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 as below.

 

LOW

MEDIUM

HIGH

KOTA KINABALU

130

186

231

128

201

216

201

195

171

190

186

216

150

191

186

SANDAKAN

126

218

306

220

263

351

260

230

330

311

308

486

280

314

498

TAWAU

233

171

231

173

186

186

131

21

243

128

306

201

77

231

207

LAHAD DATU

120

183

231

180

96

156

160

141

168

130

126

141

80

105

110 

KENINGAU

13

46

65

18

31

36

18

46

51

23

41

46

26

41

39

a)   Determine the total sum of square 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

b)   Determine the F-Statistics and P-Value for income, zonal location and interaction between income and zonal location. 6 marks

Answer 1

a)

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:

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

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

$SS_{total}=799206$

$SS_E=799206-377754=421452$

b)

F-statistics:

For income:

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

$MS_A=SS_A/df_A$

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

$MS_A=21678/2=10839$

$MS_E=SS_E/df_E$

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

$MS_E=421452/60=7024$

$F_A=10839/7024=1.5431$

p-value$=0.3186$

For zonal location:

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

$MS_B=SS_B/df_B$

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

$MS_B=324351/4=81088$

$F_B=81088/7024=11.5444$

p-value$=0.0218$

For interaction between income and zonal location:

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

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

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

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

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

p-value$=0.6081$