Submit Your Answers To The Following Questions Using The Ano

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): a. What are the independent variables and their levels? What is the dependent variable? b. State all null hypotheses associated with independent variables and their interaction. Also suggest alternative hypotheses. c. 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? d. Calculate the mean square for 1) gender, 2) marital status, 3) interaction between gender and marital status, and 4) error or within variance. e. Calculate the F ratio for 1) gender, 2) marital status, and 3) interaction between gender and marital status. f. Identify the critical Fs at alpha = .05 for 1) gender, 2) marital status, and 3) interaction between gender and marital status. g. If alpha is set at .05, what conclusions can you make?

Paper For Above instruction

The given problem pertains to a two-way Analysis of Variance (ANOVA) examining the impact of gender and marital status on happiness scores. This research explores whether these categorical independent variables influence the continuous dependent variable—happiness scores—as well as whether their interaction produces significant effects. To systematically analyze these, we need to address each question step by step.

Identification of Variables and Their Levels

The independent variables in this study are:

  • Gender: with two levels—male and female.
  • Marital Status: with three levels—married, single never married, divorced.

The dependent variable is the happiness score, which is a continuous measurement obtained from the participants.

Null and Alternative Hypotheses

For ANOVA, hypotheses are formulated as follows:

  • Null Hypotheses (H0):
  • H0 for gender: There is no difference in happiness scores between males and females.
  • H0 for marital status: There is no difference in happiness scores among the different marital status groups.
  • H0 for interaction: There is no interaction effect between gender and marital status on happiness scores.
  • Alternative Hypotheses (Ha):
  • Ha for gender: There is a significant difference in happiness scores between males and females.
  • Ha for marital status: There are significant differences in happiness scores among the different marital status groups.
  • Ha for interaction: There is a significant interaction effect between gender and marital status on happiness scores.

Degrees of Freedom (df)

Calculations for degrees of freedom are based on the number of groups and total sample size:

  • Gender: df = number of groups - 1 = 2 - 1 = 1
  • Marital Status: df = 3 - 1 = 2
  • Interaction (Gender Marital Status): df = (number of gender groups - 1) (number of marital status groups - 1) = 1 * 2 = 2
  • Error (Within): df = total observations - total number of groups = 100 - (2 * 3) = 100 - 6 = 94

Calculations of Mean Squares (MS)

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

  • 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 = SS for error / df for error = 864.82 / 94 ≈ 9.204

F Ratios Calculation

F ratios are computed by dividing the mean square of each factor or interaction by the mean square of error (within variance):

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

Critical F Values at Alpha = .05

Critical F values are obtained from F distribution tables at alpha = 0.05, with degrees of freedom:

  • For gender (df1=1, df2=94): approximately 3.94
  • For marital status (df1=2, df2=94): approximately 3.07
  • For interaction (df1=2, df2=94): approximately 3.07

Conclusions

Comparing the calculated F ratios to the critical F values:

  • Gender: F ≈ 7.41 > 3.94; thus, we reject H0, indicating a significant effect of gender on happiness scores.
  • Marital Status: F ≈ 6.92 > 3.07; we reject H0, suggesting marital status significantly influences happiness.
  • Interaction: F ≈ 2.28

In conclusion, both gender and marital status significantly affect happiness scores, while their interaction does not appear to have a statistically significant impact at the 0.05 significance level. These findings suggest that interventions aimed at improving happiness may need to consider gender and marital status independently but not necessarily their combined effects.

References

  • Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics. Sage Publications.
  • Tabachnick, B. G., & Fidell, L. S. (2013). Using Multivariate Statistics (6th ed.). Pearson.
  • Hinkle, D. E., Wiersma, W., & Jurs, S. G. (2003). Applied Statistics for the Behavioral Sciences. Houghton Mifflin.
  • Gravetter, F. J., & Wallnau, L. B. (2017). Statistics for the Behavioral Sciences. Cengage Learning.
  • Wilks, S. S. (2011). Statistical Methods in the Atmospheric Sciences. Academic Press.
  • Everitt, B. (2002). The Cambridge Dictionary of Statistics. Cambridge University Press.
  • McDonald, J. (2014). Handbook of Biological Statistics. Sparky House Publishing.
  • Laerd Statistics. (2016). Two-way ANOVA in SPSS. Retrieved from https://statistics.laerd.com/
  • Warner, R. M. (2013). Applied Statistics: From Bivariate Through Multivariate Methods. Sage Publications.
  • Anderson, D. R., Sweeney, D. J., & Williams, T. A. (2011). Statistics for Business and Economics. Cengage Learning.

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