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BA 555 Practical Business Analysis
Agenda
 Review of Statistics


Confidence Interval Estimation
Hypothesis Testing
 Linear Regression Analysis



Introduction
Case Study: Cost of Manufacturing Computers
Simple Linear Regression
1
The Empirical Rule (p.5)
1. Approximately 68% of the observations will fall within 1 standard deviation of the mean.
2. Approximately 95% of the observations will fall within 2 standard deviations of the mean.
3. Approximately 99.7% of the observations will fall within 3 standard deviations of the mean.
99.7%
99.7%
95%
95%
68%
0.15%
x  3s
  3
x  3s
3
0.15%
  3
3
2.35%
13.5%
68%
34%
34%
13.5%
x  s x 34%x  s 34% x  2s13.5%x  3s
x  2s 13.5%
 x22 s   x s  x x  s2   3
x  2s
1 0
1
2
3
 
  2
  2
  2
0
1
1
2
2
2.35%
2.35%
0.15%
2.35%
0.15%
x  3s
  3
3
2
Review Example

Suppose that the average hourly earnings of
production workers over the past three years were
reported to be $12.27, $12.85, and $13.39 with the
standard deviations $0.15, $0.18, and $0.23, respectively.



The average hourly earnings of the production
workers in your company also continued to rise
over the past three years from $12.72 in 2002,
$13.35 in 2003, to $13.95 in 2004.
Assume that the distribution of the hourly earnings
for all production workers is mound-shaped.
Do the earnings in your company become less and
less competitive? Why or why not.
3
Review Example
Year
Industry
average
Industry
std.
2002
12.27
0.15
2003
12.85
0.18
4.73%
13.35
4.95%
2.77
2004
13.39
0.23
4.20%
13.95
4.50%
2.43
%
increase
Company
average
%
increase
12.72
Z score
3
4
The Empirical Rule
 Generalize the results from the empirical rule.
 Justify the use of the mound-shaped
distribution.
99.7%
95%
68%
0.15%
2.35%
13.5%
34%
34%
13.5%
2.35%
0.15%
x  3s
  3
x  2s
xs
x
xs
x  2s
  2
 
 
  2
3
2
1

0
x  3s
  3
1
2
3
5
Sampling Distribution (p.6)
 The sampling distribution of a statistic is
the probability distribution for all possible
values of the statistic that results when
random samples of size n are repeatedly
drawn from the population.
 When the sample size is large, what is the
sampling distribution of the sample mean /
sample proportion / the difference of two
samples means / the difference of two sample
proportions?  NORMAL !!!
6
Central Limit Theorem (CLT) (p.6)
 If X ~ N(,  ), then X ~ N( 
2
X
,
 
2
X

2
n
)
Sample 1 : X 11, X 12 , , X 1n  X 1
Sample 2 : X 21, X 22 , , X 2 n  X 2
Sample 3 : X 31, X 32 , , X 3n  X 3
Sample 4 : X 41, X 42 , , X 4 n  X 4
:::::::
7
Central Limit Theorem (CLT) (p.6)
 If X ~ Any distribution with the mean , and
variance 2, then X ~ N(    ,   n ) for large n.
2
X
2
X
Sample 1 : X 11, X 12 , , X 1n  X 1
Sample 2 : X 21, X 22 , , X 2 n  X 2
Sample 3 : X 31, X 32 , , X 3n  X 3
Sample 4 : X 41, X 42 , , X 4 n  X 4
:::::::
8
Summary: Sampling Distributions
 The sampling distribution of a sample mean
 The sampling distribution of a sample
proportion
 The sampling distribution of the difference
between two sample means
 The sampling distribution of the difference
between two sample proportions
9
Standard Deviations





