track and regular-track participants on math achievement scores had a positive, medium (Cohen, 1988) effect size, d = .62, indicating that participants who took the fast track scored higher on average. The differences between fast-track and regular-track participants on visualization test scores also had a positive, medium (Cohen, 1988) effect size, d = .57, indicating that individuals who took the fast track scored higher on average. The differences in high school grades were non-significant; therefore, the direction of the effect size should not be interpreted (Morgan et al., 2020).
D7.9.3 Comparing Outputs 9.2 and 9.3
Output 9.2 presents an independent-samples t test while Output 9.3 presents a Mann-Whitney U test, both used to determine whether fast- or regular-track participants differ in their visualization, math achievement, and grades. This section compares the two outputs and discusses when to use the Mann-Whitney U test.
D7.9.3.a Compare the Results of Outputs 9.2 and 9.3
Both Output 9.2 and Output 9.3 yielded similar outcomes: the differences in math achievement and visualization test scores were statistically significant. The t test, which measures differences in means, showed that fast-track participants had higher average math scores (M = 14.76 vs. M = 10.75) and higher average visualization test scores (M = 6.43 vs. M = 4.26) than regular-track participants. The M-W U test, which compares ranked orders, showed that fast-track participants typically ranked higher in math scores (MR = 45.10 vs. MR = 32.11) and visualization test scores (MR = 43.65 vs. MR = 33.32) than regular-track participants. Both findings showed medium effect sizes. Similarly, both statistical tests indicated that school grades did not differ significantly between the two academic tracks. One other item of note: while both the independent-samples t and the M-W U used a 2-sided significance test, the M-W U is referred to as being asymptotic, meaning the p-value was estimated rather than calculated (Morgan et al., 2020).
D7.9.3.b M-W Compared to t Test
According to Morgan et al. (2020), the primary difference between the independent-samples t test and the Mann-Whitney U test is the treatment related to the dependent variable: normal/scale variables that meet the required assumptions (such as normality) use the independent-samples t test, while ordinal variables, or variables that markedly violate the parametric assumptions, use the M-W U test. As a result, each test reports different outputs: the independent-samples t test centers on differences in means and standard deviations, while the M-W U centers on differences in mean ranks and interquartile ranges (Morgan et al., 2020).
D7.9.4 Reviewing Paired Samples t Test
The following section answers questions related to Output 9.4 from Morgan et al. (2020).
D7.9.4.a Describe the Paired Samples Meaning
The paired samples in Output 9.4 between mothers' and fathers' education are a within-subjects test to determine if there is any statistically significant relationship between a participant’s parent’s education level, e.g., do the variables tend to move together?
D7.9.4.b Explain Output 9.4 Results
The paired-samples t test indicated that the participants’ fathers had, on average, significantly more education than their mothers, t(72) = 2.40, p = .019, d = .28. The effect size was categorized as small (Cohen, 1988).
D7.9.4.c Correlation Versus t Test Outcomes
The correlation (r = .68) describes the strength and direction of the relationship between two variables, i.e., father and mother’s education. The Pearson correlation of r = .68 indicates that participants with highly educated fathers tended to also have highly educated mothers, and vice versa. However, the correlation does not indicate which variable is higher than the other; it only shows the extent to which the two variables move together. The paired-samples t test evaluates whether the average difference between the variables is statistically significant. If, as in this case, the researcher knows which variable was compared to the other, the sign of the t value (e.g., negative or positive) can also provide the direction of the difference.
D7.9.4.d Checking Different r and t Outputs
The result r = .90 would be an even stronger indication that the education levels of the participant’s father and mother moved together; t = 0 would indicate that there was no average difference in the education levels between the two parents.
D7.9.4.e Checking Another Different r and t Outputs
An r = 0 would indicate no linear relationship found between the father's and mother’s education levels; t = 5.0 would indicate that the average was significantly different between the father's and mother’s education levels, assuming the p-value was statistically significant.
D7.9.5 Comparing Parametric and Nonparametric Two Related Samples
The following section compares Output 9.4 (parametric paired-samples t) with Output 9.5 Wilcoxon test, using the outputs from Morgan et al. (2020).
D7.9.5.a Comparing the Results
Output 9.4 only calculates the difference between father and mother’s education, while 9.5 also calculates the differences between visualization and visualization 2 test scores. For comparison between father and mother’s education, the paired-samples t test is statistically significant, and indicates that fathers had, on average, more education than mothers, t(72) = 2.40, p = .019, d = .28, with a small effect size (Cohen, 1988). Similarly, Wilcoxon signed-rank tests indicated that, among 73 participants, fathers had more education (27) than mothers (21), while 25 had equal levels of education (z = -2.09, p = .037, r = -.24). The Wilcoxon effect size r = -.24 indicates a small effect size (Cohen, 1988).
D7.9.5.b Selecting the Right Statistic
According to Morgan et al. (2020), the paired-samples t test and the Wilcoxon test are both designed to assess differences in a dependent variable associated with a one-factor independent variable with two levels, categories, groups, or samples (within-subject). Additionally, the paired-samples t test is used when the dependent variable is scale-level and approximately normal, while the Wilcoxon test is used when the dependent variable is ordinal, or the assumptions of normality are markedly violated.
References
Barrett, K. C., Leech, N. L., Gloeckner, G. W., & Morgan, G. A. (2025). IBM SPSS for introductory statistics: Use and interpretation (7th ed.). Routledge. https://doi.org/10.4324/9781003355908Links to an external site.
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum Associates.
Morgan, G. A., Barrett, K. C., Leech, N. L., & Gloeckner, G. W. (2020). IBM SPSS for introductory statistics: Use and interpretation (6th ed.). Routledge.