4-10
3.80
0.50
5.40.
Reconsider
the keyboard
experiment in Problem 5.30. Suppose that this experiment had been
conducted in Mean
three blocks, with each
replicatet for
a block.
Assume that the observations in the data table are
Standard
H0
given
in
order,
that
is,
the
first
observation
in
each
cell
comes
from
Treatment Difference
DF
Error
Coeff=0
Prob
> |t| the first replicate, and so on. Reanalyze
the
experiment
in0.71
blocks and estimate
the 0.0448
variance component for blocks. Does it appear
1 vsdata
2 as a factorial
1.60
1
2.24
1 vsblocking
3
3.20 useful in1this experiment?
0.71
4.48
0.0008
that
was
STATISTICS
402B
1 vs 4
2 vs 3
2 vs 4
3 vs
4 Size
Key
3.00
1.60
1.40
-0.20
1
1
1
1
0.71
0.71
0.71
0.71
Mushy
4.20
0.0012
2.24
0.0448
Feel
1.96 Keyboard
0.0736
Homework
Set#4
-0.28
0.7842
Small
31
33
35
Distance
has a statistically significant
effect
Medium
36
35 on mean
33 focus time.
1. Problem
Large 4.22 (Montgomery)
37
34
33
Spring 2016
Crisp
40
41
36
36
40
38
41
42
39
4.22.
The effectbelow
of five
different
ingredients
(A, B,byC,including
D, E) onthereaction
time
a chemical
process
is E
The ANOVA
identifies
a very
small impact
blocks in
theof
analysis.
In fact,
the MS
being
studied.
Each
batch
of
new
material
is
only
large
enough
to
permit
five
runs
to
be
made.
actually increases from 3.50 in Problem 5.30 to 3.97 with the inclusion of the blocks due to the reduction of
Furthermore,
run of
requires
approximately
1 1/2
hours,the
so MS
only
five runs can be made in one day. The
the residual each
degrees
freedom
from 12 to 10.
Because
E is greater than the MS Blocks , the variance
experimenter
decides
to
run
the
experiment
as
a
Latin
square
so
that day and batch effects can be
component estimate for blocks is zero.
systematically controlled. She obtains the data that follow. Analyze the data from this experiment (use α =
0.05)
andExpert
drawOutput
conclusions.
Design
Response:
Speed
ANOVA for selected factorial model
Analysis of variance table [Classical sum of squares - Type II]
Sum Batch
of
1 Mean2
Source
Squares1
df A=8 Square
B=7
Block
2.33
2
1.17
E=2
Model
140.00 2
5C=11 28.00
A=9
A-Key Size
12.33 3
2 B=4
6.17
B-Keyboard Feel
117.56 4
1 D=6 117.56
C=8
AB
10.11 5
2 E=4
5.06
D=2
Residual
39.67
10
3.97
Cor Total
182.00
17
Day
3
D=1
A=7
C=10
E=6
B=3
F 4
Value
C=7
D=3
7.06
E=1
1.55
29.64
B=6
1.27
A=8
5
E=3
B=8
D=5
A=10
C=8
p-value
Prob > F
0.0045
0.2583
0.0003
0.3213
significant
The Minitab output below identifies the ingredients as having a significant effect on reaction time.
2. Problem 5.41 (Montgomery)
5.41.
The C. F. Eye Care company manufactures lenses for transplantation into the eye following
cataract surgery. An engineering group has conducted an experiment involving two factors to determine
their effect on the lens polishing process. The results of this experiment are summarized in the following
ANOVA display:
Source
Factor A
Factor B
Interaction
Error
Total
DF
----2
6
11
SS
--4-34
96.333
122.167
10.000
118.667
MS
0.0833
96.3333
6.0833
---
F
0.05
57.80
3.65
P-value
0.952
<0.001
---
(a) The sum of squares for factor A is 0.1666.
(a)(b)The
of squares
is factor A in the experiment
.
Thesum
number
of degreesfor
of factor
freedomAfor
is 2.
Thenumber
number of
freedom
for factor
B is 1.A is
(b)(c)The
ofdegrees
degreesof of
freedom
for factor
(d) The mean square for error is 1.666.
(c)(e)The
of degrees
of freedom
factor test
B isstatistic is 0.1.
An number
upper bound
for the P-value
for the for
interaction
Themean
engineers
usedfor
3 levels
(d)(f)The
square
errorofisfactor A in this experiment.
.
(g) The engineers used 2 levels of factor B in this experiment.
.
.
(e) The p-value (or its upper bound) for the intercation test is
(f) The engineers used
levels of factor A in the experiment.
(g) The engineers used
levels of factor B in the experiment.
(h) There are
.
5-71
replicates in this experiment.
(i) Would you conclude that effect of factor B depends on the level of factor A?(yes/no)
.
(j) Give a resaon for your answer in the above question:
(k) An estimate of the standard deviation of the response variable is
1
.
.
3. A medical experiment is run to determine the side effects on children when they take various dosages
of a drug administered by different methods. A twoway factorial with 4 dosages (0.5, 1.0, 1.5, and 2.0
milligrams) and 3 methods of administering (oral, extended release, intravenous) is used in a completely
randomized design with each treatment combination replicated twice. The response variable is the amount
of a certain substance present in the liver after 24 hours. The data are:
Method
1
2
3
0.5
0.414
Dosage
1.0
1.5
0.541 0.592
2.0
0.672
0.312
0.423
0.575
0.610
0.537
0.513
0.595
0.709
0.451
0.580
0.573
0.623
0.572
0.622
0.613
0.695
0.554
0.597
0.650
0.751
Use JMP to analyze these data to determine effects of dosage and methods of administration. Extract
numbers from the JMP output to present your own written answers to the parts below. Assume the model
yijk = µij + ijk , i = 1, . . . , 3; j = 1, . . . , 4; k = 1, . . . , 2 where ijk are iid N (0, σ 2 ).
(a) Estimate the cell means µij and report these in a table. Obtain a scatterplot of the cell means with
Dosage on the x-axis. Join the means corresponding to each Method by line segments.
(b) Construct an analysis of variance table and test the hypothesis of no interaction and each main effects
using the p-values and α = .05. State conclusion from each test and state how you would proceed on
the basis of this analysis.
(c) Does the graph in part a) support your conclusion from the test for interaction? Discuss.
(d) Use the LSD procedure with α = .05 to compare differences among Method means and/or Dosage
means. State your conclusions specific to this problem.
(e) The hypothesis that the average effects of dosage is linearly related to the level of dosage can be tested
by constructing a contrast of the dosage means to fit an orthogonal polynomial. Give the value of a
t-test/F-test statistic and the corresponding p-value for doing this and state your conclusion.
(f) Calculate the 95% confidence interval (entirely by hand) for the difference in means for Method 1 and
Method 2 at the dosage level of 0.5 mg. Show work.
(g) Calculate the residuals and obtain plots to check adequacy of the model.
Note: Need to present written answers to each part, with calculation shown for parts that you are required
to do hand computation. Use the JMP output to obtain numbers for answering other parts. Attach edited
JMP output when you use the JMP output to extract numbers as part of the analysis. The JMP data files
side-effects.jmp, and 4-22.jmp are available to download.
Due Monday, March 21st, 2016 (turn-in at the beginning of class)
2