Question

Is the interaction variation significant in case of the following information concerning mileage based on

different brands of gasoline and cars?

Answer 1

Two Way ANOVA

$H_0:$ there is no interaction between different brands of gasoline and cars

$H_a:$ there is an interaction between different brands of gasoline and cars

factor A is cars

factor B is brand of gasoline

F Statistic:

$F_{AB} = MS_{AB} / MS_E=2.71$

where:

Mean Square:

MS_AB = SS_AB / DF_AB = 2.71

DF_AB = (a - 1)(b - 1) = 6

Sum of Squares:

SS_AB=Σ_i^aΣ_j^bn_i,j(Ȳ_i,j - Ȳ_i - Ȳ_j + Ȳ)² = 16.25

Error Mean Square:

MS_E = SS_E / DF_E = 1

SS_E=Σ_i^aΣ_j^bΣ_k^ni,j(Y_i,j,k - Ȳ_i,j)² = 12

critical value:

$F_{crit}(df_1,df_2)=F_{crit}(6,12)=2.9961$

where error df₂:

DF_E = n - a*b = $24-3\cdot4=12$

Since $F<F_{crit}$ we accept null hypothesis. There is no interaction between different brands of gasoline and cars.