Population standard deviation  X or simply  .
Sample standard deviation s X or simply s .
Standard deviation of sample means (aka. standard error)  X
Standard deviation of sample proportions (aka. standard error)  p̂
Relationships:
o
o
X 
 pˆ 
X
n

sX
n
p(1  p)

n
: ˆ X or s X
pˆ (1  pˆ )
: ˆ pˆ or s pˆ
n
10
Statistical Inference: Estimation
Population
Research Question:
What is the parameter value?
Sample of size n
Tools (i.e., formulas):
Point Estimator
Interval Estimator
11
Confidence Interval Estimation (p.7)
12
Example 1: Estimation for the
population mean
 A random sampling of a company’s weekly
operating expenses for a sample of 48 weeks
produced a sample mean of $5474 and a
standard deviation of $764. Construct a 95%
confidence interval for the company’s mean
weekly expenses.
Example 2: Estimation for the population proportion
13
Statistical Inference: Hypothesis
Testing
Population
Research Question:
Is the claim supported?
Sample of size n
Tools (i.e., formulas):
z or t statistic
14
Hypothesis Testing (p.9)
15
Example
 A bank has set up a customer service goal
that the mean waiting time for its customers
will be less than 2 minutes. The bank
randomly samples 30 customers and finds
that the sample mean is 100 seconds.
Assuming that the sample is from a normal
distribution and the standard deviation is 28
seconds, can the bank safely conclude that
the population mean waiting time is less than
2 minutes?
16
Setting Up the Rejection Region
Type I Error
 If we reject H0 (accept Ha) when in fact H0 is
true, this is a Type I error.
 False Alarm.
17
The P-Value of a Test (p.11)
 The p-value or observed significance level is
the smallest value of a for which test results
are statistically significant.
“the conclusion of rejecting H0 can be reached.”
18
Regression Analysis
 A technique to examine the relationship
between an outcome variable (dependent
variable, Y) and a group of explanatory
variables (independent variables, X1, X2, …
Xk).
 The model allows us to understand (quantify)
the effect of each X on Y.
 It also allows us to predict Y based on X1, X2,
…. Xk.
19
Types of Relationship
 Linear Relationship
 Simple Linear Relationship
 Y = b0 + b1 X + e
 Multiple Linear Relationship
 Y = b0 + b1 X1 + b2 X2 + … + bk Xk + e
 Nonlinear Relationship
 Y = a0 exp(b1X+e)
 Y = b0 + b1 X1 + b2 X12 + e
 … etc.
 Will focus only on linear relationship.
20
Simple Linear Regression Model
population
Y  b 0  b1 X  e
True effect of X on Y
Estimated effect of X on Y
sample
Key questions:
1. Does X have any effect on Y?
2. If yes, how large is the effect?
3. Given X, what is the estimated Y?
Yˆ  bˆ0  bˆ1 X
21
Least Squares Method
 Least squares line: Yˆ  bˆ0  bˆ1 X
 It is a statistical procedure for finding the “best-
fitting” straight line.
 It minimizes the sum of squares of the deviations of
the observed values of Y from those predicted Yˆ
Deviations are minimized.
Bad fit.
22
Case: Cost of Manufacturing
Computers (pp.13 – 45)
 A manufacturer produces computers. The goal is to
quantify cost drivers and to understand the variation
in production costs from week to week.
 The following production variables were recorded:





COST: the total weekly production cost (in $millions)
UNITS: the total number of units (in 000s) produced
during the week.
LABOR: the total weekly direct labor cost (in $10K).
SWITCH: the total number of times that the production
process was re-configured for different types of
computers
FACTA: = 1 if the observation is from factory A; = 0 if
from factory B.
23
Raw Data (p. 14)
Case
FactA
1
2
3
4
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
1
0
1
1
1
0
0
0
1
0
1
1
1
0
1
0
0
1
0
1
1
1
0
1
1
1
1
0
1
1
1
Units
(1000)
1.104
1.044
1.020
0.986
0.972
1.005
0.953
1.083
0.978
0.993
0.958
0.945
1.012
0.974
0.910
1.086
0.962
0.941
1.046
0.955
1.096
1.004
0.997
0.967
1.068
1.041
0.989
1.001
1.008
1.001
0.984
Switch
8
12
12
6
13
11
10
9
9
12
12
13
7
10
11
7
11
10
9
11
12
9
8
13
6
11
16
10
9
7
10
Labor
(10,000)
5.591181
6.836490
5.906357
5.050069
4.790412
5.474329
5.614134
6.002122
5.971627
5.679461
4.320123
5.884950
4.593554
4.915151
4.969754
5.722599
6.109507
5.006398
6.141096
5.019560
5.741166
4.990734
4.662818
6.150249
6.038454
4.988593
6.104960
4.605764
5.529746
4.941728
6.456427
Cost
(million)
1.155456
1.144198
1.141490
1.119656
1.124815
1.137339
1.121275
1.153224
1.119525
1.134635
1.119386
1.113543
1.132124
1.131238
1.104976
1.151547
1.127478
1.114058
1.140872
1.111290
1.159044
1.127805
1.130661
1.127073
1.141041
1.140319
1.130172
1.135118
1.121326
1.124284
1.115016
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
41
42
43
44
45
46
47
48
49
50
51
52
1
0
1
0
0
1
0
1
1
1
0
1
1
1
1
0
1
1
1
1
1
0
0
1
0
0
1
0
0
0
0
0
0
0
0
1
1
0
0
0
How many possible regression models can we build?
1.012
0.974
0.910
1.086
0.962
0.941
1.046
0.955
1.096
1.004
0.997
0.967
1.068
1.041
0.989
1.001
1.008
1.001
0.984
0.981
0.944
0.967
1.018
0.902
1.049
1.024
1.044
1.018
0.937
0.942
1.061
0.901
1.078
1.030
0.981
1.011
1.016
1.008
1.059
1.019
7
10
11
7
11
10
9
11
12
9
8
13
6
11
16
10
9
7
10
12
11
10
9
9
11
11
7
9
11
9
11
7
9
10
8
10
9
9
11
13
4.593554
4.915151
4.969754
5.722599
6.109507
5.006398
6.141096
5.019560
5.741166
4.990734
4.662818
6.150249
6.038454
4.988593
6.104960
4.605764
5.529746
4.941728
6.456427
7.058013
4.626091
4.054482
5.820684
4.932339
5.798058
5.528302
6.635490
5.617445
5.275923
2.927715
6.750682
5.029670
7.005407
4.885713
6.362366
6.261692
5.677634
6.630767
6.930117
6.415978
1.132124
1.131238
1.104976
1.151547
1.127478
1.114058
1.140872
1.111290
1.159044
1.127805
1.130661
1.127073
1.141041
1.140319
1.130172
1.135118
1.121326
1.124284
1.115016
1.124353
1.116318
1.128517
1.150238
1.094061
1.143793
1.145135
1.142156
1.140285
1.114418
1.115774
1.154069
1.105335
1.153367
1.146934
1.130423
1.130929
1.136349
1.140616
1.154121
1.142435
24
Simple Linear Regression Model (pp.
17 – 26)
 Question1: Is Labor a significant cost driver?

This question leads us to think about the
following model: Cost = f(Labor) + e.
Specifically, Cost = b0 + b1 Labor + e
 Question 2: How well does this model
perform? (How accurate can Labor predict
Cost?)

This question leads us to try other regression
models and make comparison.
25
Initial Analysis (pp. 15 – 16)
 Summary statistics + Plots (e.g., histograms + scatter
plots) + Correlations
 Things to look for



