Business Statistics

p 1.

he ai

a i ic e a

f ig ifica ce a e .

e

i

The c

e ie ce f

e defi e Y a i

he

he

Y =

ch a defi i i

ie , a

f0 1 a d

be

f

g

he hi g , i

a iab e defi ed i

cce e . F

e a

heha fac if

ch a

a , he Y i

e, if e a e dea i cg i i h e ,

n

∑X

i be he

i

f he e i g

ea d e

(i.e.

he

e a) d

i =1

he ce h

ec

he

be

f

cce e (if

e de

e a

cce ). F

i g a c i 10 i e , if ai i c ded a 1 ( cce ), he i a i ai ( Y = 0 )

e e h

igh

cc

ai ( Y = 10 ). Of c

h ai i

a he

i

e i e

ib e. S if i ,

he di

Bef e

ib i

hi

f hi

.I

he eade Re

i g

i de e de

e ie ce,

: head

he e e f head a d

a e

a e f

0

ec

e gi eede he

he

ef e hhei

i g

defi i g Y a a a d ic gi a. A

fi e h

,

e)? We

ea

e i

:

ib e,e fa

e, if

hai fa i

ha he e

f

i ca ied i

ac

f

e

a e ca ed i de e de . Agai , hi

ea

b ec .

e ee ahae dea e i g he e

a iab e each f

a iab e, e

ab e eade ,

ae

i gec i

b ai ed ha e

ch e e edge

he a iab e Y,

babi i

e he f

i ag ee ha i i

he fi

a ead ,

a

f

ai ( ccefai

a d he e babi i

a iab e, i.e. Y i a

a iab e?

a e i de e de : e e

a d

fY

di he ib i he a

e hab eai ed i

cce i e h a e

a d

af a

heife e

ha

he i h h

e

gh he a e

a iab e i i e f a a d

e e he defi i i

a e i

e

cc

ha a i

ia , he

ib e a e f , i.e. e

e gi e he f

ba ed

,a h

fcce e i

f a d

a iab e. T c

ha i

hi

e,

hich

i e a k = 0,1,..., n .

hich i

f b ai i g each

e

be

e e i ai

ec

ha e che e e

he

I i c ea ha he a d

a he d e

e, e c

e a

i

i h he

he a e 0 1 di

ib i

f . The

27

di

ib i

bi

f he a d

ia di

ib i

a iab e ca Y i ed he Be

) a d f e de

i di

ib i

ed B( , , I ) .defi i i

(

i a f

:

n P (Y = k ) =   p k (1 − p )n − k , 0 < p < 1, k = 0,1,..., n . k I

he

d ,

f b ai i g

e ha e a cce e i

N e ha hi

e i

i de e de

ide a ge e a

a iab e . Y i e ih

e ed he

ecif

ide ica

a e

he a e

f

a e

a iab e

) each

ca

ih

a e

babi i

(

f a d

ib i

f hi

(

2,

e,

ei e

a be fi i e

he e

e a di c e e

i fi i e):

m

∑p

P ( X = xi ) = pi , 0 ≤ pi ≤ 1 a d

babi i

ca a iabae e a d he

a di 1,

he

ia .

ha a a d

a e

ha i

he defi i i

d f each f he e a Se , .ge e a

he a d

ega di g

i

, i = 1,2,...

i =1

Each i e ge e a di

e

di

e d

i

e

2

ib i

e,

ih

i ia bei g he

eadi g

i g hi

i di

i gi

ib i

i

ef

2

.I c babi i

f

he f

be

de e

ae ,

ahe bi

i

f a d

i i g he

f ia eeded X each ia .

One can find a br ief description of the life and work of Jacob Bernoulli at http://en.wikipedia.org/wiki/Jacob_Bernoulli

28

ch

.I ia

a e babi i

cce be ef i achie ed i a be

cce

.F ib i

. The de c ibe he

ic a a

i he de i ed

ch a a iab e, i

(a

he

he ec

a e

a da d di

ib i

e gi e a fe

cc

ia di

e

e f aefe f

a d Be

f ia ca ied

e ce f Be cce e

i

) defi ed ab f di

The ega i e bi be

he a iab e

be a e a

he

ib i a

e a d

e ca defi e he a d

ib i

addi i

i

achie e

. f he

 k − 1 s (k −s)  p (1 − p )  s −1

P( X = k ) = 

E a

e:

S

e ha he

ie

i 0.001 ,

babi i a fa

hich

defec i e ( ife e

a d ie

ef

fa

he

a ce: P ( X

i

Th

i e,

ce

d ci g a defec i e

e h ai

ai

a d ea

fa

ai

ffice

he

fac

ed i d ci

e i ei dei e de

c

ha i

a

, idea

babi i

e ec

i

f he

ie

a

ha defec he fi i e

e ec ed.

He e = 10, = 1, a d

F

ei e

d ced. If a

d ci

he 10 h i e

i g

ha f

he ha he

he i e

f i

ha

e ie ce e

ib e). S ai

ea

fac

= 0.001.

 10 − 1 ==10)  − 0.0011=(1 0.001)(10−1) 1 − 1 

0.000991 .

,

P ( X ≤ 10) = 0.008955 , hich i a e a i e e e ,

a

e ei he be idea f he e e

he e h i e i

be . Thi g

ea

a " cg "

f deficie cie d e

i defec i e. C f defec i e i highe ha

e

,

ha

e a e dea i g

hich e c

e

a

i ha

i e

i e e e ,

ef ec ea i , i ce he a ead e dic a e he dec a ed

ha i i i eha he ei e

d ced i 1000.

29

.

Wi hi

he c

e

i de e de

f a bi

ia e

e i e ,i

ia ca be c a ified a a

a iab e c

he

be

f ia

hich he

cce a fai

c

e, a ge

e f each f e ic a d

i he ficce

P(k ) = pqk −1 f

=1,2,3, .,

he e =1 E a

e.

F

da a f

U i e i

heDe a f Tech

a ea. A c

a

e

g ,i i

ch a hi f

ca dida e f e ce. Wha i

a

e

a egi de

e Scie ce a d Ma age e , W c a ha 83% f

i e e ed i de e

a d

i

fC

? Wha

h c

de

he

babi i

f e ec i

c

a egi

a a ea . S

be

e f

i g

ef

e ha

de

ha he fi de h

a

ba

ha e ab i hed a

d be e

a e ch

e a

e ec ed i f

ec ed bef e fi di g a

a a a ea? Le ' ca c a e hec

e

di g

babi i ie : Pic i g a

de

f

a

a a ea a

he fi

e

P(1) = 0.17 ⋅ 0.831−1 = 0.17 , Needi g

e ec i

fi d

ch ea

P(2) = 0.17 ⋅ 0,832−1 = 0.1411 Needi g h ee e ec i

bef e fi di g a

de

f

a

a a ea

P (3) = 0.17 ⋅ 0,833−1 = 0.1171 The

babi i ie f

f

, fi e, i , e e a d eigh

e ec i

bef e fi di g a

a ca dida e a e 0.0972, 0.0806, 0.0669, 0.0555 a d 0.0461 e

30

ec i e .

I i

ac i cide ce ha

i a e

ed ha a e. F e

ha

e ca ied

e ge e a e

hi

e

i

eed

cha ce

f

ch a c

c

he e eigh ca c a i

a e he

babi i

a e i 0.05. The ef e,

aae a a

i i

e fe e

e e

(ai

fe

e

a i ic , i

e ee e

ec

i ha

i h he fac

e ec i

a i icia

.I

a ea

he

a ) ig ifica .

.

The h

e ge

babi i c

e ic fa

he ai

be f

babi i

be f

f cc

di

ib i

e ce

cce e ( ) i

ee e

,

i

ef

f de e

hei g a i h e ec i

f hich a e

i i g he

e ace e I .

, ih

cce e a d (

e ace e , f

a

) f hich a e faie .

 s   − s  k  n − k  P(k ) =     n    E a S

e (Ac e , 2006): e ha a

e

ce c

ae a d be

bi e a i e a a dea e hi i

ide a i ch

e f

a da d f be i

ec ed

,

5 he 10

he If 2.

ha i

hebabi i

afe ,

i be f

f each 10 a e i

d

ee

afe

f 10 a d ha f ec ed f f he 10 ca

ha a ea

1

i ea d

afe The . 5 ca he

ae

hef 5 ca

a da d ?

 2   (10 − 2 )   2  8  2! 8!  1   ( 5 − 1)        =  1  4  = 1!1! 4! 4! = 5 = 0.556 P (1) = 10! 9  10   10   5  5 5!5!  

P (2) =

10 2 2  8   2    ( − )    3  1   ( 5=− 2 ) =  1= =   10   10   5  5    

2! 8! 1!1! 3!5! 10!

2 9

0.222

5!5! 31

i.e. he e e i

i ed

ab e f

c

i P (1) + P (2) =

babi i

(i.e. he ca

e

a

0.556 + = 0.222

fac

e

)

i

i acce ab e, i ce he e i a ig ifica

0.778. I i

f ie

ha

high i e ih

ch a d f

fi di g ei he defec i e ca . (

F

af

e

i

f ie ( he

e ce f a d

a iab e

{ X n },

i g

e

( , , )

a

i h he f

ib i

de e

. T c

di

a e (k

ib i

he e ), c

he e each

ide he

ha a B e

: n ⋅p = const

for → ∞n

i di

ib i

. Le X

= lim X n .

