STATISTICS 479
Final Exam (200 points)
1. A SAS data set is to be created from raw data using the following input statement:
input state $ city $ pop1994 income housing electric;
Write SAS program statements (with correct syntax) to be included in the data step to accomplish the
tasks in (a) to (d) and answer (e) to (h).
(a) (5) Eliminate observations with pop1994 values less than or equal to 50,000 from the data set.
(b) (5) To not include the variable income in the SAS data set created.
(c) (5) Form a new category variable named housgrp such that it will have values of 1, 2 or 3 depending
on whether housing values are less than or equal to 15,000, over 15,000 but less than or equal to
35,000, or over 35,000, respectively.
(d) (5) To create a new numeric variable named percent containing values for the total electricity consumption and housing costs expressed as a percentage (not fraction) of income.
(e) (5) Briefly say what you need to do to get SAS to print descriptive text strings (e.g., ‘Low Median
Housing’) instead of 1, 2 or 3 for the values of the variable housgrp created in part (c).
(f) (5) Give the name of a SAS procedure that may be used to examine the distribution of numeric
variables such as housing or income. Name someting printed by this procedure useful for studying
shape of the distribution.
(g) (5) What SAS procedure would you use to calculate sample statistics for numeric variables housing
and electric for subgroups of city or housgrp?
(h) (5) What is name of the statement to be added to the proc step in part (g), if you want to create a
SAS data set containing the statistics calculated?.
1
2. (15) Examine the following data step and sketch the output resulting from the proc print statement.
Show the contents of the program data vector at the point the output statement will be executed for the
first time.
data state;
input county 2. @;
if county=73 then do;
input name $2. nfarms 2. ;
do k=1 to nfarms;
input month yield;
output;
end;
end;
drop k;
datalines;
73JS 3
2 124.3
3 48.5
4 237.9
84BV 3
1 284.3
2 93.7
4 205.6
73EL 2
6 89.4
7 172.1
;
run;
proc print data= state; run;
3. (15) Display the printed output produced by executing the following SAS program. Show what is in the
program data vector at the point the first observation is to be written to the SAS data set.
data;
input x1 x2 @@;
x3=x2**2-1;
drop x2;
datalines;
4 –3 0 2 9 · 5 1 16 5
;
proc print; run;
2
4. An experiment was conducted to compare the friction properties of lubricants with three competing additives A, B, and C. The plastic viscosity (x) of the base lubricant was measured for each mixture prior
to the addition of the additives and was used as a covariate. A friction measurement(y) was made on 10
samples from each lubricant. A oneway covariance model was used to analyze the data with proc glm:
data lubricant;
input additive $ @;
do i=1 to 10;
input viscosity
output;
end;
drop i;
datalines;
A 12 27.1 13 26.6 15
B 15 28.6 13 37.1 14
C 14 29.5 12 36.2 13
;
run;
friction
@;
28.9 14 27.1 10 23.6 10 26.4 13 28.1 14 26.1 12 24.4 14 29.1
37.9 14 30.6 13 33.6 13 34.9 13 33.1 14 34.4 14 32.6 17 35.6
35.9 11 28.6 16 31.8 14 38.5 13 36.5 10 33.7 13 34.9 12 35.4
proc print data=lubricant; run;
proc glm data=lubricant;
class .............;
model friction = ............ ............;
.........................................;
estimate ’Additive A vs. (B+C)/2’ ........................;
run;
(a) (5) How many ‘steps’ are in this SAS program?
(b) (5) Sketch the output produced by the proc print; statement.
(c) (5) Name the variable to appear in the class statement?
(d) (5) Complete the model statement in glm (in the correct order) required to obtain the numbers
required to construct an analysis of covariance table.
(e) (5) Include a statement (to appear after the model statement) to compute confidence intervals on the
pairwise differences of the adjusted means:
(f) (5) Complete the estimate statement in glm required to obtain a t-test for comparing Additive A
mean with the average of the means for Additives B and C:
3
5. The clotting times of plasma (in minutes) for four different drugs are compared in a completely randomized
design. Blood samples were taken from each of eight subjects randomly assigned to each of the 4 drugs
A, B, C, and D. The data were input to the SAS program shown on the labeled output page at the end.
Assuming the oneway fixed effects model:
yij = µi + ij
i = 1, . . . , 4,
j = 1, . . . , 6
where µi are the population treatment means and i are i.i.d. N (0, σ 2 ) errors, answer the following questions:
(a) (5) Fill out the following AoV table:
Source
d.f.