Features of Data (e.g., data range, outliers)
 do not want to extrapolate outside data range because
the relationship is unknown (or un-established).
 Summary statistics and graphs.
Is the assumption of linearity appropriate?
Inter-dependence among variables? Any potential
problem?
 Scatter plots and correlations.
26
Correlation (p. 15)
 r (rho): Population correlation (its value most likely is unknown.)
 r: Sample correlation (its value can be calculated from the
sample.)
 Correlation is a measure of the strength of linear relationship.
 Correlation falls between –1 and 1.
 No linear relationship if correlation is close to 0. But, ….
r = –1
r = –1
–1 < r < 0
–1 < r < 0
r=0
r=0
0<r<1
0<r<1
r=1
r=1
27
Correlation (p. 15)
Sample size
Cost
Cost
Units
0.9297
(
52)
0.0000
Units
0.9297
(
52)
0.0000
Switch
-0.0232
(
52)
0.8706
-0.1658
(
52)
0.2402
Labor
0.4520
(
52)
0.0008
0.4603
(
52)
0.0006
Switch
-0.0232
(
52)
0.8706
Labor
0.4520
(
52)
0.0008
-0.1658
(
52)
0.2402
0.4603
(
52)
0.0006
0.1554
(
52)
0.2714
0.1554
(
52)
0.2714
P-value for
Is 0.9297 a r or r?
H0: r = 0
Ha: r ≠ 0
28
Fitted Model (Least Squares Line)
(p.18)
Regression Analysis - Linear model: Y = a + b*X
Dependent variable: Cost
Independent variable: Labor
Standard
T
Parameter
Estimate
Error
Statistic
Intercept
Slope
1.08673
0.00810182
P-Value
0.0127489
85.2409
0.0000
0.00226123
3.58293
0.0008
Analysis of Variance
Source
Sum of Squares
Df Mean Square
Model
0.00231465
1
0.00231465
b
or
b
?
b
0
0
0
b1 or
b1 b1? 0.00901526
Sb0
Residual
50 0.000180305
Total (Corr.)
0.0113299 Sb151
Yˆ  1.08673  0.0081X
Estimate - 0
T Statistic 
Standard Error
F-Ratio
12.84
P-Value
0.0008
H0: b1 = 0
Ha: b1 ≠ 0
** Divide the p-value by 2
for one-sided test. Make
sure there is at least weak
evidence for doing this step.
Degrees of freedom = n – k – 1, where n = sample size, k = # of Xs.
29
Hypothesis Testing and Confidence
Interval Estimation for b (pp. 19 – 20)
Q1: Does Labor have any impact on Cost → Hypothesis Testing
Q2: If so, how large is the impact? → Confidence Interval Estimation
Regression Analysis - Linear model: Y = a + b*X
Dependent variable: Cost
Independent variable: Labor
Standard
T
Parameter
Estimate
Error
Statistic
Intercept
Slope
1.08673
0.00810182
Source
Model
b1
b0
Residual
Total (Corr.)
P-Value
0.0127489
85.2409
0.0000
0.00226123
3.58293
0.0008
Analysis of Variance
Sum of Squares
Df Mean Square
0.00231465
1
0.00231465
S
Sb0
b1
0.00901526
50 0.000180305
0.0113299
51
F-Ratio
12.84
Degrees of freedom = n – k – 1
k = # of independent variables
P-Value
0.0008
30
Analysis of Variance (p. 21)
Analysis of Variance
Sum of Squares
Df
Mean Square
Source
0.00231465
0.00901526
0.0113299
Model
Residual
Total (Corr.)
1
50
51
0.00231465
0.000180305
F-Ratio
P-Value
12.84
0.0008
- Not very useful in simple regression.
- Useful in multiple regression.
n
Syy = SS Total =
(y
i 1
i
 y ) 2  0.0113299.
n
SSR =SS of Regression Model =  ( yˆ i  y ) 2  0.00231465.
i 1
n
SSE = SS of Error =
(y
i 1
i
 yˆ i ) 2  0.00901526.
SS Total = SS Model + SS Error.
31
Sum of Squares (p.22)
Syy = Total variation in Y
SSE = remaining variation that can
not be explained by the model.
SSR = Syy – SSE = variation in Y that has been explained by the model.
32
Fit Statistics (pp. 23 – 24)
Source
Analysis of Variance
Sum of Squares
Df
Mean Square
Model
0.00231465
Residual
0.00901526
1 0.002315
50 0.000180
Total (Corr.)
0.0113299
51
F-Ratio
P-Value
12.84
0.0008
Correlation Coefficient = 0.45199
0.45199 x 0.45199 = 0.204295
R-squared = 20.4295 percent
R-squared (adjusted for d.f.) = 18.8381 percent
Standard Error of Est. = 0.0134278
SS Model
SS Re sidual

 1
SSTotal
SSTotal
SSResidual

 MS Re sidual
n  k 1
33
Prediction (pp. 25 – 26)
 What is the predicted production cost of a given week, say,
Week 21 of the year that Labor = 5 (i.e., $50,000)?
 Point estimate: predicted cost = b0 + b1 (5) = 1.0867 +
0.0081 (5) = 1.12724 (million dollars).
 Margin of error? → Prediction Interval
 What is the average production cost of a typical week that Labor
= 5?
 Point estimate: estimated cost = b0 + b1 (5) = 1.0867 +
0.0081 (5) = 1.12724 (million dollars).
 Margin of error? → Confidence Interval