= 0,1,.... We a ha X ha a P i

e e he defi i i

i e he a e

babi i

ca ed P i

a iab e a i g a e k

Xi a a d di

).

f hi

a d

= 0,1,...) a d hei

i defi ed a f

a iab e i i

ece a

babi i ie . The P i

:

P( X = k ) =

λk

e−λ

k! he e ei

he ba e f he a

i he he f c i !i

be ab

he fac

f cc e ia

a e ce

ga i h

(e = 2.71828...)

fa e e

he

babi i

gif e hich b i

f

λi a i i e ea be , e a he e ec ed be f cc e ce e e ha cc d i g a gi e i e a . F i a ce, if he e e cc a e age 5 i e e i ea d a e i e e ed i he babi hei e ef cc i g i e i a 10 i ei e a, d e a P i di ib i i h λ = 10 5 = 50 a de .

32

fa

Re

i ed c

di i

f

he P i

di

ib i

be a

ied:

The probability that an event occurs in a short interval of time or space is proportional to the size of the interval. The

babi i

ha

The

babi i

ha a gi e

i de e de The

e e

cc eI a

be

fe e

a i e a i c

e

e .

cc i i a gi e i e a i

f he e he i e a begi .

babi i

f a gi e

i de e de

f he

The P i

di

e

bee e f be

ib i

ecified i e a c

i

cc

fe e

ca a

be

i g

ha

cc

edf

he

ch a di a ce, a ea

ai i g f

e ice i f

de ed a a a d

a iab e

e a gi e i e a i ed

i be

e. F

fa iP h ai

e di

ha i e a . fe e e a

i

he

e, he

be

f

a e ca h egi e ca be ib i

i hi e

i

e

a

λ. The a d

a iab e e a i ed

f he e a d

di c e e a iab e . B de c ibed a

a

ea i ed a

be

f cc Th

,f

e ca

be

i

ha he a e e e e ca a

a

ee ha

fa a d

gi e he e

e, a

: he ea i a i f ehia e ca ed

e he ea

e a ca

ab e he

he e de

d , e c.)

e e

The ). e e ia cha ac e i ic f ea e. Thi

he ea

a iab e gi e ab

f a ea ad

be

e aa e

ed i ce i e e , ec

fi a d

a

e hi cg i

be . Va iab e

( ea

( ch a

be

e ce f

a

be . F e a

be

i h he defi i i ece a

c ea

a

i i e ea

ega i e ea

ea

fa ha e

a iab e a e a

a iab e ca

ea

ha be

be . Th

ee a

he a e ib e

e, acc di g

i hich i

a e a d he i e ih

d

each f he e a e .

a , a di c e e a iab e a i fie

he f

i g:

33

Ha a c

ab e

be

Ha di c e e (di c Ha a

ea

i

ab e

C

f

ib e a e ,

) babi i

be

ee c

feach

e di c e e a iab e a e c

cc

e ce

f a e e ).

I c

a

di c e e a d

diffe e

Ta e a

c

The

ci

ea

ed a d

ab e ( h

babi i

P babi i

f

a iab e , c

i

c

f chi d e ,

a d

ed (e.g. heigh ,

i fi i e)

ib e a e cha ge i ca

N

i

(i.e. be

be

f

a iab e ha e

cha ac e i ic :

S ch a iab e a e

C

ec i e a e

ib e a e.

i a c ibed a

be

i

f e,

i ge a e

bea c ibed

a d

a iab e a e defi ed b a f

he f

eed, e c.)

aib ee ,

a

ca

. I ha

eigh ,

a ge

f a e . ( ), ca ed he de

ci

i

i g cha ac e i ic :

1) f ( x ) ≥ 0 , +∞

2)

∫

f ( x ) dx = 1 .

−∞

Th

, he <

2

(

2

, ha i

1

babi i )c

2

e

ha d

e a d

a iab e X aae eabe

he i eg a

f he cf i

( )

ee

∫ f ( x)dx

a1

F e N

34

he e i ec

e ie i

f i eg a ca c ega i e a d e

ide a a d

a iab e

,

ea e

ha i hifa

e

e ha hehia e f

a1. di

ib i

a d

e he i e a [ 1,

a2

P( a1 < X < a2 ) =

1

he i e a

[a, b] (Fig. 1).

( )

1/(b a)

a

Fig e 1 U if

The de f

i

babi i

f

ci

i ga a

b

de

i

, ( ), f he

ica f

f

if

ci

di

he i e a [a, b].

ib i

he i e a b], [a,ha

he

:

 1 for ∈ [a , b]  f[ a ,b ] ( x ) =  b − a for x ∉ [a, b] 0

I i ea

h

ha hi i ade

i

f

ci

, i.e., ha

he

e ie :

( x ) > 0, beca e a < b, a d

1) f [ a ,b ]

35

∫

2)

b

a

f[ a ,b ] ( x )dx =

1 b a−

1

b

∫ =dxb a

−= ( b a) 1 . −

a

N e ha if e ha e i e a [c, d] a d [e, f]

ch ha b h a ei he a ge

[a, b] a d ha e he a e e g h, he

he

X a e a a fe

he a e ahe

a ei

he i e a [c, d] i

be

e ie

fc

he e

i

di

e ie ). F

i

di

he

ib i

e ai

e ia di

S

be

e he

I

ib i de

i

a iab e

babi i

fi a i ga

i

ci

if f

(a

ci

,

i h di c e e di

ha a e

ef

be

a

ff

a

ci

ib i

, he be

de i g ec

ic a d

e de c ibe

ib i

ih f cia

e f he .

.

e ce

fa e e

ee

i a f

f

i .

e ca defi e i

he e a e a

a e ,a

f cc

. The i e be f

f de ib i

ha big. Be

O e ided e

di

haa a ch d

he i e a [e, f].Thi i ca ed

U i g he e

c

babi i

cc

i a i e aa Phai

e ce ha a e

e ia di

ib i

.

:

λ e − λ x for λ , x > 0 for x ≤ 0 0

f ( x) = 

E a

e

The i e f be ih

hich a a ic a

ee b ea d

achi e

he ef e, he

i babi i

e a e bef e b ea i g d

ha e a

a a e e λ = 2 ( he e he

ha he

36

achi e

)i

c

i

i

e ided e

e ia di

f i e a e h). Wha i f

a ea eh

? We e

ha a iab e i h a e ided e

he

(i e ib i babi i

i e,

e ia di

ib i

i h a a e e λ = 2, a e aa e g ea e ha 1, h

he

i ei

ea

ed i

.

Th

, +∞

P( X ≥=1)

Thi i

ae

high

cha ce ha he

2

−2 x

∫

babi i . We

achi e b ea

−2

e= =dx

1

e

d a

d

0.1353 .

ha ha eea

i hi a h

i ce i he c

e e

e he e i

e he f a a e e λ. If he a a e e

he

he he

e

aa ee

Bef e a

e

e i g he

e ia di

a

e. The e

a e

hich

e i

ib i

,

e

de

e( a i ia

he a

e. Gi e

igh a

ch a di

ib i

ib i

The

i

i a

ca ed he Ga ib i

e

ia di

a

ai he

e ic

e

ea

e

a ia ce. M e

i a f he c ib i

, he

e?

he a a e e λ i

f he e

he

ec ed a e f a e, he

e e be ha he a ia ce b e ed i

e ca e ihea a e ge f a eha

i he

ec ed a e a d a ia ce i

a di

ea

a ed b ce

a e age . We a

a a

h e. A babi i

. The e f hea i

ai

ec ed a e

e ei e e a i e, he

e a ea

he

ec ed a e a ad ia ce f e

e , if a a

ec ed a e a d a ia ce f he a

he e

di

e,

f hi ,

e ica cha ac e i ic f he e i e a

a e ,

ae ai

a eλ

e) i cha ac e i ic f he a iabi i

de c ibe a a d

( ea ) a d e

he c

ec ea

he e

ea i a a

ib i

he

ec ed a e i a

fa a

di

ab

e e

babi i

died, i 1 0.1353 = 0.8647). The

P( X ≥ 1) = 0.6065. Wha , he , d e hi

e

ch g ea e

( he

i ai . The i

(e i a

) f

e f he a a e e

f

e . i

di

. We

i de

ib i

i

he

e ae a a e ec

a di e

ib i

,a

hi

.

37

.

A

he i

di

ib i

a .I

c

de

i i

f

babi i ci

−

f ( x; r, c ) = x

A e a i e , he ga

a di

a a e e α= a da i

e

r −1

i a

ib i

de e di g

38

i he ga

a

:

for x ≥ 0 and r, c > 0

ca be a a e e i ed i

e

f a ha e

e e ca e a a e e β = 1/c, ca ed he a e a a e e :

i i e i ege , he

B h aa eei ai

ib i

x c

c r Γ( r )

g ( x; α , β ) = If

di

(PDF) i defi ed a f

a ec

he i a i

.

βα

Γ(α )

xα −1e− β x for x ≥ 0

Γ()n = ( n − 1)! . , beca

e ei he ca be

ec

e ie

Fig. 2. I

ai

f he Ga

a PDF f

ai

aa ee

=

a e

a d

c =θ . I ca be h ga

a di

ha he ib i

e ided e

e ia di

i ha a e e

a

,

a e

x=

ch a e

x

in

a di

a i

he PERT

+ 4 xsp +

ae e

ica i

N

, i

f hi The

i idea f

e h d,

ec i e

eh

e e

a di

he c

f ie , he e

he ba i e f e ec ed a e i e i a ed a

2  xmax − xmin  , he e  6  

i e

i e a d e i ic i

. The e e i a e a e he e ib i

ce

de c ibe he d a i

f he a a e e

ec ed a e a fda

,

s2 = 

i i ic i e,

d ai

f he ga

de i g he i cae

he e he e

a d he a ia ce a xmax

i e f a gi e ac i i

ecia ca e f he

λ

f he d a i ac i f i ie

6

a d

a

ib i

he e i a i

i a

1

r = 1, c = N e ha he ga

ib i .

f a di

ib i

f he i e acf ai i . f. F

a iab e i ca ed he fi

a a e

a iab e. di a

a e f he

e h

f e

( = 1, 2, ...) f a a d

de f hi

mk =E X(

ec ed

a iab e.

