Answer to Question #203772 in Statistics and Probability for Raqib

Question

A researcher wants to understand how an annual mortgage payment (in Ringgit) depends on

income level and zonal location allowing for interaction. The data are shown below.

a.Specify the null and alternative hypotheses to test whether there is interaction between

income and zonal location. (2 Marks)

b. Determine the total sum of square, sum of square for income (factor A), sum of square

for zonal location (Factor B), sum of square for the interaction between income and zonal

location, and error sum of square. (10 Marks)

c. Determine the mean square for income (factor A), mean square for zonal location (Factor

B), mean square for the interaction between income and zonal location, and mean square

error. (4 Marks)

d. Determine the F-Statistics and P-Value for income, zonal location and interaction between

income and zonal location. (6 Marks)

e. At the 5% significance level, can you conclude there is interaction between income and

zonal location?

Answer 1

a.

The model:

$Y=\beta_0+\beta_1X_{A}+\beta_2X_B+\beta_3X_{AB}+\varepsilon_i$

$H_0:$ $\beta_1=0$ , an annual mortgage payment does not depend on income level

$\beta_2=0$ , an annual mortgage payment does not depend on zonal location

$\beta_3=0$ , an annual mortgage payment does not depend on interaction between

income and zonal location

$H_a:$ $\beta_1\ne0$ , an annual mortgage payment depends on income level

$\beta_2\ne0$ , an annual mortgage payment depends on zonal location

$\beta_3\ne0$ , an annual mortgage payment depends on interaction between

income and zonal location

b.

The sum of square income (factor A):

$SS_A=bn\displaystyle{\sum^a(\overline{x}_i-\overline{x})^2}$

$\overline{x}=169$

$SS_A=5\cdot5((140-169)^2+(163-169)^2+(196-169)^2)=$

$=21678$

The sum of square for zonal location (factor B):

$SS_B=an\displaystyle{\sum^b(\overline{x}_j-\overline{x})^2}$

$SS_B=5\cdot3((185-169)^2+(300-169)^2+(182-169)^2+$

$+(142-169)^2+(36-169)^2=324351$

The sum of square for the interaction between income and zonal location:

$SS_{AB}=SS_{cells}-SS_A-SS_B$

$SS_{cells}=n\sum\sum(\overline{x}_{ij}-\overline{x})^2$

$SS_{cells}=5\cdot125918=377754$

$SS_{AB}=377754-21678-324351=31724$

Error sum of square:

$SS_E=SS_{total}-SS_A-SS_B-SS_{AB}$

$SS_{total}=\sum\sum\sum(x-\overline{x})^2$

$SS_{total}=799206$

$SS_E=799206-377754=421452$

c.

For income:

$MS_A=SS_A/df_A$

$df_A=a-1=3-1=2$

$MS_A=21678/2=10839$

$MS_E=SS_E/df_E$

$df_E=ab(n-1)=3\cdot5(5-1)=60$

$MS_E=421452/60=7024$

For zonal location:

$MS_B=SS_B/df_B$

$df_B=b-1=5-1=4$

$MS_B=324351/4=81088$

$F_B=81088/7024=11.5444$

For interaction between income and zonal location:

$MS_{AB}=SS_{AB}/df_{AB}$

$df_{AB}=df_A\cdot df_B=2\cdot4=8$

$MS_{AB}=31724/8=3965$

d.

For income:

$F_A=\frac{MS_A}{MS_E}$

$F_A=10839/7024=1.5431$

p-value$=0.3186$

For zonal location:

$F_B=\frac{MS_B}{MS_E}$

$F_B=81088/7024=11.5444$

p-value$=0.0218$

For interaction between income and zonal location:

$F_{AB}=\frac{MS_{AB}}{MS_E}$

$F_{AB}=3965/7024=0.5646$

p-value$=0.6081$

e.

​From F-table critical value for $\alpha=5\ \%$ :

$F_{2,4}=6.9443$

Since $F_A<F_{2,4},F_B>F_{2,4},F_{AB}<F_{2,4}$ , we can conclude that:

an annual mortgage payment does not depend on income level,

an annual mortgage payment depends on zonal location,

an annual mortgage payment does not depend on interaction between

income and zonal location.