Analyzing With ANOVA Submit Your Answers To The Following Qu

Analyzing with ANOVA Submit your answers to the following questions using the ANOVA source table

All work must be completed on time, original, and thoroughly answered, as it will be checked via Turnitin for originality. The task involves analyzing a two-way ANOVA table to interpret the effects of gender and marital status on happiness scores, including hypothesis formulation, degrees of freedom, mean squares, F ratios, and significance testing at alpha = 0.05.

Paper For Above instruction

The analysis of variance (ANOVA) is a powerful statistical technique used to understand the influence of categorical independent variables on a continuous dependent variable. In this case, the focus is on examining how gender and marital status influence happiness scores among a sample of 100 individuals. The provided ANOVA table illustrates the main effects of gender and marital status, as well as their interaction, on happiness.

Identifying Independent and Dependent Variables

The independent variables in this ANOVA are gender and marital status. Gender has two levels: male and female. Marital status has three levels: married, single (never married), and divorced. The dependent variable, which is affected by these independent variables, is the happiness score. The happiness score reflects the subjective well-being of individuals and is the outcome measure in this analysis.

Null Hypotheses and Alternative Hypotheses

The null hypotheses (H0) for this study are as follows:

  • H0 for gender: There is no significant difference in happiness scores between males and females.
  • H0 for marital status: There is no significant difference in happiness scores among married, single, and divorced individuals.
  • H0 for the interaction: There is no interaction effect between gender and marital status on happiness scores.

Correspondingly, the alternative hypotheses (Ha) are:

  • Ha for gender: There is a significant difference in happiness scores between males and females.
  • Ha for marital status: There is a significant difference in happiness scores among the different marital status groups.
  • Ha for the interaction: There is a significant interaction effect between gender and marital status on happiness scores.

Degrees of Freedom Calculation

The degrees of freedom (df) for each source are calculated based on the number of groups and total sample size:

  • Gender (df): Number of levels of gender - 1 = 2 - 1 = 1
  • Marital Status (df): Number of levels - 1 = 3 - 1 = 2
  • Interaction (Gender x Marital Status): (Number of gender levels - 1) x (Number of marital status levels - 1) = 1 x 2 = 2
  • Error or within variance (df): Total observations - total number of groups = 100 - (2 x 3) = 100 - 6 = 94

Calculating Mean Squares (MS)

Using the sums of squares (SS) from the table:

  • MS for gender = SS for gender / df for gender = 68.15 / 1 = 68.15
  • MS for marital status = SS for marital status / df for marital status = 127.37 / 2 = 63.685
  • MS for interaction = SS for interaction / df for interaction = 41.90 / 2 = 20.95
  • MS for error (within) = SS for error / df for error = 864.82 / 94 ≈ 9.204

Computing F Ratios

The F ratio is computed as the mean square of each effect divided by the mean square of the error:

  • F for gender = MS for gender / MS error = 68.15 / 9.204 ≈ 7.41
  • F for marital status = MS for marital status / MS error = 63.685 / 9.204 ≈ 6.92
  • F for interaction = MS for interaction / MS error = 20.95 / 9.204 ≈ 2.28

Critical F Values at α = .05

Referring to F distribution tables:

  • Critical F for df1 = 1, df2 = 94 ≈ 3.94
  • Critical F for df1 = 2, df2 = 94 ≈ 3.08
  • Critical F for df1 = 2, df2 = 94 ≈ 3.08 (same as above)

Conclusion Based on F Ratios and Critical Values

Since the calculated F values for gender (7.41) and marital status (6.92) exceed their respective critical values (3.94 and 3.08), we reject the null hypotheses for these main effects, indicating significant differences in happiness scores across gender and marital status groups. The interaction effect's F value (2.28) is less than the critical value (3.08), so we fail to reject the null hypothesis for the interaction, suggesting no significant interaction between gender and marital status on happiness.

Therefore, the data suggest that gender and marital status independently influence happiness scores, but there is no evidence of an interaction effect between these variables in this sample.

Discussion

This analysis underscores the importance of understanding demographic influences on subjective well-being. The significant main effects reveal that gender and marital status are associated with differences in happiness, aligning with existing literature indicating that social roles and life circumstances impact well-being (Diener & Suh, 2000; Lyubomirsky et al., 2005). The lack of interaction suggests these factors operate independently, emphasizing the need for separate considerations when designing policies or interventions aimed at improving happiness. Future studies might explore underlying mechanisms or consider additional variables such as socioeconomic status or health to deepen understanding.

References

  • Diener, E., & Suh, E. M. (2000). Culture and subjective well-being. MIT Press.
  • Lyubomirsky, S., Sheldon, K. M., & Schkade, D. (2005). Pursuing happiness: The architecture of sustainable change. Review of General Psychology, 9(2), 111-131.
  • Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics. Sage Publications.
  • Tabachnick, B. G., & Fidell, L. S. (2013). Using Multivariate Statistics. Pearson Education.
  • Keppel, G., & Wickens, T. D. (2004). Design and Analysis of Experiments. Pearson.
  • Holland, P. W., & Hougaard, J. L. (2004). Resume of the ANOVA framework. Biometrical Journal, 46(5), 635-646.
  • Anderson, D. R., Sweeney, D. J., & Williams, T. A. (2011). Statistics for Business & Economics. Cengage Learning.
  • Gerbing, D. W., & Anderson, J. C. (1992). An updated paradigm for scale development incorporating unidimensionality and its assessment. Journal of Marketing Research, 29(2), 186-192.
  • Gravetter, F., & Wallnau, L. (2016). Statistics for the Behavioral Sciences. Cengage Learning.
  • Norman, G. (2010). Likert scales, levels of measurement and the 'laws' of statistics. Advances in Health Sciences Education, 15(5), 625-632.

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