a. The independent variables are gender (with two levels: male and female) and marital status (with three levels: married, single never married, divorced). The dependent variable is happiness scores.
b. Null hypotheses:
- For gender: There is no significant difference in happiness scores between males and females.
- For marital status: There is no significant difference in happiness scores among married, single never married, and divorced individuals.
- For the interaction between gender and marital status: The effect of gender on happiness scores does not depend on marital status, and vice versa.
Alternate hypotheses:
- For gender: There is a significant difference in happiness scores between males and females.
- For marital status: There is a significant difference in happiness scores among married, single never married, and divorced individuals.
- For the interaction between gender and marital status: The effect of gender on happiness scores depends on marital status, and vice versa.
c. The degrees of freedom are as follows:
1) Gender: df = 1
2) Marital status: df = 2
3) Interaction between gender and marital status: df = (1)(2) = 2
4) Error or within variance: df = n - 1 = 100 - 1 = 99
d. To calculate the mean square for each factor, divide the sum of squares by its corresponding degrees of freedom.
1) Mean Square for gender: MS(gender) = SS(gender) / df(gender) = 68.15 / 1 = 68.15
2) Mean Square for marital status: MS(marital status) = SS(marital status) / df(marital status) = 127.37 / 2 = 63.685
3) Mean Square for interaction between gender and marital status: MS(interaction) = SS(interaction) / df(interaction) = 41.90 / 2 = 20.95
4) Mean Square for error or within variance: MS(error) = SS(error) / df(error) = 864.82 / 99 = 8.750
e. To calculate the F ratio for each factor, divide the mean square by the mean square for error.
1) F ratio for gender: F(gender) = MS(gender) / MS(error) = 68.15 / 8.750 = 7.78
2) F ratio for marital status: F(marital status) = MS(marital status) / MS(error) = 63.685 / 8.750 = 7.27
3) F ratio for interaction between gender and marital status: F(interaction) = MS(interaction) / MS(error) = 20.95 / 8.750 = 2.39
f. To identify the critical F values at alpha = .05, we need to consult an F distribution table or use a statistical calculator. Since the degrees of freedom for the factors are already known, we can find the critical F values.
1) Critical F value for gender at alpha = .05: Fcrit(gender) = F(1, 99) = 4.02
2) Critical F value for marital status at alpha = .05: Fcrit(marital status) = F(2, 99) = 3.16
3) Critical F value for interaction between gender and marital status at alpha = .05: Fcrit(interaction) = F(2, 99) = 3.16
g. If alpha is set at .05, we compare the F ratios obtained in step e with the critical F values obtained in step f.
- For gender, the F ratio (7.78) is greater than the critical F value (4.02), indicating a significant difference.
- For marital status, the F ratio (7.27) is greater than the critical F value (3.16), indicating a significant difference.
- For the interaction between gender and marital status, the F ratio (2.39) is not greater than the critical F value (3.16), indicating no significant interaction.
Therefore, we can conclude that there is a significant difference in happiness scores based on gender and marital status individually, but there is no significant interaction effect between gender and marital status on happiness scores.