Submit Your Answers To The Following Questions Using 074803

Submit Your Answers To The Following Questions Using the Anova Source

Submit your answers to the following questions using the ANOVA source table below. The table depicts a two-way ANOVA in which gender has two groups (male and female), marital status has three groups (married, single never married, divorced), and the means refer to happiness scores (n = 100): What are the independent variables and their levels? What is the dependent variable? State all null hypotheses associated with independent variables and their interaction? Also suggest alternate hypotheses?

What are the degrees of freedom for 1) gender, 2) marital status, 3) interaction between gender and marital status, and 4) error or within variance? Calculate the mean square for 1) gender, 2) marital status, 3) interaction between gender and marital status, and 4) error or within variance. Calculate the F ratio for 1) gender, 2) marital status, and 3) interaction between gender and marital status. Identify the critical Fs at alpha = .05 for 1) gender, 2) marital status, and 3) interaction between gender and marital status. If alpha is set at .05, what conclusions can you make?

Paper For Above instruction

The provided ANOVA source table pertains to a study investigating happiness scores, with gender and marital status as the independent variables. This analysis aims to understand if and how these variables influence happiness, based on the data derived from 100 participants. The study employs a two-way ANOVA, which assesses the main effects of each independent variable and the interaction effect between them.

Identification of Independent Variables and Dependent Variable

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 happiness score, which reflects the level of happiness reported by participants.

Null and Alternative Hypotheses

The null hypotheses for the ANOVA 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 the three marital status groups.
  • H0 for the interaction: There is no interaction effect between gender and marital status on happiness scores.

The alternative hypotheses are:

  • H1 for gender: There is a significant difference in happiness scores between males and females.
  • H1 for marital status: There is a significant difference among the different marital status groups in happiness scores.
  • H1 for the interaction: There is a significant interaction effect between gender and marital status on happiness scores.

Degrees of Freedom and Calculation of Mean Squares

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

  • Gender: df = k - 1 = 2 - 1 = 1
  • Marital Status: df = m - 1 = 3 - 1 = 2
  • Interaction (Gender Marital Status): df = (k - 1) (m - 1) = 1 * 2 = 2
  • Error (Within): df = N - (k m) = 100 - (2 3) = 94

Using the sums of squares (SS) provided:

- Gender SS = 68.15

- Marital Status SS = 127.37

- Interaction SS = 41.90

- Error SS = 864.82

Total SS = 1102.24

Calculating the mean squares (MS) by dividing each SS by its associated df:

MS_Gender = 68.15 / 1 = 68.15


MS_Marital Status = 127.37 / 2 = 63.685


MS_Interaction = 41.90 / 2 = 20.95


MS_Error = 864.82 / 94 ≈ 9.204

F Ratios and Critical Values

The F ratios are computed by dividing each MS by the MS Error:

F_Gender = 68.15 / 9.204 ≈ 7.41


F_Marital_Status = 63.685 / 9.204 ≈ 6.92


F_Interaction = 20.95 / 9.204 ≈ 2.28

To determine whether these F values are statistically significant, we compare them to the critical F values at α = 0.05 with the respective degrees of freedom:

- For gender (df1=1, df2=94): F_crit ≈ 3.94

- For marital status (df1=2, df2=94): F_crit ≈ 3.09

- For interaction (df1=2, df2=94): F_crit ≈ 3.09

Comparing the calculated F values to the critical values:

- Gender: 7.41 > 3.94 → significant

- Marital Status: 6.92 > 3.09 → significant

- Interaction: 2.28

Conclusions

Given the results, we reject the null hypotheses for gender and marital status, suggesting that both independently influence happiness scores significantly. However, we fail to reject the null hypothesis for the interaction effect, indicating no significant interaction between gender and marital status in predicting happiness.

These findings imply that males and females differ in their happiness levels, and marital status also impacts happiness. Nevertheless, the effect of marital status on happiness does not vary significantly across genders.

References

  • Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics. Sage.
  • Tabachnick, B. G., & Fidell, L. S. (2013). Using Multivariate Statistics. Pearson.
  • Keppel, G., & Wickens, T. D. (2004). Design and Analysis: A Researcher’s Handbook. Pearson.
  • Laerd Statistics. (2017). One-way ANOVA using SPSS Statistics. https://statistics.laerd.com/spss-tutorials/one-way-anova-with-spss.php
  • Gelman, A., & Hill, J. (2007). Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press.
  • Gravetter, F. J., & Wallnau, L. B. (2017). Statistics for the Behavioral Sciences. Cengage Learning.
  • Huck, S. W. (2012). Reading Statistics and Research. Pearson.
  • Kirk, R. E. (2013). Experimental Design: Procedures for the Behavioral Sciences. Sage Publications.
  • Meyers, L. S., Gamst, G., & Guarino, A. J. (2013). Applied Multivariate Research: Design and Interpretation. Sage.
  • Stevens, J. P. (2009). Applied Multivariate Statistics for the Social Sciences. Routledge.

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