RSCH FPX 7864 Assessment 3 ANOVA Application and Interpretation

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Capella university

RSCH-FPX 7864 Quantitative Design and Analysis

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Date

ANOVA Application and Interpretation

One-way Analysis of Variance (ANOVA) provides statistical evidence about mean variations in three or more unrelated groups (Rueda, 2023). Examining student performance differences (Quiz 3 scores) in various class sections will utilize ANOVA tools on the grades.jasp data set. The analysis will perform statistical assumption testing and interpretation of results while examining possible utilization areas from the findings.

Data Analysis Plan

The analysis takes place on two grade-related factors from grades.jasp dataset and involves Section and Quiz3 variables. The section indicates the diverse class sections, including Section 1, Section 2, Section 3, and Quiz 3, demonstrating students’ scores through their number of correct answers on Quiz 3. The ANOVA test determines if the different sections affect student outcomes during Quiz 3.

It analyzes three or more independent group means to identify significant differences between their values. ANOVA can accomplish its analysis by evaluating both internal variation and group differences. An important difference between group mean, and within-group variations indicate that at least one group contains a dissimilar mean. ANOVA generates the F-statistic and p-value, establishing this conclusion (Rueda, 2023).

Component	Statement
Research Question	Does the analysis show whether the Quiz 3 score results between different class sections have statistical significance?
Null Hypothesis (H₀)	A difference in mean scores on Quiz 3 across the class sections does not exist.
Alternative Hypothesis (H₁)	A minimum of one between the classes exhibits a meaningful distinction in their Quiz 3 score average.

 

Testing Assumptions

The p-value from Levene’s Test (0.060) exceeds the established alpha value of 0.05. We must maintain the null hypothesis about equal variances because the p-value measures 0.060. The requirement for homogeneous variances remains valid according to the results. The analysis should continue with standard one-way ANOVA because the results demonstrate that such homogeneity corrections are unnecessary. Here is the explanation of the analysis: 

Levene’s Test evaluated whether the different class sections had equal variance distributions because this condition is vital for conducting ANOVA analysis (Augusto, 2024). The F-value from the Test was 2.898 at df1=2 while df2=102, and the p-value reached 0.060. The obtained p-value exceeds 0.05, which preserves the null hypothesis about equal variances across groups. Standard ANOVA should be used because the condition of homogeneous variances has been confirmed. The small gap from the threshold allows this result to confirm variable equality.

When the requirement p-value exceeds the analysis, it must shift to Welch’s ANOVA as an alternative testing methodology. The study should also consider two more assumptions: normality validation through Q-Q plots or the Shapiro-Wilk test and observation independence, which should be confirmed during data collection. The success of ANOVA analysis depends on both normal data distribution across groups and independent observations since breaking these conditions may produce research results that are inaccurate or misleading. Statistical reliability and validity remain strong when these criteria are properly satisfied and maintained. The study results show sufficient merit for regular ANOVA analysis when equal variable assumptions prove accurate.

Results & Interpretation

The “Descriptives – quiz3” table provides summary statistics for the quiz3 scores across three different sections (labeled 1, 2, and 3). For each section, we get the following metrics:

• N: Number of students in each section.
• Mean: Average score on quiz 3 for that section.
• SD: Standard deviation
• SE: Standard error (smaller SE means a more precise estimate).
• Coefficient of Variation: A standardized measure of variability (SD divided by the mean) showing relative spread.

Section 1: Mean = 7.273, SD = 1.153

Section 2: Mean = 6.333, SD = 1.611

Section 3: Mean = 7.939, SD = 1.560

Students in all sections took Quiz 3 according to the data in the table, which displays their performance results. Students achieved their maximum score of 7.939 in Section 3, Section 1 students obtained 7.273 points, and Section 2 students received the lowest score of 6.333 points. Data reveals that scores in Section 2 display wider dispersion (standard deviation of 1.611) than scores in Section 1 (standard deviation of 1.153).

Statistical analysis through CV indicates that Section 2 students demonstrated a maximum score variability of 0.254, while Section 1 showed a minimum relative variability of 0.159. Standard errors remain small, which verifies that the estimated means provide accurate measurements. The table provides the necessary background information to establish important data trends for additional analysis. The data contains these standard deviations (SD) and means, which we will utilize according to your instructions later.

