Question

Use the datafile ads2.csv. A consumer products company uses direct mail marketing for its advertising campaigns. The company has three different designs (1, 2, and 3) for a new brochure designed for customers in four regions (NE = north east, NW = north west, SE = south east, and SW = south west). The company decides to test the design types by mailing samples of each design to potential customers in each region. They repeat the direct mailing campaign 3 times for each design type and region combination and record the number of responses they receive (denoted as Response). The company wonders if the design type and region have an impact on the number of responses they get.

1. What type of statistical model would you use?
2. What is/are the factors? What are its/their levels?
3. Formulate all the relevant null and alternative hypotheses.
4. Decide whether to use a model with or without interaction term. Run the appropriate model and interpret results. What is the value of the test statistic for each hypothesis tested? What are the statistical decisions at a significance level alpha = 0.05?
5. Check all model assumptions. Are the model assumptions satisfied?
6. Test pairwise differences between factor levels at the significance level 0.05
7. show all your work, results, figures and interpretation in the Word document. At the end of the Word document, create a section entitled “Appendix: R Syntax”, and copy and paste all the R codes you used for each question part
8. 
   A B C 1 Design 2 Response 250 350 3 4 219 5 375 6 260 7 345 8 200 9 365 10 232 11 320 12 13 14 15 16 17 18 19 20 21 22 Region 1 NE 1 NW 1 SE 1 SW 1 NE 1 NW 1 SE 1 SW 1 NE 1 NW 1 SE 1 SW 2 NE 2 NW 2 SE 2 SW 2 NE 2 NW 2 SE 2 SW 2 NE 2 NW 2 SE 2 sw 3 NE 3 NW 3 SE 3 SW 222 345 400 525 390 580 420 512 385 365 465 23 510 24 379 25 567 26 275 27 340 28 200 29 310 30 3 NE 289 335 234 295 31 32 33 34 35 36 37 3 NW 3 SE 3 sw 3 NE 3 NW 3 SE 3 SW 289 327 210 287
   

Answer 1

a) There are two factors Design and Region. So we can use Two way ANOVA model.

b) Factor 1 : Disign , it has Three (3) Levels.

Factor 2 : Region, it has Three (4) Levels.   

c) Null hypotheis for two factors

Ho1: There is no significant effect of designs on number of responses.

  Ho2: There is no significant effect of regions on number of responses.

Alternatived hypotheis for two factors

H11: There is significant effect of designs on number of responses.

H12: There is significant effect of regions on number of responses.

d)

Data :

Design	Region	Response
1	NE	250
1	NW	350
1	SE	219
1	SW	375
1	NE	260
1	NW	345
1	SE	200
1	SW	365
1	NE	232
1	NW	320
1	SE	222
1	SW	345
2	NE	400
2	NW	525
2	SE	390
2	SW	580
2	NE	420
2	NW	512
2	SE	385
2	SW	365
2	NE	465
2	NW	510
2	SE	379
2	SW	567
3	NE	275
3	NW	340
3	SE	200
3	SW	310
3	NE	289
3	NW	335
3	SE	234
3	SW	295
3	NE	289
3	NW	327
3	SE	210
3	SW	287

Here we use Two way Anova without Interaction model,

summary(a)
Df Sum Sq Mean Sq F value Pr(>F)
as.factor(d[, 1]) 2 236337 118169 83.42 5.56e-13 ***
as.factor(d[, 2]) 3 94638 31546 22.27 8.80e-08 ***
Residuals 30 42495 1416   

By using P-value , we reject both the null hypotheses.(since P- value is less than 0.05).

Conclusion :There is significant effect of designs on number of responses and There is significant effect of regions on number of responses.

e) From this plot , we can say that residuals are normally distributed and all assumptions are met. There is no model defect.

f)

2000 0.000.00000000000 Normal Q-Q 3 160 N 240 0 Standardized residuals -1 -2 -3 प 020 -2 -1 0 1 2 Theoretical Quantiles aov(d[, 3] - as.factor(d[, 1)) + as.factor(d[, 2]))

f)  

Pairwise differences between factor levels at the significance level 0.05

Designs diff lwr upr p adj
2-1 167.916667 130.03784 205.79549 0.0000000 ------P- value is less than 0.05. Therefor Design 2 and Design 1 has different effect on responce.
3-1 -7.666667 -45.54549 30.21216 0.8723492   ------P- value is not less than 0.05. Therefor Design 2 and Design 1 has same effect on responce.
3-2 -175.583333 -213.46216 -137.70451 0.0000000 ------P- value is less than 0.05. Therefor Design 2 and Design 1 has different effect on responce.

$`as.factor(d[, 2])`
Region diff lwr upr p adj
NW-NE 76.000000 27.75772 124.2422836 0.0009529   ------P- value is less than 0.05. Therefor Region NW and Region NE has different effect on responce.
SE-NE -49.000000 -97.24228 -0.7577164 0.0454415 ------P- value is less than 0.05. Therefor Region SE and Region NE has different effect on responce.
SW-NE 67.666667 19.42438 115.9089503 0.0033716   ------P- value is less than 0.05. Therefor Region SW and Region NE has different effect on responce.
SE-NW -125.000000 -173.24228 -76.7577164 0.0000005 ------P- value is less than 0.05. Therefor Region SE and Region NW has different effect on responce.
SW-NW -8.333333 -56.57562 39.9089503 0.9651365 ------P- value is not less than 0.05. Therefor Region SW and Region NW has same effect on responce.
SW-SE 116.666667 68.42438 164.9089503 0.0000016   ------P- value is less than 0.05. Therefor Region SW and Region SE has different effect on responce.

g) “Appendix: R Syntax”, and copy and paste all the R codes you used for each question part :

d=data.frame(read.csv(file.choose()))
d
a=aov(d[,3]~as.factor(d[,1])+as.factor(d[,2]),data=d)
a
summary(a)
plot(a)
TukeyHSD(a)