Business Mathematics
and Statistics
A general set (population) – objects or phenomenon whose
attributes are interested in totality.
Target - a representative chosen by the General set
(population) proportion.
Mathematical statistics - that presents the science of data
analysis techniques that assist the available data to assess the
different part of all relevant population (general population)
characteristics.
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Feature
The population and sample unifying characteristic
phenomenon or event can take on different meanings.
Variable
• Size, measured on the character and changing
along with the sample members.
• Quantitative: discrete (revenue expenditure),
continuous (temperature, height).
• Qualitative:-rank
(busy
place,
condition
assessments) nominal (sex, place of residence).
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Population
Size
N
Average
m
Target
Data
n
n
x
=
1
∑ xi
n i =1
x gr =
Quantitative
Dispersion
σ2
s2 =
1
k
∑x
*
i
⋅ fi
Quantitative
Qualitative
n i=1
Qualitative
1 n (x − x )2
∑ i
n −1 i=1
Quantitative
Qualitative
2
*
1 k
h
2
s =
f i (xi − x gr ) −
n −1 ∑
12
i=1
2
gr
Standard
deviation
Probability/compar
ative rate
Correliation
rate
σ
p
(tikimybė)
ρ
s = s2
Comparative rate
υ=
f
n
r (formulas will be provided later)
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Qualitative
Quantitative,
nominal
4
Numerical characteristics calculation with
MS Excel program
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Histogram
Quantifying the distribution of attribute
values ​describe the population density function,
which graphically depicts the density function
graph.
Quantitative variable graphically depict the
distribution of the sample histogram.
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Histogram
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s
Įt !$A$1:$A$20I
345
Bin
614
3
4
5
6
7
8
9
10
11
946
1596
209
423
250
1002
1611
539
1627
12
13
14
15
16
17
18
100
387.1
674.2
961.3
1248.4
1535.5
1822.6
Frequency
387.1
8
674.2
961.3
5
3
1
1248.4
1535.5
1822.6
9
Histogram
o
3
• Freque11cy
H1stogram
Input
Įnput Range:
ii
OK
.El!
387.1
t:Įelp
==.;;-r
iJ
674.2
961.3 1248.4 1535.5
1822.6
Bin
19
20
21
22
23
8
Normal (Gauss) distribution
One of the ways to "roughly" to check whether the
quantitative attribute values ​distributed according to
population normal (Gauss) distribution is the asymmetry
coefficient g1 and g2 divide the excesses of the following
mean square error.
g1
6
n
≤ z 1 + P ir
2
g2
24
n
≤ z 1+ P
2
z (1 + P) / 2 - a normal distribution (1 + P) / 2 rows quintile, which is
taken from the 1st quintile table, P-probability (confidence level),
the basis of which conclusions are drawn about the population
characteristics. So, if P = 0.95, a = 1.96 z0.975.
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p
Zp
0,50
0,51
o -)
, _
0,53
0,5-1
0,55
0,56
o-7
,
.
0,58
0,59
0,60
0,6 1
0,62
0,63
0,6-1
0,65
0,66
0,67
0,68
0,69
0,70
0,71
0,72
0,73
0,7-1
0,0000
0,02 51
0,0502
0,0753
0,100-1
0,1257
0,1510
O,176-1
0,2019
o1 1r)
O, _ ) j -1
0,2793
0,3055
0,3319
0,3585
0,3853
0,-1125
0,-1399
0,-1677
0,4959
0,52-1-1
0,553-1
0,5828
0,6128
0,6-13-1
·--
p
Zp
0,75
0,76
0,77
0,78
0,79
0,80
0,81
0,82
0,83
0,8-1
0,85
0,86
0,87
0,88
0,89
0,900
0,905
0,910
0,915
0,920
o' 9_1 0,930
0,935
0,9-10
0,9-15
0,67-15
0,7063
0,7389
0,7721
0,806-1
0,8-116
0,8779
0,915-1
0,95-12
0,99-15
1,036-1
1,0803
1,126-1
1,1750
1 116-)
1,2816
1,3106
1,3-108
1, .Jn
.
-1
l,.Į051
1,-1395
1,-1758
1,51-11
1,55-18
1,5982
·--
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p
0,950
0,955
0,960
0,965
0,970
0.975
0,980
0,985
0,990
0,991
0,992
0,993
0,99-1
0,995
0,996
0,997
0,998
0,999
Zp
1"6-1-19
1"695-1
1"7507
1"8119
1"SSOS
1 .9600
' o- s
- :;
_l
2"170 1
2,,326-1
) 6 -6
_ ";t
-:o
2"-1089
) -1--, J
2"5121
_' " --I -8
2"6521
2"7.Ji S
2"8782
3"0902
10
Quintile zp finding in MS Excel
For example: quintile row, when
P=0,95:
p=(1+P)/2=(1+0,95)/2=0,975
Write quintile row
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Checking Excel program, or a quantitative attribute
values d
​ istributed according to Gauss's law
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General population characteristics
Evaluation of characteristics of the sample
Attribute values ​of the numerical characteristics of the
sample based on variable values, measured in two
main ways:
• Written in numeric attribute values ccharacteristics
confidence intervals;
• Examinated hypothesis.
