statguruSASwriteup

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1. Answer the following questions about study design

(a) What are the basic principles of experimental design? Briefly explain.

(b) Write down at least three specific di↵erences between observational study and experimental study. Briefly explain the di↵erences with specific examples.

(c) What is the di↵erence between nested design and split plot design?

2. For a fixed-e↵ect Randomized Complete Block Design (RCBD), answer the following questions.

(a) Show that

aX

i=1

bX

j=1

(yij � ȳ..)2 = b aX

i=1

(ȳi. � ȳ..)2 + a bX

j=1

(ȳ.j � ȳ..)2

+ aX

i=1

bX

j=1

(yij � ȳi. � ȳ.j + ȳ..)2

(b) E(MSE) = � 2

(c) E(MSBlock) = � 2 +

a P

�2j b�1

3. Chemical process. A 23 experiment was performed to improve the yield of a chemical process. Two replicates of a completely randomized experiment were run. The results are shown in the following table.

Treatment Replicate Combination I II

(1) 90 93 a 74 78 b 81 85 ab 83 80 c 77 78 ac 81 80 bc 88 82 abc 73 70

(a) Compute the estimated main e↵ect of A. Hint: Effect = Contrast n2k�1

(b) Compute the sum of squares of ABC. Hint: SS = (Contrast)2

n2k

(c) What is the degrees of freedom for error?

(d) Write down the mathematical expression of the model with assumptions.

(e) Write down the null and alternative hypothesis for testing main e↵ect A.

4. Confounding. Consider a 33 factorial design and answer the following questions.

(a) Provide a scheme for confounding interaction e↵ect A2B2C so that treatment combinations can be arranged in three blocks. Show the blocks with treatment combinations that go to them.

(b) Suppose there is only one replication of the experiment. Provide a plan on how you will compute F ratio to test the main factor e↵ects.

(c) What is the degrees of freedom for interaction e↵ect AB.

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5. In an experimental study, yields of an electric equipment were recorded on di↵erent oven temperatures and at di↵erent times. For this the researchers set a temperature first and randomly taken out an equipment from the oven at di↵erent times. The whole process is replicated three times. Now answer the following questions.

Replication I Replication II Replication III

Temperature 580 600 620 640 580 600 620 640 580 600 620 640

time 5 217 158 229 223 188 126 160 201 162 122 167 182 10 233 138 186 227 201 130 170 181 170 185 181 201 15 175 152 155 156 195 147 161 172 213 180 182 199

(a) What kind of design is it?

(b) The experiment could be performed randomly assigning temperature and a time on an equip- ment. Give a reasoning why the researchers opted a temperature of the oven first and then selected equipments at di↵erent times.

(c) Write down mathematical equation for the model to analyze the data. Include all the as- sumptions and descriptions of the notations used. What impact does this decision have on the analysis?

(d) Exam-data.csv file contains the data from this experiment. Analyze the data to identify if temperature or baking time has any e↵ect on the yields. Present ANOVA table in your analysis.

exam-data.csv

rep	temp	time	response
1	580	5	217
1	580	10	233
1	580	15	175
1	600	5	158
1	600	10	138
1	600	15	152
1	620	5	229
1	620	10	186
1	620	15	155
1	640	5	223
1	640	10	227
1	640	15	156
2	580	5	188
2	580	10	201
2	580	15	195
2	600	5	126
2	600	10	130
2	600	15	147
2	620	5	160
2	620	10	170
2	620	15	161
2	640	5	201
2	640	10	181
2	640	15	172
3	580	5	162
3	580	10	170
3	580	15	213
3	600	5	122
3	600	10	185
3	600	15	180
3	620	5	167
3	620	10	181
3	620	15	182
3	640	5	182
3	640	10	201
3	640	15	199