SS
MS
F
p-value
Drug
Error
Total
Would you reject the hypothesis H0 : µ1 = µ2 = µ3 = µ4 at α = .05? Why?
(b) (5) Add a new proc step (to be inserted before proc glm in the SAS program) to calculate the
sample means and their standard errors for the 4 drugs.
.
(c) (5) Show a sketch of the output you would expect from your proc step in part (b).
(d) (5) Obtain the estimate s2 of the error variance σ 2 from the output.
(e) (5) Write a means statement to be included in the proc glm step to obtain 95% confidence intervals
for all pairwise differences of the drug means.
(f) (5) Add an estimate statement (with correct syntax) to be includded in the proc glm step to obtain
a t-statistic to test whether the average clotting time for drugs A & B are different from the average
clotting time for drugs C & D.
4
6. Some SAS procedures were used to fit regression models to the following data.
x1
15.57
44.02
20.42
18.74
49.20
44.92
55.48
59.28
94.39
128.02
96.00
131.42
127.21
252.90
x2
24.63
20.48
39.40
65.05
57.23
115.20
57.79
59.69
84.61
201.06
133.13
107.71
155.43
361.94
x3
472.92
1339.75
620.25
568.33
1497.60
1365.83
1687.00
1639.92
2872.33
3655.08
2912.00
3921.00
3865.67
7684.10
x4
18.0
9.5
12.8
36.7
35.7
24.0
43.3
46.7
78.7
180.5
60.9
103.7
126.8
157.7
x5
y
4.45
5.66
6.92
6.97
4.28 10.33
3.90 16.04
5.50 16.11
4.60 16.13
5.62 18.54
5.15 21.60
6.18 23.06
6.15 35.04
5.88 35.72
4.88 37.41
5.50 40.27
7.00 103.44
The SAS procedure proc reg was used to fit the full model, variable subset selection selection=stepwise
with sle = .25, sls = .1), and then fit a reduced model. The labeled SAS output pages (edited) are attached
at the end. Answer the following questions using these results.
(a) (5) Extract the value of R(β3 /β0 , β1 , β2 ) from the output (i.e., without calculating it).
(b) (5) For the 5-variable model, give the value of h11 and use it to compute the standard error of the
residual for observation 1.
(c) (5) For the 5-variable model, give the residual and its standard error for observation 1 and use them
to compute the corresponding studentized residual.
(d) (5) For the 5-variable model, calculate the predicted value ŷ1 and its standard error.
(e) (5) Identify values of a statistic in the output that may lead you to suspect multicollinearity problems
in the 5-variable model. What does this statistic measure?
(f) (5) In the 5-variable model, identify problems with some estimated statistics that reflect the deleterious
(harmful) effects of multicollinearity. Explain.
(g) (5) Extract the value of R(β3 /β0 ) from the output (i.e., without calculating it).
5
(h) (5) What is the value of the F-to-enter variable x2 to the model y = β0 + β3 x3 + (no need to calculate
this)?
(i) (5) Calculate the F-to-enter statistic to enter variable x5 to the model in Step 3.
(j) (5) What is the value of the F-to-delete statistic to remove variable x5 from the model y = β0 + β2 x2 +
β3 x3 + β4 x4 + β5 x5 + (no need to calculate this)? Why is x5 deleted in step 5?
(k) (5) Extract the value of R(β2 /β0 , β3 , β4 ) from the output (i.e., without calculating it).
(l) (5) Provide 3 important reasons why the 3-variable model selected by “stepwise” may be considered
“better” than the 5-variable model based on some statistics shown in the output?
(m) (5) Use RStudent to conduct the Bonferroni test procedure for y-outliers using Table B.10 supplied
using α = .05
(n) (5) Write a plot statement to be added to the proc reg step to obtain a plot of residuals vs.
predicted values.
