Analysis of Variance (ANOVA)

Analysis of Variance (ANOVA) is a set of statistical methods used to link two or more samples. More crucially, the two mean variances on two factors (independent variables) among split groups are compared using the two-way ANOVA. According to Mayers, the primary goal of a two-way ANOVA is to determine whether there is an interaction between the two independent factors and the dependent variable (2013). In this scenario, for example, the two-way ANOVA is used to examine whether there is an interaction between gender and happiness among married, single never married, and divorced people. Therefore, the independent variables are the marital status (divorced, married, and single never married) and gender (male/female), and the dependent variable is the level of happiness. Alternatively, an interaction between gender and marital status in as far as level of happiness is concerned can also be determined through a two-way ANOVA. The independent variables and the dependent variables as well as their levels are illustrated below;

a).

Dependent variable

Level of happiness

Independent Variables level

Marital status (divorced, married, and single never married) 2

Gender (male and female) 1

b).

The hypotheses of an ANOVA are,

H1: The means are not all equal.

H0: μ1 = μ2 = μ3 = μ4

Therefore, the population means of both marital status and gender are equal. In addition, there is no interaction between the two factors. In alternate hypothesis, the two factors gender and marital status correspond to bring about the dependent factor. This is congruent with what Nesselroade, and Cattell aver that the means are always not all equal in alternate hypothesis (2013). The authors further affirm that in an ANOVA, there is always no difference in means in the null hypothesis.

c).

The following are the df,

1. Gender =1

2. Marital status = 2

3. Marital status and gender interaction 1*2 = 2

4. Error or within variance = 94

d). Mean Square

Gender

MS= ss/df

MS= 68.15/1

= 68.15

Marital Status

MS= SS/df

=127.37/2

=63.68

Marital status and gender interaction

=41.90/ (1*2)

=20.95

Error or within variance

=864.82/94

=9.20

e). F ratio

F=MSB/MSE

Where;

MSE= SSE/ (N-k), and

MSB= SSB/ (k-1)

Alternatively,

F= MS for each row/ MS for the error source

Therefore,

Gender

F ratio

F= 68.15/9.20

=7.40

Marital Status

F= 63.68/9.20

= 6.92

marital status and gender interaction

F= 20.95/ 9.20

= 2.27

f). Critical Fs

An F-table is used to find critical values for an ANOVA hypothesis. The value of alpha is 0.05 also referred to as a 5 percent level of significance. Using the two types of degrees of freedom, and the value of alpha (α=0.05), the following are critical Fs;

Gender

Two types of degrees of freedom are, (1, 99)

Critical F is 3.94

Therefore, at the level of probability, the result is significant since value of F is equal or greater than critical value (7.40 ≥3.94).

Marital Status

Two types of degrees of freedom are (2, 99)

Critical F= 3.09

The result is significant at the level of probability as 6.92 ≥ 3.09.

Interaction of marital status and gender

(2, 99) are the degrees of freedom

Critical value for 2.27 for the 0.05 significance level is 3.09.

g). conclusions

In conclusion, both gender and marital status are significant but the interaction is not significant. This means that gender factors are not equal and also the marital status factors are not equal too. Secondly, the F ratio obtained for both marital status and gender are likely to occur by chance with a p