STA 6208 – Exam 3 – Spring 2011 – PRINT Name ___________________________
Conduct Individual Test/CI’s at  = 0.05 significance level, and all simultaneous tests/CI’s @ experimentwise rate 0.05
Q.1. A randomized complete block design is conducted with 3 (fixed) treatment in 3 (random) blocks.
yij     i  b j  eij
p.1.a. E(yij) =
p.1.e. Derive

i, j  1, 2,3 b j ~ NID 0,  b2


eij ~ NID 0,  2
p.1.b. V(yij) =
 

 b   e 
j
p.1.c. COV(yij , yi’j) =


V y i , COV y i , y i ' , V y i  y i '

ij
p.1.d. COV(yij , yij’) =
(SHOW ALL WORK)
Q.2. A latin-square design is conducted to compare 5 types of packaging for a food product, in 5 stores, over a 5 week
period. The experiment is set up so that each package appears once at each store, and once over each of the 5 weeks.
p.2.a. Write the statistical model, assuming fixed packages and random stores and weeks, yij is sales for store i, week j.
p.2.b. Complete the following Analysis of Variance Table based on the data below:
Store\Week
1
2
3
4
5
Wk Mean
1
100
140
60
110
90
100
2
120
150
40
120
70
100
3
110
160
50
130
80
106
4
90
145
70
90
110
101
5
80
155
30
100
100
93
df
Sum Sq
Mean Sq
F
F(0.05)
St Mean
100
150
50
110
90
100
Pack Mean
109
92
102
100
97
100
ANOVA
Source
Package
Yes / No
Store
#N/A
#N/A
Week
#N/A
#N/A
Error
#N/A
#N/A
#N/A
#N/A
Total
Package Effects?
30250
#N/A
p.2.c. Compute Tukey’s and Bonferroni’s Minimum significant differences for comparing all pairs of packaging effects:
Q.3. A repeated measures design is conducted to compare 3 treatments over 4 equally space time points. A random
sample of 60 subjects are selected and randomized, so that 20 receive treatment A, 20 receive B, and 20 receive C.
p.3.a. Complete the following ANOVA table.
ANOVA
Source
df
SS
Treatments
4000
Subject(Trt)
22800
Time
1260
TrtxTime
MS
F(.05)
Sig Effect?
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
84
Error2
Total
F
30538
#N/A
p.3.b. Assuming no significant Time by Treatment Interaction, compute Bonferroni’s and Tukey’s Minimum Significant
Differences for comparing all pairs of treatment means.
p.3.c. Assuming no significant Time by Treatment Interaction, compute Bonferroni’s and Tukey’s Minimum Significant
Differences for comparing all pairs of time means.
Q.4. An experiment is to be conducted to compare 4 cooking temperatures and 3 mixtures of alloys on strength
measurements of steel rods. The cooking period is 2 hours, so that only 4 cooking periods can be conducted on a
business day (the temperatures are randomly assigned to periods). The experimenter decides she will assign the 3
mixtures at random to the 3 positions in the oven (experience implies there are no position effects), separately for the 4
runs on a given day. She repeats the experiment over 3 days (randomizing separately on each day). Her assistant
provides her the following sequences of random numbers for temperature and mixtures. Give the assignment of
treatments to experimental positions (in each cell, enter Ti/Mj where i=Temperature level, j=Mixture level).
Temp
Ran#
Mix
Ran#
Mix
Ran#
Mix
Ran#
1
0.96
1
0.10
1
0.29
1
0.91
2
0.91
2
0.54
2
0.64
2
0.55
3
0.16
3
0.58
3
0.43
3
0.83
Day1
Pos1
4
0.22
1
0.59
1
0.76
1
0.43
1
0.21
2
0.70
2
0.65
2
0.69
Day1
Pos2
2
0.34
3
0.79
3
0.56
3
0.19
Day1
Pos3
3
0.28
1
0.26
1
0.97
1
0.98
4
0.27
2
0.74
2
0.28
2
0.79
Day2
Pos1
1
0.75
3
0.72
3
0.06
3
0.36
2
0.23
1
0.83
1
0.83
1
0.26
3
0.37
2
0.17
2
0.76
2
0.78
Day2
Pos2
4
0.20
3
0.56
3
0.64
3
0.11
Day2
Pos3
Day3
Pos1
Day3
Pos2
Day3
Pos3
Per1
Per2
Per3
Per4
Q.5. A 23 factorial experiment is conducted to determine the main effects and interactions among 3 factors
(presence/absence) on taste quality for frozen dinners. The following table gives the design, mean, and standard
deviation (SD) for the 8 combinations of factor levels. There were 4 replicates per treatment.
(1)
a
b
c
ab
ac
bc
abc
-1
1
-1
-1
1
1
-1
1
-1
-1
1
-1
1
-1
1
1
-1
-1
-1
1
-1
1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
-1
1
1
1
-1
-1
-1
-1
1
1
40
50
42
38
53
47
40
50
2
3
1
2
2
1
3
2
p.5.a. Give the +1/-1 levels for the ABC Interaction.
p.5.b. Compute l A 
lA = _____________
n
k y ,
i 1
i
i
SSA 
r
2
l
n  A
2
where ki  1
Test H0: No Factor A effect
SSA = _________________ Test Statistic = _________________ Rejection Region: __________
Q.6. A randomized block design is conducted to compare t=3 treatments in b=4 blocks. Your advisor gives you the
following table of data from the experiment (she was nice enough to compute treatment, block, and overall means for
you), where: TSS 
Blk\Trt
1
2
3
4
TrtMean
TSS
658
 Y  Y 
1
20
10
28
10
17
2
2
22
13
25
12
18
3
24
16
34
14
22
BlkMean
22
13
29
12
19
p.6.a. Complete the following ANOVA table:
Source
Treatments
Blocks
Error
Total
df
SS
MS
F_obs
F(.05)
Reject H0: No Effect?
p.6.b. Compute the Relative Efficiency of having used a Randomized Block instead of a Completely Randomized Design
RE 
2
sCR
(r  1) MSB  r (t  1) MSE

2
sRBD
 rt  1 MSE
RE(RB,CR) = ___________
Sample Size per treatment for equivalent Std. Errors of difference of Means _________
p.6.c.. Compute Tukey’s minimum significant difference for comparing all pairs of treatment means:
Tukey’s W = _______________________________
p.6.d. Give results graphically using lines to connect Trt Means that are not significantly different: T1 T2 T3