100(1-a)% prediction interval:

( xg  x )2
1
yˆ  ta / 2 Standard Error of Est. ) 1  
,
n n  1)variance of X )
100(1-a)% confidence interval:
yˆ  ta / 2 Standard Error of Est. )
( xg  x )2
1

,
n n  1)variance of X )
34
Prediction vs. Confidence Intervals
(pp. 25 – 26)
X
3.0
4.0
5.0☻
6.0
Predicted
Y
1.11103
1.11913
1.12724☻
1.13534
95.00%
Prediction Limits
Lower
Upper
1.08139
1.14067
1.09098
1.14729
1.09988☻
1.15459☻
1.10804
1.16263
95.00%
Confidence Limits
Lower
Upper
1.09874
1.12332
1.11105
1.12722
1.12267☻
1.1318☻
1.13113
1.13954
95% Prediction and Confidence Intervals for Cost
Cost ($ million)
1.17
☺
1.15
☺
☺
☺
1.13
1.11
☺
1.09
☺
2.9
3.9
4.9
5.9
6.9
Labor ($10,000)
Variation (margin of error) on both ends seems larger. Implication?
7.9
35
Another Simple Regression Model:
Cost = b0 + b1 Units + e (p. 27)
Regression Analysis - Linear model: Y = a + b*X
Standard
T
Parameter
Estimate
Error
Statistic
Intercept
0.849536
0.0158346
53.6506
Slope
0.281984
0.0157938
17.8541
P-Value
0.0000
0.0000
Analysis of Variance
Source
Model
Residual
Total (Corr.)
Sum of Squares
0.00979373
0.00153618
0.0113299
Df Mean Square
1
0.00979373
50 0.0000307235
51
F-Ratio
318.77
P-Value
0.0000
Correlation Coefficient = 0.929739
R-squared = 86.4414 percent
R-squared (adjusted for d.f.) = 86.1702 percent
Standard Error of Est. = 0.00554288
95% Prediction and Confidence Intervals for Cost
Cost ($ million)
1.17
1.15
A better model?
Why?
1.13
1.11
1.09
0.9
0.94
0.98
1.02
1.06
Units (1000)
1.1
1.14
1.18
36
Statgraphics
 Simple Regression Analysis
 Relate / Simple Regression
 X = Independent variable, Y = dependent variable
 For prediction, click on the Tabular option icon and
check Forecasts. Right click to change X values.
 Multiple Regression Analysis
 Relate / Multiple Regression
 For prediction, enter values of Xs in the Data Window
and leave the corresponding Y blank. Click on the
Tabular option icon and check Reports.
37
Normal Probabilities
z
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
.00
.0000
.0398
.0793
.1179
.1554
.1915
.2257
.2580
.2881
.3159
.3413
.3643
.3849
.4032
.4192
.4332
.4452
.4554
.4641
.4713
.4772
.4821
.4861
.4893
.4918
.4938
.4953
.4965
.4974
.4981
.4987
.01
.0040
.0438
.0832
.1217
.1591
.1950
.2291
.2611
.2910
.3186
.3438
.3665
.3869
.4049
.4207
.4345
.4463
.4564
.4649
.4719
.4778
.4826
.4864
.4896
.4920
.4940
.4955
.4966
.4975
.4982
.4987
.02
.0080
.0478
.0871
.1255
.1628
.1985
.2324
.2642
.2939
.3212
.3461
.3686
.3888
.4066
.4222
.4357
.4474
.4573
.4656
.4726
.4783
.4830
.4868
.4898
.4922
.4941
.4956
.4967
.4976
.4982
.4987
.03
.0120
.0517
.0910
.1293
.1664
.2019
.2357
.2673
.2967
.3238
.3485
.3708
.3907
.4082
.4236
.4370
.4484
.4582
.4664
.4732
.4788
.4834
.4871
.4901
.4925
.4943
.4957
.4968
.4977