ANOVA analyzes section mean differences (between-group variance) about score diversity within each section (within-group variance). A large between-group variance compared to within-group variance indicates that the group means differ substantially (Rueda, 2023).

RSCH FPX 7864 Assessment 3 ANOVA Application and Interpretation

The table includes: 

• The Sum of Squares value shows the data variability that results from section differences and random variations across both groups and sections. The table uses Type III Sum of Squares, a method for partitioning variance in ANOVA often used when groups have unequal sizes (Mabuie, 2023).
• The number of groups and sample size determine the degrees of freedom for the analysis. 
• The calculation of the F-statistic requires a Mean Square because it represents the Sum of Squares divided by df. 
• ANOVA calculates the test statistic by dividing the Mean Square (sections) by the Mean Square (residuals). 
• We can determine the probability of observing the result through the p-value because it tells us the likelihood of finding this outcome when groups lack genuine differences.

The calculated F-statistic value (10.951), together with its p-value (< 0.001), indicates significant differences among at least two of the sections regarding quiz3 scores. The p-value under 0.05 leads to rejection of the null hypothesis, which states that section means are equivalent. The sections demonstrate differences in their quiz three score averages because their average values do not match.

The instructions specify testing the homogeneity assumption condition, which requires equal variances between groups. SD from the descriptive table (1.153, 1.611, 1.560) shows no major variance differences, and the coefficients of variation (0.159, 0.254, 0.196) indicate average relative dispersion levels. Levene’s test is typically used to determine homogeneity violations. Still, the ANOVA results (as directed by instructions) show that homogeneity is intact, so “None” can be selected as the homogeneity correction (Augusto, 2024). We require post hoc tests after this significant ANOVA result to identify section differences since the next table appears.

The Post Hoc Tests – section provides data on Tukey post hoc test outcomes following a significant ANOVA to determine the group pair differences. The Tukey test examines every pair of group means despite factoring in total error rate calculations during the assessment process (family-wise error) (Nanda et al., 2021). The “Standard” version of the Tukey test appears in this table because the instruction indicated using this test variant when homogeneity assumptions do not cause violations. The table compares sections 1 vs. 2, 1 vs. 3, and 2 vs. 3.

RSCH FPX 7864 Assessment 3 ANOVA Application and Interpretation

• Mean Difference: The difference between the means of the two sections.
• SE: Standard error of the mean difference.
• t: The t-statistic for the comparison.
• pTukey: The adjusted p-value for the comparison was corrected for multiple tests (family-wise error for three comparisons).

The Tukey test provides detailed information about which sections differ from each other. It is essential to note that P-values are adjusted for comparing a family of 3 (since there are 3 pairwise comparisons: 1 vs. 2, 1 vs. 3, 2 vs. 3).

Section 1 vs. Section 2: A significant mean difference of 0.939 exists between Section 1 (7.273) and Section 2 (6.333), with a statistical significance value of 0.021. The statistical test indicates that this outcome is significant because p < 0.05. Section 1 (mean = 7.273) scores higher than Section 2 (mean = 6.333). 

Section 1 vs. Section 3: The mean difference amounts to -0.667 while maintaining a p-value of 0.159. The calculated p-value exceeds 0.05; therefore, the difference remains statistically insignificant. The scores from Section 1 (mean = 7.273) are equivalent to scores from Section 3 (mean = 7.939). 

Section 2 vs. Section 3: The relative mean difference amounts to -1.606 while maintaining a p-value of less than 0.001. The statistical difference is confirmed by the p-value, which is less than 0.05. Section 3 (mean = 7.939) scores higher than Section 2 (mean = 6.333). Section 2 shows significantly lower quiz three scores than all other sections, yet Section 1 and Section 3 maintain their non-significant differences.