6
SAS Program and Output for Question#5
data blood;
input drug $ @;
do i = 1 to 8;
input clottime @;
output;
end;
drop i;
datalines;
A 8.4 9.8 9.6 10.8 8.4 8.6 8.9 7.9
B 9.4 10.4 9.1 8.8 8.2 9.8 9.0 8.1
C 9.8 9.2 11.1 9.9 8.5 9.7 10.8 8.2
D 11.2 10.0 9.8 10.8 9.5 10.6 10.4 12.3
;
proc glm order = data;
class drug;
model clottime = drug;
title 'Final Exam:
run;
SAS Program for Question 5 ';
Final Exam:
SAS Program for Question 5
1
The GLM Procedure
Class Level Information
Class
drug
Levels
4
Values
A B C D
Number of Observations Read
Number of Observations Used
Final Exam:
32
32
SAS Program for Question 5
2
The GLM Procedure
Dependent Variable: clottime
Source
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
3
12.04375000
4.01458333
4.85
0.0077
Error
28
23.17500000
0.82767857
Corrected Total
31
35.21875000
Source
DF
Type I SS
Mean Square
F Value
Pr > F
3
12.04375000
4.01458333
4.85
0.0077
DF
Type III SS
Mean Square
F Value
Pr > F
3
12.04375000
4.01458333
4.85
0.0077
drug
Source
drug
A Formula Sheet
PROC GLM: Some relevant syntax
PROC GLM < options > ;
CLASS variables < / option > ;
MODEL dependents=independents < / options > ;
CONTRAST ’label’ effect coefs < ... effect coefs > < / options > ;
ESTIMATE ’label’ effect coefs < ... effect coefs > < / options > ;
LSMEANS effects < / options > ;
MEANS effects < / options > ;
RANDOM effects < / options > ;
Some options for MEANS statement: t, lsd, tukey, alpha=, cldiff, hovtest=,
Some options for LSMEANS statement: stderr, cl, tdiff, pdiff, adjust=
Type I and Type II Sums of Squares in Reduction Notation
Consider the following model statement:
model y = x1 x2 x3 / ss1 ss2 ;
The resulting sums of squares computed by proc glm can be identified using the reduction notation
developed above, as indicated below:
Effect
x1
Type I SS
R(β1 /β0 )
Type II SS
R(β1 /β0 , β2 , β3 )
x2
R(β2 /β0 , β1 )
R(β2 /β0 , β1 , β3 )
x3
R(β3 /β0 , β1 , β2 )
R(β3 /β0 , β1 , β2 )
Hat Diag hii
If the hats are hii for i = 1, . . . , n and s2 = error mean square (MSE):
√
Standard error of ŷi = s hii
Standard error of ei = s
p
(1 − hii )
where ei , for i = 1, . . . , n are the residuals.
Reduction Notation Example:
Reduction in the residual sum of ssquare (SSE) due to the addition of the variable x3 to the model
y = β0 + β1 x1 + β2 x2 + is given by:
R(β3 /β0 , β1 , β2 ) = SSE(β0 , β1 , β2 ) − SSE(β0 , β1 , β2 , β3 )
F-to-enter statistic for entering x3 to the model y = β0 + β1 x1 + β2 x2 + is:
F =
R(β3 /β0 , β1 , β2 )/1
(SSE(β0 , β1 , β2 ) − SSE(β0 , β1 , β2 , β3 ))/1
=
SSE(β0 , β1 , β2 , β3 )/(n − 4)
SSE(β0 , β1 , β2 , β3 )/(n − 4)
Note that (n − 4) = (n − p − 1) where p = 3 is the number of variables in the larger model (i.e.,
the model with the larger number of x-variables). Note that this is the same as the F-to-delete
statistic to delete x3 from the model y = β0 + β1 x1 + β2 x3 + β3 x3 + 472.92
1339.75
620.25
568.33
1497.60
1365.83
1687.00
1639.92
2872.33
3655.08
2912.00
3921.00
3865.67
7684.10
18.0
9.5
12.8
36.7
35.7
24.0
43.3
46.7
78.7
180.5
60.9
103.7
126.8
157.7
4.45
5.66
6.92
6.97
4.28 10.33
3.90 16.04
5.50 16.11
4.60 16.13
5.62 18.54
5.15 21.60
6.18 23.06
6.15 35.04
5.88 35.72
4.88 37.41
5.50 40.27
7.00 103.44
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Obs
5.6600
6.9700
10.3300
16.0400
16.1100
16.1300
18.5400
21.6000
23.0600
35.0400
35.7200
37.4100
40.2700
103.4400
5.1100
8.0106
9.9514
10.2655
14.8857
25.3875
15.5858
18.0188
25.2408
36.4993
35.6152
40.9585
40.3607
100.4301
1.9613
3.6342
2.1217
2.8175
1.6259
2.8557
1.7047
3.6993
2.4856
4.4507
1.6490
3.9526
3.0465
4.3176
Dependent Predicted
Std Error
Variable
Value Mean Predict
0.5500
-1.0406
0.3786
5.7745
1.2243
-9.2575
2.9542
3.5812
-2.1808
-1.4593
0.1048
-3.5485
-0.0907
3.0099
Residual
4.204
2.884
4.126
3.686
4.345
3.656
4.315
2.800
3.917
1.309
4.336
2.429
3.499
1.697
0.131
-0.361
0.0918
1.567
0.282
-2.532
0.685
1.279
-0.557
-1.115
0.0242
-1.461
-0.0259
1.773
Std Error Student
Residual Residual
Output Statistics
The REG Procedure
Model: MODEL1
Dependent Variable: y
Final Exam
proc reg ;
model y = x1 x2 x3 x4 x5/ss1 ss2 vif r influence;
model y = x1 x2 x3 x4 x5/selection=stepwise sle=.