∞ k

a iab e i he e

) =xdF ∫ x k

−∞

 ∑ xik pi  i ( )=∞  ∫ x k f ( x ) dx  −∞

(1) (2)

where: X - random variable, E(X) - expected value of the random variable X, p - the probability function, f - a density function. Patterns (1) and (2) should be used for a random variable with probability distributions of discrete and continuous type, respectively. For k = 1, we obtain the formula for the expected value, so the expected value can

39

be treated as the ordinary first moment m1.

Similarly, the variance can be presented as a special case of a socalled central moment of a random variable. The central moment of order k (k = 1, 2, ...) of the random variable X is the expected value of the function k

[ X − E ( X )]

he e: X he

, i.e.:

Pa e di

f ci

, f a de

(1) a d (2) h

ib i

d ce ea

e

d

a

e

e

hi e

ae e I

e

he e a e e ha hi e a

40

he a ia ce a d, he ef e, he

a

ib i

babi i

e

i a

. The f

c

a da

h ce

a

e

i

ef

i ge

a iab e ca be e e i

f heec

ef : µ 2 di a

ib i

e ed b

d

de ce

eadi a f a

e

= m2 − m The eade i e c

a d ce

2 1

a

a da

e

c

e a

di i g i h be

aged

he b e ie

ee di

chec fa

ib i

ha

igh diffe e .

ac ica a he

fa a d

a ic a

di

af

ih

ec i e .

i .

a i . I fac ,

babi i

a iab e

e,e

f a di

a iab e ,

.

a a d

i he f

e

. The f

( a ia ce) i

ci

e . The hi d ce

e he a

a

ec ed a e f he a d

f

d be ed f

i ca c a i g he Ce

i

f di c e e a d c

The ca e = 2c ec

a iab e, ( ) e

a d

babi i

e f

ica i fa babi i

f a ec de a d di

a ead ch a di

ib i di

ib i

ic a d ea

a

,f ib i i

ciaa

e, i ca be a

e i .hi B i

ge a ea

hich he e a e f

he hich ei

he Ca ch di

ib i

e

e

ed ha ai : .I

a ic a ,

ec ed a e. O e ( i h de

i

f

ci

( )=1/(π(1+ 2)). I i

e

. I i

e ha

ecifie a a d c

e

f

h ai i g a

he a ec

edge f he

f ea

a iab e? The ge e a a

e i

di c

i

edge f he fi

: he

ac ica i

f

e

he

e

f

de e

E ce ca fi d he e hich ca be ai e

ha

ed a a

e

ic a

a i g ha if heae i

i ai

he a

i

gh i

he

e,

e

f

e i . U i g he "de c i i e

f

, a d e ce

he e

f he e a e

. P ea e efe

de

f

fficie ecif a a d

ec ed a e, di e i

. Ca c a i

ibi

ega i e, a h

ae cif ec

i

a iab e. N e ha he e i a he e a

f he di

e

de

,

a i ic "

e

a a

e

i f

,

ec i e a e he i

ie

edge f he fi

ia e defi i i

f

he ec

f e ded

eadi g. A he e d f hiec i c

ed

di

ib i

di

ib

i

,f

he c

e

f

babi i

f

e ie ce f he eade

he af e e i

e

a e he

ed di c e e a d c

i

.

Ma

c i

de

i

E (

ec ed a e Va ia ce (ce e

f fi

f ec

d

a de )

de )

P ( X = x=1 )

T

x2 )

ia

(Be

n P (Y = k ) =   p k (1 − p ) n − k , k 

i )

0 < p < 1, k = 0,1,..., n

The ega i e bi

1

+ 2(1 )

(1 )(

1

)2

2

1 − p , for 0 < p < 1

i Bi

p; =P ( X =

 k − 1 s (k −s) P ( X = k ) =  s − 1  p (1 − p )

(1 )

p s1− p

s (1 − p )2

ia

41

e

Ge

P (k ) = pq k −1

e

1

1− p

p

p2

ic H

 s   − s    P (k ) =  k  n − k    n   

e ge e ic

P i

P( X = k ) =

[a, b]

e− λ

λ e − λ x for λ , x > 0

e

f ( x) = 

a



 (  − 1)

λ

λ

a+b 2

(b − a ) 2

1

1

λ

λ

12

2

for x ≤ 0

0

ia Ga

k!

ns ( −n )( s− ) 2

 1 for x ∈ [a, b]  f[ a ,b ] ( x ) =  b − a 0 for ∉ [a , b]

U if

E

λk

ns

g ( x;,α β ) =

βα

Γ(α )

xα −1e− β x

α

α

β

β2

for x ≥ 0

A he e d, f a d de ia i

ei

d ce e a

a iab e . A σ

a be

he

a d

a ic a

a iab e

a da di ed

i g he f

Y = Thi

eai

a e0a d

42

a

f

a d

a da d de ia i

fea

e f he

ec ed a e

i g

a d

eai

X −m σ

a eYi 1.

ef

i he

a a d

a ee ec i hed

e a da d

Lec

e III. The

The

adi

a di

ib i

a ea . I i a

ib i

.

i a e

e e i

ca ed he Ga

ia di

e gi ee i g. I fac , hi i a fa i

di

( ca e). The

ib i

ih

a di

ib i

c

e. A a ead

c

i

di

ib e f

ea

a d

i

f

e i ib i

a

he

di

ca i

a da d

e

de

babi i

, a ic a

f i fi i e

a a e e : he a e age ( e a da d de ia i

a

ib i

ib i

a di

ib i f

a di

ib i

i

a

, defi ed b ib i

i a

)a d a

e. Si ce hehe ga h f

e e b e a be , i i

ed, he

ib i

h ic a d

f he di

a da d de ia i

ci

di i

f e ca edhe be

be

g

he c a

f

.

Hi he

a di

( e i ed i fa f

ib i

he ec

a fi

i a i g ce ai bi he de e

P babi i

i

a

ib i

e h d f ea

i

a e a

a d di

ec

c

i

ed i

e

he

i

.

ei e

he

,

.

a

h c ai ed ha he had ed hi

he be c e fJ a

g i

di e

i

i ge

a di

ib i he (

he

a di

f

a i ).

a di

ib i

ib i

fe ,

a

had a

,

h c i ed he

a di

di a ib i

he be , beca

e a i dica e ha

ib i

fa

f

f i e

babi i

b Cha e S. Pei ce, F a ci Ga

cha ac e i ic

Pa a e e

he a a ei

. The a e f he

f hi g ha e a cia

,

e f

i1872 f e

ea 1875. Thi

ee

.

face f a be

i de e de

i

e

e

ica The

ade g ea ad a ce i 1809 b a

The a e f he c e

a ge . Thi

The A a

ae ,

d ced b Lege d e i 1805. Ga ib i

f

ca ed he de M i e La ace a e i a di

e h d i ce 1794, a di

ib i

ed b La ace i hi b

ed he

The i

f The D c i e f Cha ce , 1783) hei c ia di

(1812) a d i

La ace

e e ed b de M i i ae a ic e i 1773

d edi i

a d Wi he ei

gge

hi e ece

ib i

a i

ih

d ced Le i a

d he

ha he a ec die

f ic

e a ha e a a he diffe e ab

20%da fa e

h

he

:

43

The e

μ

ec ed a e

2

σ

Va ia ce M da a e (

μ

de)

μ

Media S e K

e i

The de

0 0 i

f

ci

X

~

f

ci

σ (e

de ia i

.T

f he

ica , he

 (µ,σ ) (

aa ee

ai

i g de

X

ei e

f

ci

ha e i

44

h

be

ih

.

ead μaa da d

e f a Ga a iab i de eX

 (m, σ ) ,( he hich

ea

1 σ 2π

fe h

d a a e he e i

ha he a d

 ( x − µ )2   2σ 2  

exp  −

ia ed

he a e

e he he he ec

:

f ( x) = I

~

he a ia ce), i

ib i

f he a d

a e gi e be ca ef

a da d de ia i he f

a di

: a ia ce σ2) i a e a

i ae

a iab e X ha

A

a de

di

ib i

i hi

i

f

. Ab e

ci

a da d de ia i

a da d de ia i ig a a

ae

a d ab

he

ea

de he g a he h de f

ff

he a e age, ab

99.7%

e). N e ha he de di e

e ic ab

68% f he a ea

i

f

i hi ci

h ee f

a e f he i

f

ci

95.5%

i

i hi

a da d de ia i

he

a di

( he h ee

ib i g ea i e

ec ed a e.

The i f ec i

i

i

i hi

e

a da d de ia ihe fea .

.

We ha e defi ed a d di c e e a d a iab e .) The c e f

ci

de he i

a i e di

f a d

B defi i i

a iab e b

a iab e )

i g he f

ib i

ci f

babi i

(f

ci

c

a

i

f

ci

(f

a d

(cdf) ca be

ed

defi e b h

a iab e.

