In the thread for each short-answer discussion thestudent will post short answers to the prompted questions. The answers must demonstratecourse-related knowledge and support their assertions with scholarly citations in the latest APAformat. Minimum word count for all short answers cumulatively is 200 words.For each thread thestudent must include a title block with your name, class title, date, and the discussion forumnumber; write the question number and the question title as a level one heading (e.g. D1.1Variables) and then provide your response; use Level Two headings for multi part questions (e.g.D1.1 & D1.1.a, D1.1.b, etc.), and include a reference section.
Discussion Thread: Correlation and Regression
Respond to the following short answer questions in Chapter 8 from Morgan, Leech, Gloeckner, & Barrett textbook:
D6.8.1 Why would we graph scatterplots and regression lines?
D6.8.2 In Output 8.2, (a) What do the correlation coefficients tell us? (b) What is r2 for the Pearson correlation? What does it mean? (c) Compare the Pearson and Spearman correlations on both correlation size and significance level; (d) When should you use which type in this case?
D6.8.5 In Output 8.5, what do the standardized regression weights or coefficients tell you about the ability of the predictors to predict the dependent variable?
Reply:
The student must then post 1 reply to another student’s post. The reply must summarize thestudent’s findings and indicate areas of agreement, disagreement, and improvement. It must besupported with scholarly citations in the latest APA format and corresponding list of references.The minimum word count for Integrating Faith and Learning discussion reply is 250 words.
D.5.7.1.a Meaning of “Count” and “Expected Count”
When creating cross-tabulations, the count represents the actual frequency observed in each category, while the expected count is the number predicted if the two variables were independent. As Morgan et al. (2013) describe, expected values are calculated from the marginal totals and assume no relationship exists between the variables.
D.5.7.1.b Difference Between Them
A large gap between observed and expected counts signals that independence may not hold. Rather than leaving this as speculation, Morgan et al. (2013) emphasize that chi-square testing provides a formal method to determine whether the difference is statistically meaningful.
D.5.7.2 In Output 7.1
D.5.7.2.a Statistical Significance of Chi-Square
The reported Pearson chi-square was χ²(1, N = 75) = 3.65, p = .056. Since the p-value is greater than .05, the result is not significant. This aligns with Morgan et al. (2013), who explain that a nonsignificant chi-square indicates we cannot reject the null hypothesis and that any differences may have occurred by chance.
D.5.7.2.b Expected Values and Importance
The SPSS output confirmed that all expected frequencies were ≥ 5, meaning the assumption was satisfied. Harris (2021) explained that this condition matters because if too many cells fall below the threshold, the chi-square test may no longer approximate the theoretical distribution accurately, leading to unreliable results.
D.5.7.3 In Output 7.2
D.5.7.3.a Calculation of the risk ratio
The risk ratio divides the probability of an outcome in one group by the probability in another. In the SPSS output, students with low math grades had a 70% chance of not taking Algebra 2, compared with 45.7% for higher-achieving students. Morgan et al. (2013) note that the resulting ratio of 1.53 shows that students with lower grades were about one and a half times more likely to avoid Algebra 2.
D.5.7.3.b Calculation of the Odds Ratio
Whereas risk ratios focus on probability, the odds ratio compares the odds of an event between groups. In Output 7.2, the odds ratio was 2.77, meaning students with low math grades had nearly three times the odds of not taking Algebra 2 compared with students with higher grades. Harris (2021) highlighted that odds ratios are especially useful when analyzing dichotomous outcomes because they provide a standardized measure of association.
D.5.7.3.c Practical Importance of the Odds Ratio
By framing results in terms of “how much more likely” an outcome is, odds ratios allow findings to be communicated in practical terms. For example, Harris (2021) explained that when educators see that low-achieving students are almost three times more likely to skip Algebra 2, they can design targeted interventions for this population.
D.5.7.3.d Limitations of Odds Ratio
Although useful, odds ratios have limitations. Gnardellis et al. (2022) pointed out that when outcomes are very common, odds ratios can exaggerate associations. They also warn that ORs are less intuitive than risk ratios, which can cause misinterpretation when findings are shared outside of statistical circles.
D.5.7.4 In Output 7.3
Most Appropriate Statistic
Since both father’s and mother’s education are ordinal variables with three levels, Kendall’s tau-b is the most appropriate statistic. Morgan et al. (2013) note that while phi is only valid for 2×2 tables and Cramer’s V treats data as nominal, tau-b takes ordering into account and therefore fits best in this context.
Interpretation of Results
The reported tau-b of .572, with p < .001, indicates a strong, positive, statistically significant association. Morgan et al. (2013) explain that such a result shows higher father’s education is closely related to higher mother’s education.
Why Tau-b and Cramer’s V Differ
Tau-b incorporates the ordinal nature of the variables, while Cramer’s V measures only nominal association. As a result, the values differ, with tau-b offering a more nuanced interpretation of ordered relationships.
D.5.7.5 In Output 7.4
D.5.7.5.a Appropriate Value of Eta
The eta statistic of .328 was reported because one variable (academic track) is nominal and the other (number of math courses taken) is scale-level. Morgan et al. (2013) emphasize that eta is the correct statistic for measuring associations when combining nominal and continuous data.
D.5.7.5.b High or Low Value
With an eta of .328, the association falls in the medium-to-large range. Harris (2021) interpreted such values as practically important, meaning the relationship between track placement and number of math courses is meaningful in real-world educational settings.
D.5.7.5.c Description of Results
The eta value translates to an eta-squared of about .11, indicating that academic track explains roughly 11% of the variance in math courses taken. According to Morgan et al. (2013), this suggests that students in the fast track were substantially more likely to complete additional math courses, a result that carries both statistical and practical significance.
References
Gnardellis, C., Notara, V., Papadakaki, M., Gialamas, V., & Chliaoutakis, J. (2022). Overestimation of relative risk and prevalence ratio: Misuse of Logistics Modeling. Diagnostics, 12(11), 1-10. to an external site.
Harris, J. K. (2021). Primer on binary logistic regression. Family Medicine and Community Health, 9(1), 1-7. to an external site.
Morgan, G. A., Leech, N. L., Gloeckner, G. W., & Barrett, K. C. (2013). IBM SPSS for introductory statistics: Use and interpretation (5th ed.). Routledge.