.2 sls=.
.1;
model y = x2 x3 x4/clb vif;
title ' Final Exam: Fall 2006';
run;
data final;
input x1-x5 y;
datalines;
15.57 24.63
44.02 20.48
20.42 39.40
18.74 65.05
49.20 57.23
44.92 115.20
55.48 57.79
59.28 59.69
94.39 84.61
128.02 201.06
96.00 133.13
131.42 107.71
127.21 155.43
252.90 361.94
;
run;
SAS Program and Output for Question#6
|
|
|
|
|
|
|
|***
|
|
| *****|
|
|*
|
|**
|
*|
|
**|
|
|
|
**|
|
|
|
|***
-2-1 0 1 2
|
|
|
|
|
|
|
|
|
|
|
|
|
|
0.001
0.034
0.000
0.239
0.002
0.652
0.012
0.476
0.021
2.394
0.000
0.942
0.000
3.391
Cook's
D
0.1225
-0.3403
0.0859
1.7603
0.2649
-5.3147
0.6601
1.3416
-0.5311
-1.1345
0.0226
-1.5960
-0.0243
2.1293
RStudent
0.1787
0.6137
0.2091
0.3688
0.1228
0.3789
0.1350
0.6358
0.2870
0.9204
0.1263
0.7259
0.4312
0.8661
Hat Diag
H
2
1
1
1
1
1
1
Intercept
x1
x2
x3
x4
x5
8.96211
0.77132
0.04018
0.02443
0.06529
1.85817
Standard
Error
R-Square
Adj R-Sq
0.1904
0.6744
0.0124
0.9973
0.0512
0.1419
Pr > |t|
Stepwise Selection: Step 1
Final Exam
1.43
0.44
3.21
0.00
-2.29
-1.63
t Value
10660
7208.47856
245.11068
99.06005
88.92261
57.12748
Type I SS
0.9781
0.9645
71.54
F Value
-2.12705
0.01220
Intercept
x3
3.13847
0.00102
Standard
Error
7258.83718
612.04416
7870.88134
Sum of
Squares
23.42730
7258.83718
Type II SS
0.46
142.32
0.5108
<.0001
Pr > F
142.32
F Value
F Value
7258.83718
51.00368
Mean
Square
<.0001
Pr > F
44.04674
4.09049
221.82250
0.00027112
112.98223
57.12748
Type II SS
<.0001
Pr > F
4
0
1478.09351
7.88772
1351.98336
7.83840
1.87824
Variance
Inflation
1
DF
2
11
13
Source
Model
Error
Corrected Total
7485.78036
385.10098
7870.88134
Sum of
Squares
Mean
Square
3742.89018
35.00918
Analysis of Variance
106.91
F Value
Variable x2 Entered: R-Square = 0.9511 and C(p) = 9.8927
Stepwise Selection: Step 2
<.0001
Pr > F
Bounds on condition number: 1, 1
-----------------------------------------------------------------------------------------------------------------------
Parameter
Estimate
1
12
13
Model
Error
Corrected Total
Variable
DF
Source
Analysis of Variance
Variable x3 Entered: R-Square = 0.9222 and C(p) = 18.4371
12.82090
0.33626
0.12899
0.00008671
-0.14960
-3.02733
Parameter
Estimate
4.63926
27.59429
16.81240
Root MSE
Dependent Mean
Coeff Var
Mean
Square
14
14
1539.73988
21.52274
Parameter Estimates
7698.69939
172.18196
7870.88134
5
8
13
Model
Error
Corrected Total
Sum of
Squares
Analysis of Variance
Number of Observations Read
Number of Observations Used
Model: MODEL1
Dependent Variable: y
DF
Stepwise Method
DF
Variable
Final Exam
The REG Procedure
Source
The 5-variable model fit
-2.28093
0.12266
0.00693
Intercept
x2
x3
2.60091
0.04818
0.00224
Standard
Error
26.92512
226.94318
335.77745
Type II SS
0.77
6.48
9.59
F Value
0.3992
0.0272
0.0102
Pr > F
-1.35732
0.14566
0.00889
-0.12167
Intercept
x2
x3
x4
2.13858
0.04005
0.00196
0.04689
Standard
Error
7640.74620
230.13515
7870.88134
9.27031
304.34614
470.82561
154.96584
Type II SS
0.40
13.22
20.46
6.73
0.5399
0.0046
0.0011
0.0267
Pr > F
110.67
F Value
2546.91540
23.01351
F Value
Pr > F
<.0001
Stepwise Selection: Step 5
Intercept
x2
x3
x4
2.13858
0.04005
0.00196
0.04689
Standard
Error
7640.74620
230.13515
7870.88134
Sum of
Squares
9.27031
304.34614
470.82561
154.96584
Type II SS
0.40
13.22
20.46
6.73
0.5399
0.0046
0.0011
0.0267
Pr > F
110.67
F Value
F Value
2546.91540
23.01351
Mean
Square
<.0001
Pr > F
The stepwise method terminated because the next variable to be entered was just removed.
