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 alternate 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?
Source Sum of Squares (degrees of freedom [df]) Mean Square Fobt. Fcrit.
Gender 68.15 ? ? ? ?
Marital Status 127.37 ? ? ? ?
Gender * Marital Status (A x B) 41.90 ? ? ? ?
Error (Within) 864.82 ? ? NA NA
Total 1102.24 99 NA NA NA
Please Note: The table that you see in the assignment has been slightly modified from the one presented in the module notes since it is beyond the scope of this unit to have students calculate p values. Instead you are asked to calculate the F value and compare it to the critical F value to determine whether the test is significant or not.
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.
a. The independent variables are Gender (with two levels: male and female) and Marital Status (with three levels: married, single never married, and divorced). The dependent variable is happiness scores.
b. Null hypotheses:
- Null hypothesis for Gender: There is no difference in happiness scores between males and females.
Alternate hypothesis for Gender: There is a difference in happiness scores between males and females.
- Null hypothesis for Marital Status: There is no difference in happiness scores among the different marital status groups.
Alternate hypothesis for Marital Status: There is a difference in happiness scores among the different marital status groups.
- Null hypothesis for the interaction between Gender and Marital Status: The effect of Gender on happiness scores does not depend on the Marital Status.
Alternate hypothesis for the interaction between Gender and Marital Status: The effect of Gender on happiness scores depends on the Marital Status.
c. Degrees of freedom:
- Gender: df = 1
- Marital Status: df = 2
- Interaction between Gender and Marital Status: df = 2
- Error or within variance: df = 93 (total df - sum of df for the independent variables)
d. Mean square:
- Mean square for Gender: Mean square = Sum of Squares / df = 68.15 / 1
- Mean square for Marital Status: Mean square = Sum of Squares / df = 127.37 / 2
- Mean square for the interaction between Gender and Marital Status: Mean square = Sum of Squares / df = 41.90 / 2
- Mean square for Error or within variance: Mean square = Sum of Squares / df = 864.82 / 93
e. F ratio:
- F ratio for Gender: F ratio = Mean square for Gender / Mean square for Error
- F ratio for Marital Status: F ratio = Mean square for Marital Status / Mean square for Error
- F ratio for the interaction between Gender and Marital Status: F ratio = Mean square for interaction / Mean square for Error
f. Critical Fs at alpha = .05:
- Critical F for Gender: Look it up in the F-distribution table with df numerator = 1 and df denominator = 93.
- Critical F for Marital Status: Look it up in the F-distribution table with df numerator = 2 and df denominator = 93.
- Critical F for the interaction between Gender and Marital Status: Look it up in the F-distribution table with df numerator = 2 and df denominator = 93.
g. Based on the calculated F ratios and the critical Fs at alpha = .05, if the calculated F ratio for any analysis (Gender, Marital Status, or interaction) is greater than the critical F value, then we reject the null hypothesis and conclude that there is a significant difference. If the calculated F ratio is less than or equal to the critical F value, then we fail to reject the null hypothesis and conclude that there is no significant difference.