• (30 points) A bank is studying the impact of its advertisement channels on number of new Savings accounts opened during a specific period. The followings channels are considered:
A: Printed adverb sent via direct mail with two levels of paper quality: standard (-1) and premium (+1).
B: Email adverb with two different designs: standard (-1) and fancy (+1).
C: Advertisement on Google search engine with two different methods: unpaid location promotion (-1) and paid location promotion (+1) (location is the rank of the advertisement shown in a search result).
D: Advertisement on radio at two different time slots: regular (-1) and premium (rush hour) (+1).
E: Advertisement on TV at two different time slots: daytime (-1) and evening (+1).
F: Advertisement on bus sides, at two different routes: local bus (+1) and express bus (-1).
Bank’s marketing team created several combinations of above advertainment channels and each advertisement combination is applied to a geographic location with known population. Here we assume “Geographic location” has NO IMPACT ON RESPONSE RATE. The team calculated the response rate which is the percentage of approximate audience that were exposed to a particular “run” and opened a Savings account during a period of 3 months when experiment was conducted. Results are as follows:
Run
A
B
C
D
E
F
ResponseRate
1
1
1
1
1
1
1
1.4
2
-1
1
1
1
-1
1
1.1
3
1
-1
1
1
-1
-1
1.23
4
-1
-1
1
1
1
-1
0.34
5
1
1
-1
1
-1
-1
1.12
6
-1
1
-1
1
1
-1
1.1
7
1
-1
-1
1
1
1
0.89
8
-1
-1
-1
1
-1
1
0.34
9
1
1
1
-1
1
-1
0.67
10
-1
1
1
-1
-1
-1
0.39
11
1
-1
1
-1
-1
1
1.19
12
-1
-1
1
-1
1
1
0.78
13
1
1
-1
-1
-1
1
0.99
14
-1
1
-1
-1
1
1
0.66
15
1
-1
-1
-1
1
-1
0.89
16
-1
-1
-1
-1
-1
-1
0.51
Assume bank team has decided to set factor E and F to be equal to two 3-way interactions of factors A, B, C and D.
a) What type of design is this? (Hint: use the design notation that shows number of factors, number of reductions, if any, that make the design “fractional” and its resolution.)
b) Find out what design generators bank team selected for factors E and F.
c) Demonstrate this design is orthogonal. (Hint: Show properties of an orthogonal design are holding).
d) WITHOUT using SAS derive: (I) defining relation, (II) resolution, (III) and confounding pattern. Show ALL your works.
e) Using SAS: (I) Find out which main factor(s) and two-way interaction(s) are significant? (II) Find the estimate of all main factors, (III) in a few sentences explain your finding to a person who doesn’t know Statistics. Use significance level of 10%. Include your SAS code.
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• (30 points) Table below presents the free height of leaf spring for a brand of trucks, in inch. In a good unit the free height should be close to 8 inches, with minimum variation. First 5 factors are considered: B: furnace temperature, C: heating time, D: transfer time, E: hold-down time and O: cooling oil temperature. The research team used E=BCD as the design generator and performed three replications as outlined in table below. Later studies showed cooling oil temperature cannot be controlled so they ignored factor O.
Free Height of leaf springs
B
C
D
E
O
Rep1
Rep2
Rep3
-1
-1
-1
-1
-1
7.78
7.78
7.81
oil temperature (O): low
-1
1
-1
-1
1
-1
8.15
8.18
7.88
-1
1
-1
1
-1
7.5
7.56
7.5
1
1
-1
-1
-1
7.59
7.56
7.75
-1
-1
1
1
-1
7.94
8
7.88
1
-1
1
-1
-1
7.69
8.09
8.06
-1
1
1
-1
-1
7.56
7.62
7.44
1
1
1
1
-1
7.56
7.81
7.69
-1
-1
-1
-1
1
7.5
7.25
7.12
oil temperature (O): high
+1
1
-1
-1
1
1
7.88
7.88
7.44
-1
1
-1
1
1
7.5
7.56
7.5
1
1
-1
-1
1
7.63
7.75
7.56
-1
-1
1
1
1
7.32
7.44
7.44
1
-1
1
-1
1
7.56
7.69
7.62
-1
1
1
-1
1
7.18
7.18
7.25
1
1
1
1
1
7.81
7.5
7.59
a) Since factor O is ignored, how many replications are available for the combination of remaining factors? What type of design is this and what is its resolution? (Hint: use the design notation that shows number of factors, number of reductions that make the design “fractional” and its resolution.)
b) What is the defining relation? What is confounding pattern of factor B?
c) Using SAS: Find the estimate of following main effects and interactions: B, C, B*C, D, B*D, C*D, E
d) Using SAS: find out which main effect(s) or 2-way interaction(s) is significant (using alpha=0.05)?
e) Using SAS plot the normal probability plot for main effects and two-way interactions. Include your SAS code and the resulting graph.
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• (30 points) In an experiment for time of filling a container, three factors are considered, each with three levels:
Machine
Container Type
Container shape
1
Plastic
Cube
2
Metal
Cone
3
Mixed
Cylinder
All three container shapes have the same capacity. Table below shows the result of the experiment for 2 replications, where content of table is “fill time” (response variable) is in seconds:
Replication 1
Replication 2
Container shape
Container shape
Machine
Container Type
Cube
Cone
Cylinder
Cube
Cone
Cylinder
1
Plastic
3.45
4.14
5.8
3.36
4.19
5.23
Metal
4.07
4.38
5.48
3.52
4.26
4.85
Mixed
4.2
4.26
5.67
3.68
4.37
5.58
2
Plastic
4.8
5.22
6.21
4.4
4.7
5.88
Metal
4.52
5.15
6.25
4.44
4.65
6.2
Mixed
4.96
5.17
6.03
4.39
4.75
6.38
3
Plastic
4.08
3.94
5.14
3.65
4.08
4.49
Metal
4.3
4.53
4.99
4.04
4.08
4.59
Mixed
4.17
4.86
4.85
3.88
4.48
4.9
a) What type of design is this? (Hint: use the design notation that shows number of factors, number of reductions, if any).
b) Is this design orthogonal? Why? Demonstrate your reason(s).
c) Using SAS: What main factor(s) or interaction(s) is significant? Include your SAS code and all results. Based on SAS outputs explain how you concluded that a factor is significant or not.
d) In a few sentences explain your findings from this analysis to a person who does not know any Statistics.
e) Using SAS output from part (c) explain why Type I and Type III sum of squares are the same.
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• (10 points)
(No SAS code to be used)
a) Create a Plackett-Burman design table for N=24 runs. You can create the design table in Excel and then copy/paste it into your submission file.
b) Consider the Placket-Burman design in the following table:
Runs
1
2
3
4
5
6
7
8
9
10
11
1
1
1
-1
1
1
1
-1
-1
-1
1
-1
2
-1
1
1
-1
1
1
1
-1
-1
-1
1
3
1
-1
1
1
-1
1
1
1
-1
-1
-1
4
-1
1
-1
1
1
-1
1
1
1
-1
-1
5
-1
-1
1
-1
1
1
-1
1
1
1
-1
6
-1
-1
-1
1
-1
1
1
-1
1
1
1
7
1
-1
-1
-1
1
-1
1
1
-1
1
1
8
1
1
-1
-1
-1
1
-1
1
1
-1
1
9
1
1
1
-1
-1
-1
1
-1
1
1
-1
10
-1
1
1
1
-1
-1
-1
1
-1
1
1
11
1
-1
1
1
1
-1
-1
-1
1
-1
1
12
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
Calculate correlation between factor 1 and interaction 4*5. Show all your work.