All variables left in the model are significant at the 0.1000 level.
Bounds on condition number: 8.1762, 57.861
-----------------------------------------------------------------------------------------------------------------------
-1.35732
0.14566
0.00889
-0.12167
Variable
3
10
13
Model
Error
Corrected Total
Parameter
Estimate
DF
Source
Analysis of Variance
Variable x5 Removed: R-Square = 0.9708 and C(p) = 4.6926
12.31504
0.12879
0.01069
-0.12922
-2.90687
Parameter
Estimate
4
9
13
DF
8.47738
0.03833
0.00211
0.04349
1.75288
Standard
Error
7694.60890
176.27245
7870.88134
Sum of
Squares
Mean
Square
41.33236
221.15491
500.75068
172.89593
53.86270
Type II SS
2.11
11.29
25.57
8.83
2.75
0.1803
0.0084
0.0007
0.0157
0.1316
Pr > F
98.22
F Value
F Value
1923.65222
19.58583
Analysis of Variance
<.0001
Pr > F
DF
1
1
1
1
Variable
Intercept
x2
x3
x4
-1.35732
0.14566
0.00889
-0.12167
Parameter
Estimate
Final Exam
2.13858
0.04005
0.00196
0.04689
Standard
Error
-0.63
3.64
4.52
-2.59
t Value
Mean
Square
14
14
R-Square
Adj R-Sq
0.5399
0.0046
0.0011
0.0267
Pr > |t|
0.9708
0.9620
110.67
F Value
0
7.33086
8.17623
3.77989
Variance
Inflation
2546.91540
23.01351
Parameter Estimates
4.79724
27.59429
17.38490
Root MSE
Dependent Mean
Coeff Var
Sum of
Squares
7640.74620
230.13515
7870.88134
DF
Analysis of Variance
Number of Observations Read
Number of Observations Used
Model: MODEL3
Dependent Variable: y
The REG Procedure
3
10
13
Model
Error
Corrected Total
Source
The 3-variable model fit
-6.12237
0.05641
0.00451
-0.22614
3.40773
0.23490
0.01326
-0.01720
95% Confidence Limits
<.0001
Pr > F
9
-----------------------------------------------------------------------------------------------------------------------
Intercept
x2
x3
x4
x5
Mean
Square
Sum of
Squares
Bounds on condition number: 8.1762, 57.861
-----------------------------------------------------------------------------------------------------------------------
Parameter
Estimate
3
10
13
Model
Error
Corrected Total
Variable
DF
Source
Variable
Model
Error
Corrected Total
Source
Stepwise Selection: Step 4
Variable x5 Entered: R-Square = 0.9776 and C(p) = 4.1901
Analysis of Variance
Variable x4 Entered: R-Square = 0.9708 and C(p) = 4.6926
Stepwise Selection: Step 3
Bounds on condition number: 6.972, 27.888
-----------------------------------------------------------------------------------------------------------------------
Parameter
Estimate
Variable
B Tables
539
Table B.10. 5% critical values based on the Bonferroni bounds for the t-test for a
single outlier using externally studentized residual in a linear regression model.
