Q:

# QUESTION 1 A researcher compares differences in positivity between participants in a low-, middle-, or upper-middle-class family. If she observes 15 participants in each group, then what are the degrees of freedom for the one-way between-subjects ANOVA?

Accepted Solution

A:
Answer:The degrees of freedom for the denominator on this case is given by $$df_{den}=df_{between}=N-K=3*15-3=42$$. Step-by-step explanation:Analysis of variance (ANOVA) "is used to analyze the differences among group means in a sample".  The sum of squares "is the sum of the square of variation, where variation is defined as the spread between each individual value and the grand mean"  If we assume that we have $$3$$ groups and on each group from $$j=1,\dots,15$$ we have $$15$$ individuals on each group we can define the following formulas of variation:   $$SS_{total}=\sum_{j=1}^p \sum_{i=1}^{n_j} (x_{ij}-\bar x)^2$$  $$SS_{between}=SS_{model}=\sum_{j=1}^p n_j (\bar x_{j}-\bar x)^2$$  $$SS_{within}=SS_{error}=\sum_{j=1}^p \sum_{i=1}^{n_j} (x_{ij}-\bar x_j)^2$$  And we have this property  $$SST=SS_{between}+SS_{within}$$  The degrees of freedom for the numerator on this case is given by $$df_{num}=df_{within}=k-1=3-1=2$$ where k =3 represent the number of groups. The degrees of freedom for the denominator on this case is given by $$df_{den}=df_{between}=N-K=3*15-3=42$$. And the total degrees of freedom would be $$df=N-1=3*15 -1 =44$$ And the F statistic to compare the means would have 2 degrees of freedom on the numerator and 42 for the denominator.