, hec

a i e di

) fa a d

ib i

a iab ei Xa f

f

ci

ci

( fe

h

e ed

di

ib i

x∈R :

f heea a iab e

FX ( x ) = P( X < x ) Ac

a i e di

ib i

f

ci

F( ) ha

he f

i g cha ac e i ic :

1) lim x →−∞ F ( x ) = 0 , lim x→∞ F ( x ) = 1 2) The f

ci

F( ) i

a

dec ea i g f

ci

x1 < x2 ,

, i.e. if

he

F ( x1 ) ≤ F ( x2,)a d 3) F( ) i a ef c

i

3

Thi

c

e a i e di

i c

f

e

ib i

ci

(he e

e efe

e a ed f

ci

heic i e .I i

h

ai

he ec

f he di

ib i

f

ci

ed i

he defi i i

i g ha ef ha ded c

igh ha d c i i if e cha ge hi ic i e a i he f i g defi i i f he c a i e di ib i f S ch a defi i i

he eade

e

i )3.

a he a ica a a

ca be f

i a ci di

ic

f he i

i

i

e,, i.e.

e

FX ( x ) = P( X ≤ x ) a

e b

e

a i ic .

45

E a

e.

Le

de e

di

ib i

P( X = 1) =

i e he c

a i e di

ib i

f

ci

f he

i

01

. p; P=( X=− 0) 1 <
p 1 . Thi i i

a ed b

he f

i g

fig e.

( )

1

1

0

N e ha i

hi di

1

ib i

f

ci

ha

i

f di c

E a

e.

Le

de e

i e he c

a i e di

he i e a [0, 1]. I i defi ed

ib i i g he f

f

c i f he i g de

 1 for ∈ [a , b]  f[ a ,b ] ( x ) =  b − a 0 for x ∉ [a, b]

46

i

i , a e

= 0 a d = 1.

if i

f

di ci

ib i

he

The ef e, i

c

a i e di

ib i

f

ci

i a f

:

( )

1

0

N e ha he c di c

i

, he h

he c di

ib i

The c ci

ib i

ci e

i gac

a i e di

babi i

di

ib i

ib i f

fa

=1i

b ai

ib i

e ed b

he f

he f

af

he c

a di ib i ( adi i a hi ge e a f a.

ib i ib i

ha f

i

ci

f

i

babi i di ib i . e ie )1)i ed 3 ab e i

babi i

di

ib i

ha i

f

. Th i

ci

e

, if i babi i

e ca c ea e he

.

ci

f he

ha he a iab e X a e aa e e

i e

f hi di a i e di

died b

a di ha

ib i

a d ie

i defi ed a f he de

he

i

a

F ( µ ,σ ) ( x ) = T

ci

haa if c

f

a i e di

babi i

f

a

e, ha a he ,b

ia e

ib i

e ca

e a e dea i g i h a c i i g ha a f c i hich ha

a i e di e a

a

f

a i e di

i . I ge e a ,

c i I i a ,f

1

1 σ 2π

a i e di de

x

∫e

−

( u − µ )2 2σ 2

du

−∞

ib i

ed b Ф),

f e i

ci

f he b i

a da d eμ= a 0d σ

47

1. If X

N (μ, σ2) ,

2

(aσ) ). Th di

ib i

he e a a d b a e ea

, a i ea

a

gi e a

f

ai

he a d

2

a iab e

a iab e ai h

a di

i ha

a

ib i

.

2

2. If X1

N (μ1, σ1 ), X2

X1 + X2

N (μ1 + μ2, σ12 + σ22), i.e. he

a iab e a

N (aμ + b,

be , he aX + b = Y

fa d

N (μ2, σ2 ), a d X1 a d X2 a e i de e de , he

ha a

a di

ib i

i fde e de

a a d

.

3. If X1, ..., X a e i de e de a d a iab e i h a a da d di ib i , he X12 + ... + X 2 ha a chi a ed di ib i ih f eed

. We

The c

e

i

e

e ce f

di

ib i

ca be a

di

ib i

.

If X i

i dih ib i

a

di

e f

ib ed

i

1i

ha aa d

a iab e

i ha

a a d

a iab e

i h he

ih

C

a iab e

e e , if Z i a a d

ih

X −µ σ

a iab e

a di ih

ib i

a da d

X =σZ + µ ,

48

a a da d

ea μ a d a ia ce σ2, he :

a da d

f

e.

ed i

Z= Zi a a d

he f

a deg ee

N(0, 1). a di

ib i

a d

a

he X i a The di a d di

ib i he

ib i

a

di

f

ci

a di .I

hi

babi i ie c

e

E a

ib ed a d f he ib i

a ,

a da d

ae i

e ca

di g

a iab e

a di

e a

e a di a

ib i

f

ib i

a di

2

ea μ a d a ia ce σ .

ih

ha bee

ab a i ed

ai

f he

a da d

ab e

de e

i e

ib i

.

e.

Le

fi d he a e f a (a> 0), f

hich a a d

a iab e X

ih

a

X  (0, σ ) a e a a e i he i e a [ a, a] i h babi i 1,− α 0 < α < 1 . We a e i g, he ef e, f he a ea de he c he e f

di

ib i

de

i

f

ci

be

ee

a d

be e

1 α, ee be

a

.

( )

1−α

α 2

α 2

a

S a da di a i i he

a

0

a

f

ec ed a e 0 a d

he

a a d

a da d de ia i

a iab e X i 1( i

a a d

a iab e

a ). S ,

49

,

N (0, 1)

Whe e: uα

Le

=

ca c a e:

S e I: P (Y

50

a−µ σ

 

P  −uα< =Y

< uα ) = 1 −

α 2

X −µ  < = −uα  1 α σ 

S e II: P (Y

< − uα ) =

α 2

51

S e III: P ( − uα<

Le

52


e he

e

f hedi

ib a i

:

FY (u )

FY ( u )

P( −uα< <αY =u )α< P− (Y α < u − =) αP(Y< − =Y α u ) > P( = P(Y< −uα−) [1 < P=(Y
u ) P (Y

u )

The ef e,

P( −u< α < Y = uα ) <2−P (Y A

e

f

ci

c

, he e a

α

a i e di

P(Y < uα ) i he a e f he c

e i

. If a a d ib i

uα ) 1

a iab e Y ha f

ci

i de

he ed

a da di ed

f he di

ib i

ab a i ed. The ca a F

α

e

a

1.36, f

f

ci

be ead he Ece

f

ib i

ib i

,i

Φ ( x ). Th :

P ( − uα< <αY =u ) α <2 P(Y − =α Φu ) −1

The a e

a i e di a di

he0,N1) ( di

2 (u ) 1

ib i

ae

i g a ead hee . ead hee

53

E a e • Le X N(5, 4). • Ca c a e P( 3
, ea da di e:

P(-3 < XN(5,4)< 20) =

Fi a , f

ci

e

ie

= P (-2 < YN(0,1) < 3,75)

hi

, Ф (3.75)

a

he diffe e ce be

ee

Ф ( 2) :

Ф(3.75) Ф( 2) = Ф(3.75) [1 Ф(2)] = Ф(3.75) + Ф(2) 1 = 0.9999 + 0.977 N e

e fi d Ф (2)f

1 ≈ 0.977. he ab e a f

0 0.01 0.02 0.03 0.04 1.9 0.971 0.972 0.973 0.973 0.974 2.0 0. 0.978 0.978 0.979 0.979 2.1 2.2 2.3

54

0.982 0.983 0.983 0.983 0.984 0.986 0.986 0.987 0.987 0.987 0.989 0.990 0.990 0.990 0.990

:

a da di ed c

ai e

2.4 0.992 0.992 0.992 0.992 0.993 2.5 0.994 0.994 0.994 0.994 0.994 2.6 0.995 0.995 0.996 0.996 0.996 2.7 0.997 0.997 0.997 0.997 0.997 2.8 0.997 0.998 0.998 0.998 0.998 2.9 0.998 0.998 0.998 0.998 0.998 3.0 0.999 0.999 0.999 0.999 0.999

The a e ca c a i

i ca ied

i

he E ce

ead hee a f

:

Le X N(5, 4). Ca c a e P( 3
55

Fi a e a The fi

.

c

e

N( , σ) i

c

ce

he

ide acce ed

a e age μ a d

a da d de ia i

he a ia ce i

f e gi e i.e. X

The ec di

d e a

ib i

hi

.I

fe

c

ce i

de

e a e dea i g

b ai

he a e f he di

0.5, he ib i

1a i

If (C

ci he

a

56

di

he E g i h ed

f ab e f he di

a di g , i i

he

f

ci

f

ci

a da d f

ci

ha

a da d ab e. Y

i g E ce . A

ca a

e ha

he b

ci

a

. Si ce

ab e

a 0. If hi

). Ifa he e f he f

d,

.

dified ab e (a d he ef e ha e

ib i

.X

ih

ea i g

ai

ib i

h chec i g ib i

ib i

ib ed

e ae a ei 0

add 0.5

a

e

i

ead he a e f he

e ab e gi e P(Z> ), i.e. 1

f he ab e

i be c

e

0, a d

a da d ab e.

ea

i

f

ci

f

e e e ai

he

a di

f fa e) i

ib i

he a

,

ia e b

a i e).

a fe

ac ed f

iha

0 ( he B

N e 3. The a i

ci

b

ca c a e he de

d i

a di a

e e ,i

he c

hi ca e he a e a

e

h

f

he

2

i h he

e' e dea i g

Ф( ). I c

σ a d. H

i h b fi di g he a e f he di

he

di

ed f

ha he a iab i eX

N ( , σ)i c

e ab e 0.5i

ead

dea i g

ai

ea

f

efe ed

ic a

ac ica

a a ce

e ie i

f he

a di

f ie . I

a i i he e

he i e a .

ib i

ae

e, hi

he

a ic a e

i

F

h N e. Le a da di ed

di

ib i

[ 2, 2]. Thi e e f i ca ed h a di

e e

e

, he e

he a d ea

ea

ha gi e

a a e fa i he i i g) ha ib i

e i i .If

e

i a he a e

e c b e ai ce ai f he

e ed c c be c

ai ed i

a da di ed

he ab e a ge, e ca e he a e b e ed c e f

ce

e f

f he a

a he i e a ia d

e (i a i ic hi a a i ae

.