.4983
.4988
.04
.0160
.0557
.0948
.1331
.1700
.2054
.2389
.2704
.2995
.3264
.3508
.3729
.3925
.4099
.4251
.4382
.4495
.4591
.4671
.4738
.4793
.4838
.4875
.4904
.4927
.4945
.4959
.4969
.4977
.4984
.4988
.05
.0199
.0596
.0987
.1368
.1736
.2088
.2422
.2734
.3023
.3289
.3531
.3749
.3944
.4115
.4265
.4394
.4505
.4599
.4678
.4744
.4798
.4842
.4878
.4906
.4929
.4946
.4960
.4970
.4978
.4984
.4989
.06
.0239
.0636
.1026
.1406
.1772
.2123
.2454
.2764
.3051
.3315
.3554
.3770
.3962
.4131
.4279
.4406
.4515
.4608
.4686
.4750
.4803
.4846
.4881
.4909
.4931
.4948
.4961
.4971
.4979
.4985
.4989
.07
.0279
.0675
.1064
.1443
.1808
.2157
.2486
.2794
.3078
.3340
.3577
.3790
.3980
.4147
.4292
.4418
.4525
.4616
.4693
.4756
.4808
.4850
.4884
.4911
.4932
.4949
.4962
.4972
.4979
.4985
.4989
.08
.0319
.0714
.1103
.1480
.1844
.2190
.2517
.2823
.3106
.3365
.3599
.3810
.3997
.4162
.4306
.4429
.4535
.4625
.4699
.4761
.4812
.4854
.4887
.4913
.4934
.4951
.4963
.4973
.4980
.4986
.4990
.09
.0359
.0753
.1141
.1517
.1879
.2224
.2549
.2852
.3133
.3389
.3621
.3830
.4015
.4177
.4319
.4441
.4545
.4633
.4706
.4767
.4817
.4857
.4890
.4916
.4936
.4952
.4964
.4974
.4981
.4986
.4990
38
Critical Values of t
DEGREES OF
FREEDOM
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
t.100
t.050
t.025
t.010
t.005
3.078
1.886
1.638
1.533
1.476
1.440
1.415
1.397
1.383
1.372
1.363
1.356
1.350
1.345
1.341
1.337
1.333
1.330
1.328
1.325
1.323
1.321
1.319
6.314
2.920
2.353
2.132
2.015
1.943
1.895
1.860
1.833
1.812
1.796
1.782
1.771
1.761
1.753
1.746
1.740
1.734
1.729
1.725
1.721
1.717
1.714
12.706
4.303
3.182
2.776
2.571
2.447
2.365
2.306
2.262
2.228
2.201
2.179
2.160
2.145
2.131
2.120
2.110
2.101
2.093
2.086
2.080
2.074
2.069
31.821
6.965
4.541
3.747
3.365
3.143
2.998
2.896
2.821
2.764
2.718
2.681
2.650
2.624
2.602
2.583
2.567
2.552
2.539
2.528
2.518
2.508
2.500
63.657
9.925
5.841
4.604
4.032
3.707
3.499
3.355
3.250
3.169
3.106
3.055
3.012
2.977
2.947
2.921
2.898
2.878
2.861
2.845
2.831
2.819
2.807
DEGREES OF
FREEDOM
24
25
26
27
28
29
30
35
40
45
50
60
70
80
90
100
120
140
160
180
200
?
t.100
t.050
t.025
t.010
t.005
1.318
1.316
1.315
1.314
1.313
1.311
1.310
1.306
1.303
1.301
1.299
1.296
1.294
1.292
1.291
1.290
1.289
1.288
1.287
1.286
1.286
1.282
1.711
1.708
1.706
1.703
1.701
1.699
1.697
1.690
1.684
1.679
1.676
1.671
1.667
1.664
1.662
1.660
1.658
1.656
1.654
1.653
1.653
1.645
2.064
2.060
2.056
2.052
2.048
2.045
2.042
2.030
2.021
2.014
2.009
2.000
1.994
1.990
1.987
1.984
1.980
1.977
1.975
1.973
1.972
1.960
2.492
2.485
2.479
2.473
2.467
2.462
2.457
2.438
2.423
2.412
2.403
2.390
2.381
2.374
2.368
2.364
2.358
2.353
2.350
2.347
2.345
2.326
2.797
2.787
2.779
2.771
2.763
2.756
2.750
2.724
2.704
2.690
2.678
2.660
2.648
2.639
2.632
2.626
2.617
2.611
2.607
2.603
2.601
2.576
39