Exam 1 Review
Chapter 1
Probability: a measure of how many likely an event is to occur
o Prob = 1, event will happen
o Prob =0, will not happen
Population: all individuals who are of interest to a researcher
Sample: the individuals who were actually co
Statistics l200 Exam #2 RI!
SP20I0 I Form Fl March 15. 20") 100 poin
Name: Student #: Rec. sect:
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Nick
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Instructions
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Statistics l200 Final Exam Ric:
F8200? I Form Fl December 13, 2007 140 points
Namaj Student #:V 7 Rec. sect:
Circle the name of 'our recitation instructor below:
Chen Ellebracht
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L iu Yau
Instructions
Carry all computations to at least three decima
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5.6
Checking the Equal Variance Assumption
113
2
2
ratio of the largest of the v treatment variance estimates to the smallest, smax /smin , does
not exceed three. The rule of thumb is suggested by simulation studies in which the true
variances i2 are spec
112
Chapter 5
Checking Model Assumptions
zit
T
3
2
b
b
b
b
1
0
1
2
b
b
b
b
b
b
b
b
b
b
b
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b
b
b
b
b
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b
3
b
b
b
bb
b
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b
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bb
b
b
b
b
b
b
b
b
Figure 5.6
E
Megaphoneshaped
residual plot
30
60
90
120
yit
are higher). Thus, unle
110
Chapter 5
Checking Model Assumptions
zit
T
3
2
b
1
0
b
b b
b
1
2
b
b
b
b
b
b
b
b
b
b
b
3
Figure 5.4
E
Residual plot for the
battery experiment
5
10
15
Run Order
the plot were to exhibit a strong pattern, then this would indicate a serious violation
5.6
Checking the Equal Variance Assumption
111
2
If an analysis is conducted under the assumptions of model (3.3.1) when, in fact, the
error variables are dependent, the true signicance levels of hypothesis tests can be much
higher than stated and the tru
5.5
109
Checking Independence of the Error Terms
zit
T
3
2
1
0
1
2
Figure 5.3
Residual plot after
data correction for the
battery experiment
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
3
E
1
2
3
4
Battery type
value has been copied incorrectly at some stage. If the
5.3
Checking the Fit of the Model
107
residuals for treatment 2 seems a little larger than the spread for the other three treatments.
This could be interpreted as a sign of unequal variances of the error variables or that the data
values having standardiz
108
Chapter 5
Checking Model Assumptions
zit
T
3
2
1
0
1
b
b
b
2
Figure 5.2
Original residual plot
for the battery
experiment
b
b
b
b
b
b
b
b
bb
b
b
3
1
2
E
3
4
Battery type
and +2, and approximately 99.7% between 3 and +3. If there are more outliers t
104
Chapter 5
Checking Model Assumptions
the adequacy of the model can be checked. Even if a pilot experiment has been used to
help select the model, it is still important to check that the chosen model is a reasonable
description of the data arising from
5
Checking Model Assumptions
5.1 Introduction
5.2 Strategy for Checking Model Assumptions
5.3 Checking the Fit of the Model
5.4 Checking for Outliers
5.5 Checking Independence of the Error Terms
5.6 Checking the Equal Variance Assumption
5.7 Checking the
643
Exercises
7. Consider the following mixed model:
+ i + Bj + Ck + m + (B)ij + ()im
Yij kmt
i
+ (B)j m + (C)km + (B)ij m + ij kmt ,
1, . . . , a, j 1, . . . , b, k 1, . . . , c,
m
1, . . . , d, t
1, . . . , r,
2
2
2
Bj N (0, B ), Ck N (0, C ), (B)ij N
639
Exercises
Table 17.15
SAS analysis of variance for the ice cream experiment
The SAS System
General Linear Models Procedure
Dependent Variable: MELTTIME
Sum of
Mean
Source
DF
Squares
Square F Value
Model
4
250538.12
62634.53
13.93
Error
28
125927.94
44
638
Chapter 17
Random Effects and Variance Components
Plot of Z*ORDER.
Figure 17.6
Plot of the
standardized residuals
against order of
observation for the ice
cream experiment
The SAS System
Legend: A = 1 obs, B = 2 obs, etc.
2 +
A

A

A
A A
Z 

A
A

636
Chapter 17
Table 17.13
Random Effects and Variance Components
SAS program for the temperature experiment
DATA TEMPR;
INPUT THERM SITE SUBJ TIME;
LINES;
1 1 1 62.16
1 2 1 61.53
: : :
:
3 2 4 304.58
;
* Note that the option TEST gives correct denominato
17.10
Using SAS Software
635
The SAS System
Plot of AVCONT*NSCORE. Symbol is value of BALE.
Figure 17.5
Normal probability
plot of the
standardized
treatment averages for
the clean wool
experiment
2 +



7


AVCONT 




 6
5


0 +2
4 

3
17.10
Table 17.14
637
Using SAS Software
SAS analysis of variance for the temperature experiment
The SAS System
General Linear Models Procedure
Source
SUBJ
THERM
SITE
THERM*SITE
SITE*SUBJ
Type III Expected Mean Square
Var(Error) + 3 Var(SITE*SUBJ) + 6 Var
632
Chapter 17
Random Effects and Variance Components
However, for illustration purposes, we ask whether the average time taken for these three
digital thermometers to register is the same whether used in the mouth or under the arm.
Thus, we will test the
634
Chapter 17
Random Effects and Variance Components
The SAS System
Plot of Z*PRED. Symbol is value of BALE.
Figure 17.4
Residuals versus
predicted values for
the clean wool
experiment, excluding
the outlier.
Table 17.12
Z 
4 +



2 +
1
4

3

4
5