57

.

Ba ic c

ce

Ge e a

: ai

cha ac e i ic a d

:

ai

i be i

e iga ed. T

ai

E a

e. The heigh

S

hich

ed a

a d

ica , a cha ac e i ic i de e

a be fi i e ( ia h a

h

e

i ed a a

e

e

a

e a i e, beca ea

a ge

be e e fe

e i a e he a e age heigh

ei

)

i fi i e.

fa h e e .

fi di g ca be ba ed

58

f

a iab e X, Y, Z, W,...

A

ab e

fee e

a e ec ed (i

e

e iga i g a a h e e i

e he heigh

aa

f a h e e . Of c

e,

he ) g fea ib e. I agi e ha

f fi e a h e e a d b ai ed he f

i g e

e

ee

M A

Mi

B

192 c

165 c

M C

Mi

183 c

D

M E

170 c

Mea

175 c

177

S a da d E

4,785394

Media

The observed values x1, x2, x3, x4 and x5 are realizations of the same variable: X – height of statistical athlete

175

S a da d De ia i

10,70047

Statistical Athlete: Sa

e Va ia ce

114,5

K

i

0,84205

S e

e

0,5142

De c i i e a i ic i E ce

S

e he

di

ib i

i h di di

f i e

f

ci

ib i

ib i

A a d

ai

f a

hich ha e ed a

f ci

ha di a

ib i

de e

F( ), i.e. hefea ci

e

F( ). Thi di

i

i ed b hi

ib i

he c ai

ai e haa di

ib i

i ca ed he he e ica

.

e i gi e ba ec he a ebabi i he heigh

di

f a d ib i

. I

a iab e hi e a

a h e e bef e e

ea

( X 1 , X 2 ,..., X n ), each e, he

ca be

ed he .

59

A a d

a

i de e de e a

e,

ei a d

e ca

he heigh

aid

a iab e a

i h he di a eib i

ha he heigh

f he f

ba

aed

he

.C

i g

a e ha

ac ica i e

i e, he a

i

e c

f

bea i g

eai

f

ide ed he e

i be

e.

A ea i a i

f ai

e a

ei

( X 1 , X 2 ,..., X n ) , de

ec

he e

ed b

X

X1

X2

M A

Mi

The a

e di

The e ci

gi e a ai

60

a d

1

f he

, x2 ,..., xn .

a i ica heigh f a a h e e

B

he

a i ic i a f

f bhee ed a e

X3

ib i

cha ac e i ic X i

f

f he h cicee

a e . Thi he i

i de e de ce. U e i

e if ( X 1 , X 2 ,..., X n ) i a ec

be i

X4

M C

a i ica

Mi

ic

e f he di

X5

D

M E

ib i

f he

a a h e.

edai a

e

e

e: a

a i icai a bi a

a iab e . S a i ic a e he ef e a d

a iab e , a d a

ch ha e a F

e a

babi i

di

ib i

e, ha i g a a d

ea

a i ic (a

, e ac

a

e ( X 1 , X 2 ,..., X n ) ,

a

abe ed a X ) a f

a

a

eha i h e ic f a d

babi i

di

a ia ce, Of c a

ea

n

,

e, he e i

a d

The a e age f

a iab e. Th ha

i g ea i

e

a iab e f

e

X a

,

,

ch a

di g

hi be

ee

he

ai

ha a

he e

ec ed a e,

a i ic X f

The a e age f

e: x

he a

ai e

a i ic X

f he

ea d

he

aa ee :

be !).

Rea i a i

he a

, e c.

he f

e a d he c

(

fhe

hich i

a da d de ia i

i

i =1

a iab ei ie f a a d ib i

ca e c ea e he

:

∑X

X =1 n X i i

i a e.

he

:a a a e e ca ed he

ec ed a e

f he

a i ic X

N e ha

e ha e a ead

di

ca be e

ib i

each f hich ha a be ha

f

he

We

a

ea

babi i

ee a

e f a ch i ic. a A bi

he

f

0 1 di

ib i

e f ei f

i de e de

. If

e de

ha a bi

cce , a d he e

ia a d

e a ia di

d de e e d

a iab e ,

cce , he ib i

he

. Each ia

he e

ia . ee

de c ibe hei he

i

ecc e i a a

he a e

f

e

e ed a

e f

a i a i ic (f babi i

a e fa a

di

ib i

ci .F

f a d e a

e,

a iab e ), i ha i

he di

de ib i

f

e?

61

S

e he

e

ai

ec ed a e a d a ia ce f he a

1

E( X ) = E 

n



n

∑=Xn

i

i =1

a d σ2. Le ' ca c a e he

ea a d a iaa ece e

ea :

11 n  1 n  En=n∑ X i ==⋅ ∑ E ( X )   i =1 i =1

1 n ∑=X i  n i =1 n

D2 ( X ) = D2 

1 2  2 D=n  

n

∑X i =1

1

i

n m

m

n

1

==⋅ 2 ∑ D 2 ( X i )

n

n σ2

2

ni=1

σ2

I de e de c

The ef e, If ( X 1 , X 2 ,..., X n ) ha e ( he a e) i

e a

a di

e (i.e. hea e i de e de ), he

 

a d X ~   m,

be ib i

ib i

Di

ib i

h a d

62

di

ib ed

.

σ, he a

ei he

f i de e de ), ha a

he eade ' a e i di

a

n

30), he ba ed

di

X i a

σ 

If ( X 1 , X 2 ,..., X n ) ha e he a e di de ia i

( ,σ) a d de c ibe a

ib i

i

ib i

i h a e age

(ie de e de c !) a d he a

ca ed ce a d

a i hei e

a iab e ha a

i ae

he a e di

he fac ha hi d e

( he

a d

e i a ge (i.e. > f a a ge

i ae a ib i e

a da d

a

a gi e ab ie

e. We d a

edge f he

f he cha ac e i ic ! f

a i ic

a a ce

he e ), a d he ef e a iab e .)

e

i

a

e ai i ica i fe e ce ( e ie

e

ef

i ga d e

a i ic (f

ci

f

χ2 If

,

1

4

,

hei

a e i de e de ,

a da d

a a d

a iab e , he

he

f

ae

χ 2 (k ) =

k

∑X

2 i

i =1

i di

ib ed acc di g

Tab e f

hi di

a d he f

ci

hechi

ib i

ia i

i i c ded i

Fig. De The χ f eed

2

di ),

ib i 0

a df

i

a e di c

a

i

ab a i ed f

σ = 1. O e

ib i a i ef

ead hee

f

ih

ci diffe e

afi d he

deg ee ae

2

ac age .

ib i

.

(ca ed he babi i :

.

ide a ai ab e

a d aa i ica

f he χ di

f f eed

be

f deg ee

P ( χ ≥ u0 ) f 2

4

Those readers who are interested in the srcin of the names used in mathematics are encouraged to have a look at the following page http://jeff560.tripod.com/c.html.

63

f

ch a ab e. F ab

diffe e

a e

f he

(

≠ 1, he f

e, i.e. σ

 χ2

P ( χ 2 ≥ z0σ 2 ) = P 

2 σ

a P χ

2

≥ u0 ) h

f hei e he

d be e aced b

 ≥ z0  . 

e 5)

(G

a da d de ia i

c.

S a i ic:

X −m n −1 S

t= i ca ed he

de

Ge e a , he

a i ic

a i ic

ih

1 deg ee

ih

1 deg ee

f f eed

f f eed

 

X ~   0,

.

i a

e

e he i

fe

σ 



n

Y ~ χ 2 with (n-1) df n −1 The e : The e he S de

S , he di he he

he a da d

di

ib i

e ce F ( ) ib i

c

be deg f ee a di

ih

f di

ib i

f

1 deg ee

( )=

e ge

hea da d

ib i

f a d a i fie

t 1 e dt 2π −∞

lim n→∞ Ft

f f eed

ci

f f eed

∫

−

ai

ih :

t2 2

a di

e ceed 30, he

a d he S de

a iab e he e

di

ib i

.I

ac ice,

he diffe e ce be ib i

ee

i

5

The derivation was first published 1908 William Sealy Gosset, while he workedof at the the t-distribution Guinness brewery in Dublin. He in was not by allowed to publish under his own name, so the paper was written under the pseudonym Student. We encourage the reader to become familiar with his biography http://en.wikipedia.org/wiki/William_Sealy_Gosset

64

i ig ifica . Thi e di

ib i

de

i

ai

h i

e b

a 30 deg ee

f he S de

di

f f eed

ib i

he ab he e f S de

. Thi e i ide

f

he g a h f he

.

Fig. De

i

f

ci

f he S de

e

e ce if de e de

di

ib i

.

