Question

A researcher reports an F-ratio with df = 2, 18 from an independent-measures research study. Based on the df values, how many treatments were compared in the study, and what was the total number of subjects participating in the study?a. 3 treatments and 22 subjectsb. 3 treatments and 21 subjectsc. 2 treatments and 19 subjectsd. 2 treatments and 20 subjects

Answer 1

Answer:

b. 3 treatments and 21 subjectcs

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"  

On this case we have 4 degrees of freedom for the numerator and 95 for the denominator.  

If we assume that we have $k$ groups and on each group from $j=1,\dots,n_j$ we have $n_j$ 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 $df_{num}=df_{within}=k-1$ where k  represent the number of groups or treatments.  So then if we solve for k we got $k=df_{num}+1=2+1=3$

The degrees of freedom for the denominator on this case is given by $df_{den}=df_{between}=N-k$.  And we can solve for N like this:

$N=df_{den}+k =18+3=21$ so we have 21 individuals in total

And the total degrees of freedom would be $df=N-1=21 -1 =20$

On this case the correct answer would be:

b. 3 treatments and 21 subjectcs