Answer to Question #280937 in Statistics and Probability for surajit das

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.