.

Le X 1 , X 2 ,... X n1 , Y1 ,..., Yn2 be f

he  ( m,σ ) di

X =

S 12 =

The

ea

1

n1

1

n1

ib i

k =1

n1

∑ (X k =1

k

Y =

k

− X)

a i ic ha e he f

a iab e

:

n1

∑X

a d

2

S 22 =

i g di

ib i

1

n2

1

n2

n2

∑X k =1

n2

∑ (Y k =1

k

k

−Y )

2

:

65

σ

X ~  ( m,

Th

n1

Y ~  ( m,

)

σ

n2

)

:

X −Y σ

Z=

n1n 2 ~  (0,1) n1 + n2

a d

W = The U

n1S 12 +n S2 σ2

2 2

~ χ 2 ith n1 + n2 − 2 df

a i ic i defi ed a :

Z

U=

U ha a S de

U=

di

i h n1 + n2

ib i

X −Y σ

n1n 2 n1 + n2

= nS n+S 2 2 1 1

W n1 + n2 − 2

2 2

− 2 df

X Y−

nn

− + nS nS + 2 22 2 1 1

1 2

n1 + n2

( n1

n2

2)

2

σ n1 + n2 − 2 N e: i i a

ed ha he

Si ce he di

ib i

a da d de ia i

i a

U

a i ic i a

ica

di

ib i

a i ic i a

66

f he U

ica

hef a,

a . Thi

ea

i ae

a d he ec

c de f

ha fa ge a a

a ee hi

a.

ha he

e ( > 30), he

(

).

I ca be h

1

p=

ha he

a i ic haS a ga

n

( n − 1) a d b =

2

a di

ib i

ih aa ee

.

2σ 2 .

a i ic i defi ed a Z

The Z di

ib i

ih aa ee

2

he χ di

ib i

= nS 2 . O e ca h p=

i h1 deg ee

6

1 2

( n − 1) a d b = f feed

1 2σ 2

,

hich i e

a i ae

.

.

Le X 1 , X 2 ,... X n , Y1 ,..., Yn be e e ce 1

he  ( m,σ ) di A a d

ha i ha a ga

if de e de

a d

a iab e

ih

2

ib i

a iab e

.

i h a F di

ib i

a i e he a ai

f

chi

a ed

a iab e :

F= i h ( 1= N e: (d e

6

1

1,

2

a

i i a

he f

=

2

1) deg ee

f f eed

ed ha he

fa da d ab e f

S12 S22 . e a a ige ha

he F di

ib i

he de

i a

).

To find out more about t he author see: http://en.wikipedia.org/wiki/George_W._Snedecor

67

Fig. De

i

If he cha ac e i ic X ha a e ec ed Di

ib i

X

a i ic a egi e i

ci a he f

f he F S edec babi i

di

ib i

E

ec ed a e

if X

~

 (0, σ )

2 i

i =1

if X

X −m n −1 S

X −Y

68

2 2

~

Va ia ce

n

 (0,1) n −1 f n−3

0

if

X

nS +n S2

f

2

k

2 1 1

ib i

σ

i

i =1

∑X

a iab e.

, he di

0

n

∑X

a d

i g ab e:

Defi i i

1 n

χ2

f

n1n 2 ( n1 + n2 − 2) n1 + n2

0

if

~

 ( m,σ ) n1 + n2 n1n 2

>3

X 1

n

∑(X n i =1

i

− m)2

~

n −1 2 σ n if X

nS 2

if X

2

~

n2

2(n − 1)σ 4

 (m,σ )

~

n2 − 1 n2 − 3 f

2( n − 1)σ 4

 (m,σ )

~

( n − 1)σ 2 if X

S12 S22

 ( m,σ )

2(n2 − 1) 2 ( n1 + n2 − 4) ( n1 − 1)(n2 − 3) 2 ( n2 − 5)

>3

f

2

>5

 (0,1)

69

Lec

e V. C

fide ce i e a .

The fac ha i i

ib e, ba ed

aa ee gi e

e

b ai i g a

he

aa ee C

a

ea a

i agi e ha

ia di

he

e i

ib i ih

,

ef

i

i ai

(ca i

6 i e i 01 h

a ee

a

i e .H

x = 0.4 , hi e he ea f ca be e

ed

a iab e

a

i e ,a di

ha ca he e

ai

i h a high S

he

ea i i i h

ib i

i

f

P( X = xk ) = pk (Qi)

70

h

ha head

he

babi i

he

ea f

ai

ae

ea . I i

he a ge [0.4, .6] a0 d

ha

, each fhe e a

eeI b ai

= 0.6. N e ha b h a

i ai

ef

ie ea

he

ea

i ibea e i .S , ab ce ai

ch a e i a e i g

hich i c de

e

! B h a ae e

ai

ha

d ea

babi i .

e he di

f he de

i x

ai

defi ei a e a

f

a a e e Q (e.g. he f

he a e a

e II). Th

ec ed a e i he e

e a ie a i

gh, i.e. e

e

f

be he

. The babi i

e I) i

( a

a

ii e

a

e e , ca c a i g he

e i a e he e

f4

babi i

i e a if1

. Defi e he a d

4 i e i 10 h

babi i

he

= 4)= P (=X = 6) 0.205078 .N

a c i 10 i e a d each e

head a e h

eb

ca c a ed a e.

d ha he

= 0.5 ihe a e a

i di id a a iab e , each de c ibi g a i g e c i h

a a e e , beca

he e i a ed a e f ha

f

e, i.e. P ( X

ch a a

head , a d 0 if ai a e h

i

e i a ea

fide ce i e i.e. a . e ca a gi e

ai

e f 10 ee e

e

e fec

e.

ed he bi

cce e i

fc d

bediffe e

i ge a

e,

ce e

he

ce ai

cce e i a a i

he c

hich c

ef

i a

ide he f

Whe

de ie

e ica a e

a a

e ai Xa i ai

f ( x, Q )

ci .

ai

a e age a d/ babi i

i de e de a ia ce). I i a f

ci

e ed ha he

Let E be a vector of observations:

( X 1 , X 2,..., X ) . E is an n - dimensional n

random variable dependent on the parameter Q. Let U (E ), U (E ) UE

()

be functions of the random variable E such that

≤ UE

. () Let α be a real number 0 < α

< 1.

If the following holds:

PU ( E( ) Q ≤

≤UE

≥ (− )) 1

α

(for continuous random variables = 1 - α), then the interval U (E)U, E( )

is

called a confidence interval for Q, and 1 − α the confidence level. Theoretically, you can build a confidence interval for each parameter of the trai t’s distribution, but in practice they are used to construct confidence intervals for the population average (Q = m) and variance (Q = Var(X)). Below we show the corresponding confidence intervals for both parameters.

Confidence interval for the population mean (Q=m) Model I. Suppose a trait in some population has a N(m, σ) distribution. Let’s assume that m - is unknown, σ is known, and the sample is small (n < 30). The estimate of the population average is the sample mean statistic X . Given these conditions, X has a N ( m,

σ

n

) distribution.

Thus, the standardized random variable U

=

X −m σ

has a N(0, 1) distribution.

n 71

( )

1α

α

α

2

2

− uα He ce, he a d

a iab e U a i fie :

P ( −u<α F

U <

a gi e α, he a e uα ca be ead f a di

ib i

uα

0

u=α−) 1 α eh ab e

f he

:

Φ (uα ) = 1 −

Si ce U ha a (0, 1) di

ib i

a d

α 2

e ha e U

=

X −m : σ n

  P  −u< α  

72



 X −m < =−uα  1 α , σ  n

 

a da i ed d

 

PX  u−

σ α

<
=−

σ

α

  1

n

α

.

Thus, for example for α = 0.05, the confidence interval is as follows (from the uα

table

= 1.96):

X

−

1, 96σ n

,X

1, 96 σ

+

.

n

This means that in 95 cases out of 100, the estimated „m” is in this range. In other

1,96σ words, the error in estimation is not greater than

n

in 95% of cases.

Example. The waiting time for a tram has been studied and the following values obtained (in minutes): 12, 15, 14, 13 , 15. Suppose that the waiting time for a tram has a normal distribution with unknown mean value (m) and a known standard deviation σ

=

2.

Construction of the confidence interval for the mean involves finding the appropriate values in tables and some simple calculations . In our example, we find that x

= 13.8 .

read the value u

α

Assuming that the confidence level is 0.95 (i.e.

= 1.96

formula, we obtain x

α

=

0.05 ) , we

from the table. Substituting these values into the

− 1.753; x + 1.753 =

probability of not less than 1 − α

=0.95

12.0469;15.5530 . Thus with a

, the desired population average lies in

this interval. M de II.

73

The trait X has a N(m, σ) distribution in some population, where neither m nor σ are known. To build a confidence interval for m, we will use the t-statistic with n-1 degrees of freedom:

t

=

X

−m

n −1

S

()

1α

α

α

2

2

− tα The a e deg ee

tα

i

f f eed

ead f

he ab e f

he S de

di

ib i

:

P ( −tα<
 

P  −t< α

 

74

tα

0

=tα − ) 1 α



 X −m < = −tα  1 α , S  n −1

 

ih

1

 

PX  t −

α

S m< < X +t n −1

=− α

S

  1

n −1 

α

.

Thus, for example for α = 0.05, the confidence interval is as follows ( t is read α

from the table for the Student distribution); e.g. for n = 26, this value is 2.056):

X−

2, 056 S 5

;X +

2, 056 S 5

.