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Confidence intervals
Quantitative trait values o
​ f average population of confidence
range
s
x−
t
1+ P
2
n
s
< m < x + t 1+ P
n
2
x - variable values ​mean, standard deviation, s, t-Student
distribution is with n-1 degrees of freedom (1 + P) / 2 successive
quintile
Interest characteristic values o
​ f the likelihood
confidence interval for the population
ν − z 1+P
2
ν (1 − ν )
n
< p < ν + z 1+P
2
ν (1 − ν )
n
z – a normal distribution N (0,1) (1 + P) / 2 successive quintile, P-confidence
level;  - are interested in the relative values ​of the variable target rate
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The size of the sample
When you are interested in a quantitative character
values m
​ ean:
z2
s2
n=
(1+ P ) / 2
∆2
z - a normal distribution N (0,1) (1 + P) / 2 successive
quintile,
P-confidence level;
s - a quantitative variable standard deviation
∆- the desired accuracy assessment
When we are interested in the qualitative character of
the distribution of values:
z (1+ P) / 2 ν ⋅ (1 − ν )
n= 2
∆2
ν– interested in the relative values ​of a qualitative variable frequency
∆ -the desired accuracy assessment
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Testing hypotheses
• Any assumption about the observed random variable or
several sizes refer to the statistical distribution hypothesis.
• The main hypothesis of the observed law of distribution
and its parameters are called main hypothesis H0. The
opposite argument is called by Alternative hypothesis H1.
• The rule, according to which hypothesis is examined
rejected (accepted), called the criterion.
• Criteria for the award is based on the idea of ​low
expectations the principle is very little chance that the
main hypothesis is false.
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Testing hypotheses
• Usually, the hypothesis tested in accordance with 0.95
probability, the probability of failure 0.05. This possibility is
also called the significance level  and bears.
• The most basic hypothesis states that there is no
difference between the groups compared parameters.
• If the alternative hypothesis states that there is a difference
(eg. H1  m1 m2) means the criteria that we use to check
the hypothesis is two-sided.
• If the alternative claim that a population parameter is
greater than or less than the other population parameter
(eg H1 m1> m2), then the criteria - one-sided.
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Type I and type II errors of testing hypothesis
• Hypotheses are tested based on a random sample
of the data, thus making it possible mistakes or
rejection.
We are
accepting
Is for real
H0
H1
H0
+
β
H1
α
+
-the first type of error. These are likely to
reject the hypothesis correct.
-the second type of error. This is likely to
P H0 H1 = β
accept a false hypothesis.
P{H1 H0}= α
{ }
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General hypothesis testing scheme.
1. Formulated the main and alternative hypotheses (H0 and H1).
2. According to the chosen criterion is calculated from the sample
data random - the criterion of statistical significance of T.
3. According to the chosen level of confidence and sample
degrees of freedom the critical value tables, which are formed
on each of the criteria selected by the criterion of critical
importance.
4. Compare the obtained with t-test statistics with a critical criterion
value. Generally, if the test statistics module is less or equal to the
critical value, then accept H0. If test statistics module is greater
than the critical value reject H0 and H1 take (the only difference
hypothesis H0  2 = C, H1 C check)
5. We conclude based on the received hypothesis testing results.
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Criterion critical value-a test of the statistical distribution row
quintile, quintile of the distribution is found in the table or MS
Excel program.
If the hypothesis (i.e. H0 m = m0) used to check the one-sided criteria (eg. H1 m>
m0), then turn tailed and a confidence level of P (Fig. b);
If the hypothesis (i.e. H0 m = m0) used to check the two-sided test (ie. H0
 m m0), then tailed a number calculated: (1 + P) / 2 (sometimes tailed series)
(Fig. a). necessary and (1-P) / 2
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Testing hypothesis:
H0 m = m0
Let P = 0.95, n = 26
one-sided test H1 m>
m0:
The column P row n-1
The column is 0.95,
and line 25, then the
quintile tp = 1.7081
(blue)
Two-sided test of H0
 m0
Column (1 + P) / 2 row
n-1
So column 0.975, line
25, then the quintile t
(1 + P) / 2 = 2.0595
(red)
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Tailed Student's distribution with MS Excel
1. Click fx and find TINV:
Assume P = 0.95, n = 26
Duplex criteria:
The Probability write 1-P
to write Deg_freedom n-1
Kvantilio t eikšmė
Assume P = 0.95, n =
26 one-sided criteria:
The Probability write
2 * (1-P)
the Deg_freedom
write n-1
Quantile ratio
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The first and second kind of error, if verified hypothesis
H0 m = 10, m = m0 H1> 10
tP
tP – Student distribution P = P consecutive quintile, P-confidence level.
tP - found in the second quintile table or MS Excel program.
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p-meaning
The probability that the criterion of statistical
significance (when H0 is true) will be not less
than the observed realization (counted) is called
p-value:
• p-value = P (t ≥ t), where H0 is correct.
• p-value - observed significance level.
If p< α, tai H0 rejected;
If p≥ α, tai H0 accepted.
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Literature
• V. Čekanavičius, G. Murauskas. Statistics and
its aapplications: Part I. Vilnius 2000;
• G. Kasnauskienė. Statistics on
business decisions. Vilnius 2010.
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