Statistical Conclusions

Student achievement in quiz 3 demonstrated statistically important differences between sections based on ANOVA analysis as indicated by an F-statistic of 10.951, which produced a p-value less than 0.001, thus denying the null hypothesis of identical means among all sections. Section 3 produced the highest mean score of 7.939, accompanied by 1.560 SD, while Section 1 achieved 7.273 mean with 1.153 deviations, and Section 2 registered the lowest score of 6.333 and 1.611 deviations.

RSCH FPX 7864 Assessment 3 ANOVA Application and Interpretation

The Tukey post hoc analysis demonstrated that Section 2 students performed at a statistically lower level than both Section 1 (p = 0.021) and Section 3 (p < 0.001). Additionally, the performance levels of Sections 1 and 3 did not differ significantly (p = 0.159). The performance results tells that Section 2 is the weakest, even though Sections 1 and 3 display equivalent outcomes.

The ANOVA and Tukey tests offer strong evidence about mean differences yet present certain weaknesses. The results depend on two assumptions of normality and independence of observations that this study failed to verify explicitly because potential violations of these conditions could affect the outcome. The results of Levene’s test (p-value = 0.060) approach the significance threshold of 0.05, thus indicating possible heterogeneity of variances that should be addressed through Welch’s ANOVA to obtain more accurate outcomes (Celik, 2020).

ANOVA fails to consider student population traits, previous knowledge levels, or teaching performance as potential reasons explaining observed differences between groups. The results in Section 2 show lower scores because the instructor has less experience. At the same time, the educational content is harder to understand than in other sections, thus requiring additional analysis to determine their specific performance differences.

Application

The medical field, particularly oncology, benefits from using one-way ANOVA as a strong method to examine differences between patient outcomes in different treatment groups. The independent variable, “Type of Cancer Treatment,” includes three treatment groups: chemotherapy, immunotherapy, and radiation therapy. The reduction of tumors measured through percentage change in size represents a proper dependent measure. Analyzing tumor reduction rates according to different cancer treatment types is essential because it determines the best cancer reduction methods oncologists can use to enhance treatment effectiveness and patient survival outcomes. 

ANOVA enables numerous advantages when applied in this situation. ANOVA helps scientists determine substantial variations in tumor reduction numbers between treatment groups to discover optimal therapy options for individual cancer diseases. If immunotherapy generates substantial tumor reduction statistics beyond chemotherapy results, scientists will use it to analyze its basic processes for broader medical applications. ANOVA helps researchers analyze how additional elements, such as patient age and cancer stage, impact treatment results by connecting these elements to the treatment type. ANOVA helps develop personalized medicine approaches by identifying specific treatment differences to help patients obtain the most suitable therapy for their condition, thus maximizing therapeutic results and resource usage in medical practice (Chatzi & Doody, 2023).

References

Augusto, C. (2024). Levene’s test for verifying homoscedasticity between groups in quasi-experiments in social sciences. South Eastern European Journal of Public Health, 2119–2125. https://doi.org/10.70135/seejph.vi.2342 

Celik, N. (2020). Welch’s ANOVA: Heteroskedastic skew-t error terms. Communications in Statistics – Theory and Methods, 1–12. https://doi.org/10.1080/03610926.2020.1788084 

Chatzi, A., & Doody, O. (2023). The one-way ANOVA test explained. The One-Way ANOVA Test Explained, 31(3). https://doi.org/10.7748/nr.2023.e1885 

Mabuie, J. (2023). Alternatives test in ANOVA with unequal variance and unequal sample size. International Journal of Innovative Science and Research Technology, 8(11). https://ijisrt.com/assets/upload/files/IJISRT23NOV1335.pdf 

Nanda, A., Mohapatra, B. B., Mahapatra, A. P. K., Mahapatra, A. P. K., & Mahapatra, A. P. K. (2021). Multiple comparison test by Tukey’s honestly significant difference (HSD): Do the confident level control type I error. International Journal of Statistics and Applied Mathematics, 6(1), 59–65. https://doi.org/10.22271/maths.2021.v6.i1a.636 

RSCH FPX 7864 Assessment 3 ANOVA Application and Interpretation

Rueda, A. (2023). Analysis of variance. Elsevier EBooks, 157–160. https://doi.org/10.1016/b978-0-323-91259-4.00099-0