This means that in 95 cases out of 100, „m” lies in this range. In other words, the error in estimation is not greater than

2.056 S 5

in 95% of cases.

Note: this length of this interval is variable, since it depends on the value of S.

Example. Suppose that in the previous example, o n the average waiting time for a tram, there was no information on the standard deviation in the population. Again, we calculate the average value x , which is 13.8. We calculate the standard deviation for the sample s tα =

= 1.3038 :

and read the value

2.7764 for α= 0.05 from the tables. We thus calculate the confidence

interval to be: x − 1.618931187; x + 1.618931187 =

probability of not less than 1 − α

=0.95

12.18107;15.41893 . Thus with a

, the desired population mean lies within

this confidence interval

M de III. F

a ge a

e ( > 30), he ce

a i i he e

ae

ha

75

σ

X →  ( m,

n

n→∞,

)f

hi e he a

f a ge be

ae

ha

S →σ . The ef e, b b i i g he he a e a da d de ia i

 

s

P  X − uα

E a

e. S

e ha i

f

a a

he e

b

e

a aged ed x

c

ai f

ai a da d de ia i e ge :

,

< < m+ X ≅u−α

n

he a a

i

ai

he

ga he

ch

f he e a

α i

s   1 α. n

ehe a e age

a da d de ia i

e da a:

i

= 34. F

ha 1 − α

= 0.95, he de i ed

f b e ai

e i a e bec

M de IV (c

e

cha ac e i ic (e.g. a d

he e he a d

76

ai

ih

,

e

i

ii

ea

i ha ie i

,a ac

babi i hi c

e

f

fide ce

e ce f he a ge

he e g h f hi i e a ha dec ea ed ig ifica

ai ai

a

i

acc di g c

he

, i.e.

d c

). e e ce c a ed a g

e , e c.), i ca be de c ibed b a

P ( X = 1) =

e e .

ai

e acc a e.

fide ce i e a f

If e e a i e a

e

he

= 13.9411 , s = 1.1791. He ce, he e i ed c fide ce i e a i

i e a . N e ha , i acc da ce be

ai i g i e

he e da a

x − 0.41142; x + 0.41142 = 13.5297;14.3525. Th e

de I i h

a iab e X a e

d a d bad, i

p, P=( X=− 0) 1 he aif ehe 1 fea

ab ecece aif a

ei

di

ib i

p, e e

a d if0

:

Th f

, if a fea

ei

i gi e b X

We fi d ha f

1 n

=

n

∑X

i

i =1

i e i a

=

ee e

p

f he e ce age f

e

a

be f

m 600 = = 0,333 , n 1800 he f ac i

f

e, a a

i ai

0.05 < <0.95 a d > 100:

m m   nmn  u−α≈

e, he

a

m ) = p. n

 m m   1− m n n  <+ P  − uα nnnn  E i ai a

b e ed

e

g

de

. I a 1800 e e e

α = 0.95, uα

= 600. F 1

m m 1 − n  n  ≅ 0,011 . Th n

de

    1 α  

1−

= 1,96. Th ,

he 95% c

fide ce e e f

i gi : 32,19% < < 34,40%.

σ).(

The ec

d

c

a i i . De e di g a da d de ia i a da d de ia i

ed e

e

ai

ie e

. I ge e a , de i i g a c i

ea . H

aa ee i a

, hi ea i

e e , if

ed

fide ce i e a f a

ea

e

f i

i g he a ia ce

e ha

he a ia ce

he fea

e

ha a

77

a di di

ib i

ib i

,

a ea

a

i ae

, a ead

a,

2

e he χ a d Z

e ca

.

M de V: S

e ha he

ai

ea

a d

ai

a da d de aiae i 2

a d di

he a

ib i

e i a ge (

i h deg 1 ee

f f eed

> 30). The

a i ic

1/2α

c1

c2



P  c1 <



{ <} c=

78

2

1

1

nS ha a σ2

:

1/2α

he e: P χ

Z=

{2 α }≥P =χ

2

 nS 2 < c=2 − 1 α σ2  c2

1 2

α .



χ2

N e: M

ab e

gi e

de i a i

f he a e f

We b ai

he f

a e i

1

i gc

e

 nS 2

< σ <2

 c2

e. C

ide agai

We ca c a e he a a da di ed χ

2

ie

e a

fide ce i e a f

P

E a

P {χ 2 ≥ a} a d he ef e he

f he f

he da a

f

he a ia ce:

de III.

e a ia ce: =1.3903. Th ea

.

nS 2  = − 1 α . c1 

ed i

a i ic ( he

ai

he b e ed a e f he

f he e

ai

) i gi e b 1.3903 * 34

= 47.2727. We e

ca c a e he a e i ed c

f c 1 a d c2: c 1= 19.0466 a d c2=50.7251. He ce, he

fide ce i e a f

he a ia ce i :

47.2727 ; 47.2727 ≡ 0.9319; 2.4819 50.725 19.0466

M de VI. A

e ha

he a

e i

b ai ed f

he

ai

fi e e

he f

i gf

 

P

1 + 

he e: uα i

ha a

a

c

e

a ge, i.e. > 30. I e a e i a i

ead f

ab e f

a di

f he

ib i

a da d de ia i

a d i

a



s <σ < uα

1−

2n he a da d

s  ≈− 1 α uα 



2n   a di

ib i

, N(0, 1).

79

E a

e. Agai ,

e

e he da a f

34, s = 1.1791. The a e uα Th

e b ai

he f

he e a

= 1.96 (f

i gc

ef

de I hi III. ca e,

he ab e f

fide ce i e a f

=

he N(0, 1) di

he

ai

ib i

).

a da d

de ia i [0.9527; 1.5467]. N e ha he c ed

a i g he f

fide ce i e a

b ai ed i

b ai a i e a e i a e f ae

i gc

he

he

e i

f b h e dhe fi e a . I

hi ca e,

fide ce i e a : [0.96535, 1.5754]. A

e i a e b ai ed i

i ia

e a

a da d de ia i

e b ai

ca

he i e a gi e ab

e ca a

be

. Thid i e b he

ee, he i e a

e.

.

A a ead aa ee

e i

ed, he

fa

ai

a ai ab e ab ai e he

he

.I i a eci i

e fc

ai

e e ,i

g ea e a

ei i f da a i

80

ee e

ec a i

i gf

e

a

ea

cia ed i h

eci i

i

de i ab e i

e he a gehe fi e a , he be e a

e i e. F

ai a

a ge. I hi ca e,

a e i ei g ea e ha acc ac f he e i a e.

he e i e

ecif i g a e e c f fide ce,

ib eb ai a a ge

N e ha i each ca e

e i a e he

e a ed i f he a i

be e e be ed ha he " ice"

acc ac i a i c ea ed a e.g. beca

ai

b ea e ae a he ha ea

e fi a e a d h

e . Ad i ed , he a

e i a ei .H

c

: e

ed ha , a

b ai i g a acc a e e i a e. L c

fide ce i e a

i d e

e

e gi ehe e

i ed f

ea e, e he a

,

e a ea

he c c a

e a f

hi

e ,

ec f i ga

a e

he e ha

ed ec e fide f ce a d

i ed e g h f he i e a , a

he

hi

diffe e ce be

ee

he igh a d ef e d f he c

edge a

de e

i e a e i ae We c

i e he ece a

i h a ede e

d i e, he ef e, fide ce i e a

i ed c

de e

f agi e

fide ce i e a . Thi a

e i e,

fide ce e e a d

i e he a

ha eci i

ia e a

e ca gi e .

e i e

e g h (acc ac ) i h he ch

e c

b ai a 1

fide ce e e

α. Le 2 be he e g h f ihee a .

M de VII U i g he f

af

he c

fide ce i e a f

2d = 2

e fi d ha he e g h f hee i a i

i ei : n F

e a

f

=

acc ac ee e ac ea

uασ n

ai

ea i I,

. He ce, he e

de

i ed a

e

uα2σ 2 . d2 e,

i g

he ab e f e i

he

de I, i ai he

σ = 2 . The a e uα = 1.96i ead

ed ha

a da d

a di

ib i

f he he e i a i g he a e age f, e.g. 30 ec . Ca c a i g

fide ce e e

ib e

= 0.5,

f

f 95%, i i

i h a acc ac

i e f 62 b e a i

d ,i

f 1i

i g he a

e fi d ha n ib e f

e. F

. We ca

ai i g i e f

hi

e

=

he ef e a a a ,

e ha e, i.e. f 5

1.962 ⋅ 2 2 0.52

= 61.47 . Th

e i a e he e,

e he

i ha

e eed a

ih

ai i i

a

e

.

81

M de VIII Si i a

, ca c a i g he e g h f he i e a f

he

=

,

hi

e

he da a f

,

i ed

e e , gi e

e) f i e bea

ef

de

ai i g f

a a

e i e,

b e ai

he ab e he a e f tα

he he i i

30ec

de

he

. We ca c a e he a e age

e

a da d de ia i

:

= 2.7764 f α = 0.05. A f he

d ( = 0.5)? Ca c a i g he e

= 53.41. He ce, i i

(54 5),i

he ea e

30 ec

e a ea

0 , he

ibee i a e he a e age d a i

heea e

e b ai

>

0. If

ed.

hich i 13.8. The , e ca c a e he a

ea

ha e d

e eed a e i a e. He ce,

a

he e a

s = 1.3038 a d ead f bef e,

t s + 1. H d

a da d de ia i

ee e

0

i g

a

: n

. Th

e ( i ca c a ed f

e ai i g

a e

n −1

ai

i i ia a

Re

tα s

=2

: 2d

f

de II, ceed e a

2 2 α 2

ece a

e i a e he a e age

da

49

i ed

e

ai i g i e f

a a

d .

