In a factorial ANOVA with df_bet(A) = 2, df_bet(B) = 4, df_wi = 30, k = _______ and N_tot = ________. A) 8; 38 B) 15; 45 C) 8; 22 D) 15; 15

Question

Quizplus · Accepted Answer

Using the formula for total degrees of freedom in a factorial ANOVA:

dfTotal = dfBetween1 + dfBetween2 + dfWithin + dfInteraction

We can fill in the given values:

dfS1U1B1bet(A)S1U1B0 = 2
dfS1U1B1bet(B)S1U1B0 = 4
dfS1U1B1wiS1U1B0 = 30

dfTotal = ?
dfTotal = 2 + 4 + 30 + dfInteraction

We can solve for the missing interaction degrees of freedom:

dfInteraction = dfTotal - 36

Now, we need to use the formula for the number of cells in a factorial design:

k = a * b

Where a is the number of levels of factor A and b is the number of levels of factor B. We are not given these values directly, but we can infer them from the degrees of freedom.

For factor A: 
dfS1U1B1bet(A)S1U1B0 = 2 
dfS1U1B0 = 1

We know that dfS1U1B0 represents the degrees of freedom for factor A, with 1 level of the factor. Therefore, dfS1U1B1bet(A)S1U1B0 must represent the degrees of freedom for the interaction between factor A and the within-subjects factor. This means that factor A is a between-subjects factor with 2 levels.

For factor B: 
dfS1U1B1bet(B)S1U1B0 = 4 
dfS1U1B0 = 1

Again, dfS1U1B0 represents the degrees of freedom for factor B, with 1 level of the factor. Therefore, dfS1U1B1bet(B)S1U1B0 must represent the degrees of freedom for the interaction between factor B and the within-subjects factor. This means that factor B is a between-subjects factor with 4 levels.

So,

k = 2 * 4 
k = 8

Finally, we can calculate the total number of observations:

NS1U1B1totS1U1B0 = k * ns 
NS1U1B1totS1U1B0 = 8 * 5 
NS1U1B1totS1U1B0 = 40

However, notice that the question is asking for the total number of degrees of freedom, which is different from the total number of observations. The total number of degrees of freedom is just the sum of all the degrees of freedom we have calculated:

dfTotal = 2 + 4 + 30 + dfInteraction 
dfTotal = 36 + dfInteraction

We do not have enough information to directly calculate the interaction degrees of freedom. However, we do know that dfS1U1B1bet(A)S1U1B0 * dfS1U1B1bet(B)S1U1B0 = dfInteraction, because these two factors are both between-subjects factors and their interaction would be the only other source of between-subjects variance.

dfS1U1B1bet(A)S1U1B0 * dfS1U1B1bet(B)S1U1B0 = dfInteraction 
2 * 4 = dfInteraction 
dfInteraction = 8

So,

dfTotal = 2 + 4 + 30 + 8 
dfTotal = 44

Therefore, the answer is B) 15; 45.

In a Factorial ANOVA with Dfbet(A) = 2, Dfbet(B) =

In a Factorial ANOVA with Df_bet(A) = 2, Df_bet(B) =