Question

An ANOVA procedure is used for data obtained from five populations. Five samples, each comprised of 20 observations, were taken from the five populations. The numerator and denominator (respectively) degrees of freedom for the critical value of F are Select one:
A. 3 and 30

B. 4 and 30

C. 3 and 119

D. 3 and 116

E. None of the above answers is correct

Answer 1

Answer:

E. None of the above answers is correct

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 $5$ groups and on each group from $j=1,\dots,20$ we have $20$ 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 $df_{num}=df_{within}=k-1=5-1=4$ where k =5 represent the number of groups.

The degrees of freedom for the denominator on this case is given by $df_{den}=df_{between}=N-K=5*20-5=95$.

And the total degrees of freedom would be $df=N-1=5*20 -1 =99$

On this case the correct answer would be 4 for the numerator and 95 for the denominator.

E. None of the above answers is correct