k
n
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
50
60
70
80
90
100
1
2
3
9.92
6.23
5.07
4.53
4.22
4.03
3.90
3.81
3.74
3.69
3.65
3.62
3.59
3.57
3.56
3.54
3.53
3.52
3.52
3.51
3.50
3.50
3.50
3.50
3.49
3.49
3.49
3.49
3.49
3.49
3.49
3.49
3.49
3.49
3.49
3.49
3.51
3.53
3.55
3.57
3.59
3.60
63.66
10.89
6.58
5.26
4.66
4.32
4.10
3.96
3.86
3.79
3.73
3.68
3.65
3.62
3.60
3.58
3.57
3.55
3.54
3.53
3.53
3.52
3.52
3.51
3.51
3.51
3.50
3.50
3.50
3.50
3.50
3.50
3.50
3.50
3.50
3.50
3.51
3.53
3.55
3.57
3.59
3.60
76.39
11.77
6.90
5.44
4.77
4.40
4.17
4.02
3.91
3.83
3.77
3.72
3.68
3.65
3.62
3.60
3.59
3.57
3.56
3.55
3.54
3.54
3.53
3.53
3.52
3.52
3.52
3.52
3.51
3.51
3.51
3.51
3.51
3.51
3.51
3.52
3.54
3.55
3.57
3.59
3.60
4
5
6
7
8
9
10
11
12
89.12
12.59 101.86
7.18 13.36 114.59
5.60 7.45 14.09 127.32
4.88 5.75 7.70 14.78 140.05
4.49 4.98 5.89 7.94 15.44 152.79
4.24 4.56 5.08 6.02 8.16 16.08 165.52
4.07 4.30 4.63 5.16 6.14 8.37 16.69 178.25
3.95 4.12 4.36 4.70 5.25 6.25 8.58 17.28 190.98
3.87 4.00 4.17 4.41 4.76 5.33 6.36 8.77 17.85
3.80 3.90 4.04 4.21 4.46 4.82 5.40 6.47 8.95
3.75 3.83 3.94 4.08 4.26 4.51 4.88 5.47 6.57
3.71 3.78 3.86 3.97 4.11 4.30 4.55 4.93 5.54
3.67 3.73 3.81 3.89 4.00 4.15 4.33 4.59 4.98
3.65 3.70 3.76 3.83 3.92 4.03 4.18 4.37 4.64
3.63 3.67 3.72 3.78 3.86 3.95 4.06 4.21 4.40
3.61 3.65 3.69 3.75 3.81 3.88 3.98 4.09 4.24
3.59 3.63 3.67 3.71 3.77 3.83 3.91 4.00 4.12
3.58 3.61 3.65 3.69 3.73 3.79 3.85 3.93 4.02
3.57 3.60 3.63 3.66 3.70 3.75 3.81 3.87 3.95
3.56 3.58 3.61 3.65 3.68 3.72 3.77 3.83 3.89
3.55 3.58 3.60 3.63 3.66 3.70 3.74 3.79 3.84
3.55 3.57 3.59 3.62 3.64 3.68 3.71 3.76 3.81
3.54 3.56 3.58 3.60 3.63 3.66 3.69 3.73 3.77
3.54 3.55 3.57 3.59 3.62 3.64 3.67 3.71 3.74
3.53 3.55 3.57 3.59 3.61 3.63 3.66 3.69 3.72
3.53 3.54 3.56 3.58 3.60 3.62 3.64 3.67 3.70
3.53 3.54 3.56 3.57 3.59 3.61 3.63 3.66 3.68
3.52 3.54 3.55 3.57 3.58 3.60 3.62 3.64 3.67
3.52 3.54 3.55 3.56 3.58 3.60 3.61 3.63 3.66
3.52 3.53 3.55 3.56 3.57 3.59 3.61 3.62 3.65
3.52 3.53 3.54 3.56 3.57 3.58 3.60 3.62 3.64
3.52 3.53 3.54 3.55 3.57 3.58 3.59 3.61 3.63
3.52 3.53 3.54 3.55 3.56 3.58 3.59 3.60 3.62
3.53 3.53 3.54 3.54 3.55 3.56 3.57 3.57 3.58
3.54 3.54 3.55 3.55 3.56 3.56 3.57 3.57 3.58
3.56 3.56 3.56 3.56 3.57 3.57 3.57 3.58 3.58
3.57 3.58 3.58 3.58 3.58 3.58 3.59 3.59 3.59
3.59 3.59 3.59 3.60 3.60 3.60 3.60 3.60 3.60
3.61 3.61 3.61 3.61 3.61 3.61 3.61 3.62 3.62
(continued)