Analyzing With ANOVA: Submit Your Answers To The Following

Analyzing with ANOVA: Submit your answers to the following

Analyze the provided two-way ANOVA table involving gender and marital status in relation to happiness scores. Identify the independent and dependent variables, formulate the null and alternative hypotheses, calculate the degrees of freedom, mean squares, and F ratios for each factor and their interaction, determine critical F values at α = 0.05, and interpret whether the results are statistically significant based on these values.

Paper For Above instruction

Analysis of variance (ANOVA) is a crucial statistical method employed to examine the differences among group means and determine whether any of those differences are statistically significant. In this context, a two-way ANOVA explores the effects of two factors—gender and marital status—on happiness scores and investigates potential interactions between those factors. This analysis provides insights into how gender and marital status independently and jointly influence happiness, an important construct in social and behavioral sciences.

Identification of Variables

The independent variables in this analysis are gender and marital status. These are categorical variables with two levels for gender (male and female) and three levels for marital status (married, single never married, divorced). The dependent variable is happiness scores, a continuous measure assigned to each individual in the study.

Formulation of Hypotheses

The null hypotheses (H₀) assert that there are no differences in happiness scores across the levels of each factor and their interaction:

• H₀ (Gender): The means of happiness scores are equal for males and females.
• H₀ (Marital Status): The means of happiness scores are equal across the three marital status groups.
• H₀ (Interaction): There is no interaction effect between gender and marital status on happiness scores.

The alternative hypotheses (H₁) posit that there are differences:

• H₁ (Gender): The means of happiness scores differ between males and females.
• H₁ (Marital Status): The means of happiness scores differ among the marital status groups.
• H₁ (Interaction): The effect of gender on happiness scores depends on marital status, indicating an interaction effect.

Calculations of Degrees of Freedom (df)

From the ANOVA table, the degrees of freedom (df) for each source are calculated as follows:

• Gender: Since gender has 2 groups, df = number of groups - 1 = 2 - 1 = 1.
• Marital Status: With 3 groups, df = 3 - 1 = 2.
• Interaction between gender and marital status: df = (number of levels for gender - 1) x (number of levels for marital status - 1) = 1 x 2 = 2.
• Error (Within): The total df = total observations - 1 = 99; thus, error df = total df - df for gender - df for marital status - df for interaction = 99 - 1 - 2 - 2 = 94.

Mean Square Calculations

Mean squares (MS) for each factor are obtained by dividing the sum of squares (SS) by their respective df:

• MS (Gender) = SS (Gender) / df (Gender) = 68.15 / 1 = 68.15.
• MS (Marital Status) = SS (Marital Status) / df (Marital Status) = 127.37 / 2 = 63.685.
• MS (Interaction) = SS (Interaction) / df (Interaction) = 41.90 / 2 = 20.95.
• MS (Error) = SS (Within) / df (Within) = 864.82 / 94 ≈ 9.204.

F Ratio Calculations

F ratios are calculated as the ratio of the mean square of each factor to the mean square error:

• F (Gender) = MS (Gender) / MS (Error) = 68.15 / 9.204 ≈ 7.41.
• F (Marital Status) = MS (Marital Status) / MS (Error) = 63.685 / 9.204 ≈ 6.92.
• F (Interaction) = MS (Interaction) / MS (Error) = 20.95 / 9.204 ≈ 2.28.

Critical F Values at α = 0.05

Using an F distribution table or calculator, the critical F values are approximately:

• For df (gender) = 1, df (error) = 94: F crit ≈ 3.94.
• For df (marital status) = 2, df (error) = 94: F crit ≈ 3.09.
• For df (interaction) = 2, df (error) = 94: F crit ≈ 3.09.

Interpretation of Results

Comparing the calculated F values with the critical F values:

• Gender: F_obt = 7.41 > 3.94 (F_crit), indicating a statistically significant effect of gender on happiness scores at α = 0.05. Therefore, we reject H₀ and conclude that males and females differ significantly in happiness scores.
• Marital Status: F_obt = 6.92 > 3.09 (F_crit), indicating a significant effect of marital status. We reject H₀ for marital status and conclude that happiness scores differ among married, single, and divorced groups.
• Interaction between Gender and Marital Status: F_obt = 2.28

Conclusion

The analysis indicates that both gender and marital status independently influence happiness scores significantly. Men and women differ in their happiness levels, and these differences are also evident across different marital statuses. However, the interaction effect is not statistically significant, implying that the influence of gender on happiness is consistent across marital status groups. These findings have meaningful implications for social scientists and policymakers aiming to understand determinants of happiness.

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