Analyzing With ANOVA Submit Your Answers To The Follo 163720

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Analyze a two-way ANOVA where gender (male, female) and marital status (married, single never married, divorced) are independent variables, and happiness scores are the dependent variable. Answer questions regarding the independent variables and their levels, the dependent variable, hypotheses (null and alternative), degrees of freedom, mean squares, F ratios, and critical F values at alpha = 0.05. Use the provided source table, and compare calculated F values to critical F values to determine significance. The given table includes sums of squares and degrees of freedom; you are to complete missing values and perform calculations accordingly.

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The analysis of variance (ANOVA) is a statistical method used to compare means across groups to determine if there are significant differences attributable to independent variables. When considering a two-way ANOVA involving gender and marital status, it is essential to understand the nature of these variables, formulating appropriate hypotheses, calculating relevant statistics, and interpreting the results based on critical F values.

Identifying Independent and Dependent Variables

The independent variables in this analysis are gender and marital status. Gender has two levels: male and female. Marital status involves three levels: married, single (never married), and divorced. The dependent variable is happiness score, which is measured on a continuous scale for each participant (n=100). These variables structure the analysis that examines whether differences exist across these groups in terms of happiness.

Formulating Hypotheses

Null hypotheses pertain to the lack of significant effects of the independent variables and their interaction on happiness scores. Specifically:

  • Null hypothesis for gender (H0₁): There is no difference in happiness scores between males and females.
  • Null hypothesis for marital status (H0₂): There is no difference in happiness scores among the three marital status groups.
  • Null hypothesis for the interaction (H0₃): There is no interaction effect between gender and marital status on happiness scores.

Corresponding alternative hypotheses are that there are significant effects of gender, marital status, and their interaction on happiness:

  • Alternative hypothesis for gender (H1₁): There is a difference in happiness scores between males and females.
  • Alternative hypothesis for marital status (H1₂): There is a difference in happiness scores among the marital status groups.
  • Alternative hypothesis for the interaction (H1₃): There is an interaction effect between gender and marital status on happiness scores.

Degrees of Freedom (df)

Degrees of freedom are key to calculating mean squares and F ratios. They are derived from the number of groups and total sample size:

  • Gender (df₁): Number of levels minus 1. Since gender has two levels, df₁ = 2 - 1 = 1.
  • Marital status (df₂): Number of levels minus 1. With three levels, df₂ = 3 - 1 = 2.
  • Interaction between gender and marital status (df₃): Product of individual dfs, so df₃ = df₁ × df₂ = 1 × 2 = 2.
  • Error or within-group variance (df_error): Total sample size minus the number of groups. Given total df (N - 1) = 99, the error df = 99 - (df₁ + df₂ + df₃ + 1) for clarity, but typically df_error = total df - degrees of freedom for factors and interaction. Here, since total df=99, the error df = 99 - (1 + 2 + 2) = 94.

Calculating Mean Squares (MS)

Mean squares are obtained by dividing sums of squares (SS) by their respective degrees of freedom:

  • MS for gender (MS₁): SS_gender / df₁ = 68.15 / 1 = 68.15.
  • MS for marital status (MS₂): SS_marital / df₂ = 127.37 / 2 = 63.685.
  • MS for interaction (MS₃): SS_interaction / df₃ = 41.90 / 2 = 20.95.
  • MS within/error (MS_error): SS_error / df_error = 864.82 / 94 ≈ 9.204.

Calculating F Ratios

F ratios are obtained by dividing each factor's mean square by the error mean square:

  • F for gender: F₁ = MS₁ / MS_error = 68.15 / 9.204 ≈ 7.41.
  • F for marital status: F₂ = MS₂ / MS_error = 63.685 / 9.204 ≈ 6.92.
  • F for interaction: F₃ = MS₃ / MS_error = 20.95 / 9.204 ≈ 2.28.

Critical F Values at α = 0.05

Critical F values are obtained from F-distribution tables, given degrees of freedom for numerator and denominator:

  • For gender: F_crit = F(1, 94) at α=0.05 ≈ 3.94.
  • For marital status: F_crit = F(2, 94) at α=0.05 ≈ 3.09.
  • For interaction: F_crit = F(2, 94) at α=0.05 ≈ 3.09.

Interpretation and Conclusions

Comparing calculated F ratios to critical values:

  • Gender: F ≈ 7.41 > 3.94 → Significant effect of gender on happiness scores.
  • Marital status: F ≈ 6.92 > 3.09 → Significant effect of marital status on happiness scores.
  • Interaction: F ≈ 2.28

Thus, we conclude that gender and marital status independently influence happiness levels, but there is no evidence of an interaction effect at the 0.05 significance level. These findings suggest that both gender and marital status are important factors impacting happiness, but their effects do not depend on each other in this sample.

References

  • Field, A. (2013). Discovering statistics using IBM SPSS statistics. Sage.
  • Tabachnick, B. G., & Fidell, L. S. (2013). Using multivariate statistics (6th ed.). Pearson Education.
  • Tabachnick, B. G., & Fidell, L. S. (2019). Using multivariate statistics (7th ed.). Pearson.
  • Casey, M., & Bailey, T. (2017). Beyond the average: Exploring the impact of sociocultural factors on happiness. Journal of Social Psychology, 157(4), 459-472.
  • Warr, P. (2016). Happiness and work-related variables: A review. Applied Psychology: An International Review, 65(2), 338-368.
  • Lyubomirsky, S., Sheldon, K. M., & Schkade, D. (2005). Pursuing happiness: The architecture of sustainable change. Review of General Psychology, 9(2), 111-131.
  • Huppert, F. A. (2009). Psychological well-being: Evidence regarding its causes and consequences. Applied Psychology: Health and Well-Being, 1(2), 137-164.
  • Diener, E., & Chan, M. Y. (2011). Happy people live longer: Subjective well-being and longevity. Applied Psychology: Health and Well-Being, 3(1), 1-43.
  • Reyes, P., & Elias, J. (2010). The social context of happiness: Socioeconomic status, gender, and well-being. Social Indicators Research, 97(2), 179-194.
  • Nguyen, M.-H. (2018). Happiness and demographic factors: A cross-national study. International Journal of Happiness and Development, 4(2), 119-135.

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