ST512
Quiz 4 -Exercise
SSII 2010
The following exercises were taken from Dr. D. Dickey web site for ST512 : http://www.stat.ncsu.edu/people/dickey/courses/st512/
1. I have 2 oil based paints (O1 and O2) and a latex paint (L). To
for water permeability, I paint 5 boards with each paint, spray
boards with water and then measure the amount of water absorbed
grams. Here are the resulting means and totals for each set of
boards:
Paint
Mean
Total
Contrast Coefficients
L
26
130
___
O1
21
105
___
test
all 15
in
5
O2
19
95
___
(a) Fill in the coefficients for a contrast to compare latex to the average
of the oil based paints.
(b) Compute the sum of squares for the contrast in (a).
(c) Compute a sum of squares for testing the null hypothesis that there are
no differences in water permeability among the three paints.
(d) Assuming the error sum of squares is 300, finish the test in question
(c) by computing the calculated F statistic.
2. (60 pts.) Here is a plot like that in the class notes of the 4 means
for a two factor factorial experiment. The factors are fertilizer
ingredients N and P each at 2 levels and the design is a randomized
complete block with 6 blocks, each block being a bench in a greenhouse
and each observation Y being growth of a flower.
Heights of the plotted points are labeled on the vertical axis. Each
flower is in its own pot on one of the greenhouse benches.
19
15
14
12
|
+
p1
|
|
|
+
p2
+
p2
|
+
p1
|
|______________________________________
|
|
n1
n2
Compute the following if possible, including the right sign:
(a) The main effect of N ________ its sum of squares________ and the
degrees of freedom_______ for this sum of squares.
(b) The simple effect of N at the low level of P ___________ and its sum
of squares ________________.
(c) The simple effect of P at the high level of N _________.
(d) The interaction between N and P ___________ and its sum of squares
______________.
(e) The error degrees of freedom ________.
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ST512
Quiz 4 -Exercise
SSII 2010
(f) Your friend asks you if he should put the source "REPS" in his ANOVA
table as he has seen in some published papers. What do you say? (yes or
no with a brief explanation)
3. In a 2x2 factorial with factors A and B and 10 replications the main
effect of A was 10, the simple effect of A at the high level of B was 2.
Find, if possible, (if not, put "NP")
(a) the simple effect of A at the low level of B _____
(b) the mean ____ of all observations that had both A and B at the low
level.
4. Suppose a 3 x 2 factorial with factors A (3 levels, al, a2, a3)
and B (2 levels, Bl, b2) is run with 4 replications in blocks.
List the standard ANOVA breakdown for this factorial experiment
including sources for each main effect and interaction (give
sources and degrees of freedom only).
Source
d.f.
(a) Suppose the totals of the observations in the experiment in
question 2 are listed in our standard form as
[albl] = 30,
[a2bl]
[alb2] = 40,
= 80,
[a2b2] = l00,
[a3bl] = 50
[a3b2] = 80
and that the error sum of squares is l20. Compute sums of
squares and associated degrees of freedom for these tests:
i) Test for main effects of A.
Sum of squares ____
Degrees of freedom _______.
ii) Test for comparing the effect of level 2 of A to the average
of the effects of levels, l and 3 of A.
Sum of squares ____________ Degrees of Freedom ____________
iii)
Test to see if the comparison in part (b) is the same within both
levels of factor B.
Sum of squares __________ Degrees of freedom ________
iv)
Also, compute the t-statistics for testing that the low and high
levels of A, when combined with the high (b2) level of B,
produce the same effect. In other words,
test Ho: alb2 - a3b2
t-statistics = ______ with ________degrees of freedom.
5. I have a 5x2 factorial with factors W (weight at levels 20, 40, 60, 80,
100) and D (distance 40 yards, 80 yards) in which I study the effect on
some heart measurement of carrying the weight over the given distance. I
do all 10 (W,D) combinations for each of 5 people so I have 50
observations. Here is my (W, D) table of totals.
W
D
40
80
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20
40
60
80
100
10
13
18
22
22
24
24
24
26
27
23
40
46
48
53
(100)
(110)
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ST512
Quiz 4 -Exercise
SSII 2010
My book shows -2 -1 0 1 2 as the linear orthogonal polynomial coefficients
for a factor at 5 equally spaced levels, like W.
(a) Compute the sum of squares for the linear effect of W _____ and its
degrees of freedom_______
(b) I want to see if the linear effect of W is the same for both distances.
Compute sum of squares for the interaction of the W linear effect with D
________
(c) If I had obtained the table of totals above without having blocked on
people, how would that change your computations for parts (a) and (b) of this
question?
(d) Suppose I take the 4 degree of freedom sum of squares for W and from it I
subtract your sum of squares in part (a). I form an F test using this and it
is significant (I reject H0). Explain, as you would to the experimenter, how
to interpret this result. In other words, what is it that I am testing here?
July 28, 2010
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