Analyzing With ANOVA: Submit Your Answers Below

Analyzing with ANOVA: Submit your answers to the following

Analyze the provided two-way ANOVA data involving gender and marital status as independent variables, and happiness scores as the dependent variable. Identify the variables and their levels, state all null and alternative hypotheses, compute degrees of freedom, mean squares, F ratios, and compare to critical F values at alpha = .05 to determine significance. Present findings in a clear, organized, and academically rigorous manner.

Paper For Above instruction

The purpose of this analysis is to examine how gender and marital status influence happiness scores among individuals, using a two-way ANOVA approach. The independent variables in this study 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 is the happiness score, which quantifies individuals’ perceptions of happiness on a numerical scale. Understanding the effects of these variables and their interaction can provide valuable insights into the determinants of happiness across different demographic groups.

Null and Alternative Hypotheses

For each independent variable, the null hypotheses are: (1) gender has no effect on happiness scores, (2) marital status has no effect on happiness scores, and (3) the interaction between gender and marital status does not influence happiness scores. Correspondingly, the alternative hypotheses posit that each of these effects does influence happiness: (1) there is a significant difference in happiness scores between males and females, (2) happiness scores vary significantly across marital status groups, and (3) there is a significant interaction effect between gender and marital status on happiness scores.

Degrees of Freedom Calculation

The total degrees of freedom (df) is 99, derived from the total number of observations minus 1 (n - 1). The degrees of freedom for each factor are calculated as follows:

  • Gender: Since there are two levels (male, female), df = 2 - 1 = 1.
  • Marital Status: With three groups (married, single, divorced), df = 3 - 1 = 2.
  • Interaction between Gender and Marital Status: df = (levels of gender - 1) (levels of marital status - 1) = 1 2 = 2.
  • Within (Error): Calculated as total df minus the sum of the above, df = 99 - (1 + 2 + 2) = 94.

Mean Square Calculations

Mean squares (MS) are obtained by dividing the sum of squares (SS) by their respective degrees of freedom:

  • Gender: MS = SS / df = 68.15 / 1 = 68.15.
  • Marital Status: MS = 127.37 / 2 = 63.685.
  • Interaction (Gender * Marital Status): MS = 41.90 / 2 = 20.95.
  • Error (Within): MS = 864.82 / 94 ≈ 9.208.

F Ratio Calculations

The F ratios are given by dividing the mean square of each factor by the mean square within (error):

  • Gender: F = MS_gender / MS_within = 68.15 / 9.208 ≈ 7.41.
  • Marital Status: F = 63.685 / 9.208 ≈ 6.92.
  • Interaction: F = 20.95 / 9.208 ≈ 2.28.

Critical F Values and Significance Testing

Using an alpha level of .05, the critical F values for the respective degrees of freedom from F-distribution tables are approximately:

  • Gender (df1=1, df2=94): F_crit ≈ 3.94.
  • Marital Status (df1=2, df2=94): F_crit ≈ 3.09.
  • Interaction (df1=2, df2=94): F_crit ≈ 3.09.

Comparing the obtained F ratios with the critical values:

  • Gender: 7.41 > 3.94 → significant effect.
  • Marital Status: 6.92 > 3.09 → significant effect.
  • Interaction: 2.28

Conclusions

Based on the F ratios and critical values, we reject the null hypotheses for gender and marital status, indicating that both variables significantly influence happiness scores. Specifically, there are notable differences in happiness between males and females, and among different marital status groups. However, the interaction effect between gender and marital status does not reach statistical significance at the 0.05 level, suggesting that the combined influence of these factors on happiness is not statistically complex within this data set. These results highlight that demographic factors such as gender and marital status independently relate to happiness, but their interaction does not add further explanatory power in this context.

References

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