Consider the following partial ANOVA table: $$\begin{array} { | l | c | c | c | c | }
\hline \text { Source of Variation } & \text { SS } & d f & M S & F \
\hline \text { Treatments } & 75 & * & 25 & 6.67 \
\hline \text { Error } & 60 & * & 3.75 & \
\hline \text { Total } & 135 & 19 & & \
\hline
\end{array}$$ The numbers of degrees of freedom for numerator and denominator, respectively, (identified by asterisks) are:
A) 4 and 15.
B) 3 and 16.
C) 15 and 4.
D) 16 and 3.

Question

Consider the following partial ANOVA table: $$\begin{array} { | l | c | c | c | c | } 
\hline \text { Source of Variation } & \text { SS } & d f & M S & F \
\hline \text { Treatments } & 75 & * & 25 & 6.67 \
\hline \text { Error } & 60 & * & 3.75 & \
\hline \text { Total } & 135 & 19 & & \
\hline
\end{array}$$ The numbers of degrees of freedom for numerator and denominator, respectively, (identified by asterisks) are:
A) 4 and 15.
B) 3 and 16.
C) 15 and 4.
D) 16 and 3.

Quizplus · Accepted Answer

The total degrees of freedom is always equal to the sum of the degrees of freedom for the treatments and error. 
Therefore, $df_{total} = df_{treatment} + df_{error}$. 
Substituting the given values, we get $19 = df_{treatment} + df_{error}$.
Also, the numerator degrees of freedom for treatments is equal to the number of treatment groups minus 1. Since this information is not given, we cannot determine the exact value of $df_{treatment}$.
However, we can use the fact that the degrees of freedom for error is equal to $df_{error} = df_{total} - df_{treatment}$. Substituting the given values, we get $df_{error} = 19 - df_{treatment}$.
From the ANOVA table, we see that $MS_{error} = \frac{SS_{error}}{df_{error}} = 3.75$. Substituting the given values, we get $60/df_{error} = 3.75$ which gives $df_{error} = 16$.
Using $df_{total} = df_{treatment} + df_{error}$ and $df_{error} = 16$, we get $19 = df_{treatment} + 16$ which gives $df_{treatment} = 3$.
Therefore, the answer is B) 3 and 16.

Consider the Following Partial ANOVA Table The Numbers of Degrees of Freedom for Numerator and Denominator