M de IX: U de acc ac

de III, he de e i

de III. We a i ic c

i ai

a aged

ae c

de II. Le '

c

ed,

ec

he he i i he ea e

82

i ed a e f ib e = 30 ec

ch

i ed a

e i ef

e he da a f e da a:

0

= 34. Fi

a gi e

e e

he e a

ef

i g

ha

he

= 0.05), e ead he a e uα = 1.96f i

h

n=

uα2 s 2 + 1. Agai , a bef e, d2

e i a e he a e age d a i

f he

f

, he ece a

x = 13.9411 a d s = 1.1791 . A

fide ce e e i 0.95 (i.e. α

ab e. The e

f he e

he a e a i

ai i g f

d ( = 0.5)? We ca c a e he a e f

he ea a

a

= 22.36. Th

,

i

hi ca e i i ec

d )f

ie

ib e

ih a

fide ce e e ( α

a gi e c

, a d he e

e i ae

i ed a

ede e

i ed

eci i

(

= 0.05), beca e e ha e a a

e i ei

= 30

e f 34

23.

M de X: I a e i g he a ca

e he f a) if

e i e e

i g

i ed

i

e

he

e i a ea

ai

i

,

e

: ag i de

f

,

he

e

i ed

a

e

i e i

uα2 p (1 − p ) : d2

n=

b) If we do not know the order of magnitude of p, we use the inequality

p (1 − p ) ≤

Le ' a hi e ce age f a

1 4

=

. Thus: n

u2 α

4d 2

.

i g he da a f he e a ef e a g de . We b e ed

de WeIV. e i a ed he = 600 e i he

e f 1800. F

1 α= 0.95 uα ei m

n

= 1.96 . He ce, he e i a e f he

=

he ea e If e

600 1800

(e.g. f

322.69. S

he a h

i.e. 33.3%. I i

ib e

fa

de

h

e i a e hi

i

5%?

e ce age f

If

= 0,333

i

die b de

he a

d

e i e1800 f

he e e he ec

he a h i g, f

e a de

) he e

ec ed a e f he

e, 30%, he de i ed a ed i

fficie

ec ed a e f he e ce age f

df

gh i i g ea e ha

aa d he

e

e b ai e i

f

hi

=

e.

de

= 384.16. The e ca c a ed,

e i ei

i g, i g a

i i dica e

e i e,

ha he

83

a

e f 1800

I b h i ai ai e a 385 e e e

84

de ,

i fficie

f

achie e he e i

e . i ed

(5%) i h a 95% c a

e. C

eci i

f

he e i a e f he

fide ce e ei ia

ec i g da a f

1800

de

a i c ea

ib e ed

da !

6.

A

c ai

.

ega di g he

a i ica h

Ah

di

ib i

f a cha ac e i ic i ca ed a

he i .

f

aa ee

he di

he i ib i

f a cha ac e i ic i ca ed a a a e ic h

he i . I

e if

ch h

he e , e aea e ic e

Ah i

hich

he i

ecifie

ega di g

he

he fea

a a e e ) i ca ed a e

e

he i

e i gi

c

ide ed

be

he be c

h

If

eh he he hi h ch a e

The f

be

decide

ib i

f a cha ac e (i c di g

he i . T

e

e if

he he a gi e h ecifi i aia h

a ea a e a i eh

e if i ic

he i fi

c ded ha he h

a ed a d he b ec

he i i

i ca eda e

f he di

fa e. He ce,

he i ) a d f

ide ed

de

ch h

he e ,

.

ece a e

e

f

.

a a e ic h

a a e ic e

H

e ica a e

e

a dd

fa

e chec

he i ca be

he i (ca ed H

0

he i (ca ed H 1),

hich

he i H

e.

0

i

i

a i ica e chec i he h

he e ,

f ig ifica ce.

i gi a ag i h

f

e i g a a a e ic h

he i ( ig ifica ce

e i g): 1.

We formulate the hypothesis H0: (Q = Q 0).

2.

We set the significance level α.

3.

Next, we observe an n - element simple sample.

4.

Calculate the realization u of the relevant U statistic (significance test).

85

5. We

f

he c i ica

a i f i g he f

a e

i gi e

0

f he

he e ec ed ≠

a i ic U f

ai :

P ( U ≥ u0 ) ≤ α

≥

0, he i H u u0 , he eeec heh u < u 0 , he e i If ea e ec he h he i H

6. If

N e: F a E a

a

e

a

e

he i i i g di

e ib i

e he e ac di f he e

ib i a i ic i

af i heic U; f

0.

a ge

ed.

e.

Let Xbe a trait with a normal distribution N ( m,1) in the population of interest, where m is unknown. We believe that the unknown average value is 0, i.e., we test the hypothesis H0: m = 0 against the alternative hypothesis H1: m ≠ 0. The only way to test our hypothesis is to compare it with a sample from the general population. Hence, a random sample of 10 items was chosen and the following results obtained: -0.30023 -1.27768 0.244257 1.276474 1.19835 1.733133 -2.18359 -0.23418 1.095023 -1.0867

86

We eed

a e

10 e e e c

f ic

he he he a

e, i a ge e

i h he h

ha if he

f b ai i g

ee

a d = 1.2924. Th

e

ea a d

h

ig ifica

he i ? T a e

(i

a )

t=

ed

de H 0 ha

h

e

e e ec H 0.

x = 0.04649

a d b ai

he a

be ie e ha

hi fac he

aa

he a e age a e f

he i a d he a e age a e ca c a ed f

diffe e ce

ha hi d e

e e e

ee

ecifici a e , a

a ce , i i

a da d de ia i

, he diffe e ce be

ch

c ai

a ci c

ha 0.05( he ig ifica ce e e ), he

We ca c a e he a

h

f b ai i g

gh be ab e

he i H0. I

babi i

b e ed i e

babi i

he

e i 0.04649. I hi

e

e ec he

a i ic i ed.

X −m n −1 S

he e:

x = 0.04649 , =1.2924, = 10, =0. The ea i a i babi i ha

hi

ha he ab

a e

each af a

de H0 i 0.912. We i e

he i i a

e, he

f f eed

i e

e a e f he ea i a i

a e (i.e. ha he

0.04649) h

f he i aic 9 deg ee

ei f

e hi a

he cha ce ( ea

a

0.114. The

e f hea i ic i g ea e he a a f

0 ha

ea i g ha if he

e f he i e ih

d) f b ai i g

0.912. Q i e a , if e e e be ha he

e i i i

ch e

1. Tech ica , he a e f hi a ec

e

1d e

a

babi i

i cahe ed

he e ec i

a eO

f he

h

i

ii

i

he i . B

ha a ha

87

h

d be he

a ee e

e

ha

be

e .F

e a

e, h e e,

i e d ha

e

hich accee

f head a d ai . H

dc 10:0 c

i

.I

ibi i

ce ai

e !I i

d ha

e .I

a e 8 head ,

e a

e h

10 ai i.e. e ai , he ce

he e iabi i c

ac ice, i i a

e

h

=2*(1+10+45)/1024≈0.11, b =2*11/1024≈0.02. Th

ai 0.05 e a d hi i

ha c e ci

ge he

acce

fa ehi

babi ei i hef 8,9

afa a a f

e ec head if 8 (

he

be

e ec H0 ( ha he c i i fai ). If he e

gi e

ha hi he e

8,9

ec ed 5 head a d 5

ea

e ec if 9 head (

, if he

e

10head

he e

ai ) a

a

a d

he i H 0 af e a a e e e , gi e de

ha a e a ea

ec a e

he e e

ee ce ai

e

ea

ee

ee

f he c i ? 7:3? 8:2? B

ed ha hi

e, i

he i ? Af e a ,

i e be

d ca c a e he

ed

h

i g a c i 10 i e ,

ab

ece a

f e ec i g a

he

ha i ba a ce be

f ig ifica ce= α 0.05. Thi

ca ed he e e

c

a ef

ai ) a

ea

f head ( ai ) a 8,e e e9

10 head (

d ai ),

e

d

e ec H0. T ei e a e: he e e f ig ifica ce a ci i e acce ed b ac i i e . The ig ifica ce e e ca be cha ged: i fac , he e e a e af aid f e ec i g H0, he hi e e h

e

he

h

h

he e he i i

fide ce i e a f

e d

i

a d

eaffi

ed ha

ha he i i a i

fide ce i e a f

ea i g

he

ai

ai

a a a e e ca a

i , he

a e age i

e

fide ce i e a . If b h e d

e a

ai

ea ba ed hi

e

be

e,

= 0, a d he

a a ae

ae ha ega i e ef ha i i

(i.e. e d e ec H i

a

α i ed, a d e

b

he a e af ahi e e . If, f

ha he a e age a ehe i

i i e igh e d

ib e ha he i i ide he c

88

ide a i

i cide a . The a e

he a e a e f 0.05. The c

ed

c

,c

f1 αi

fide ce e e fe

d be. i aI

d be i he a ge [0.02, 0.1].

A he e d f hiec i c

e he ig ifica ce e e h

fhe c

0)

ie , beca

fide ce i e a f

e0

he

ai

ea ha e he a e ig , i f α = 0.05 (i.e. i i

H0 a a ig ifica ce e e ea i

F

e

ea i g H0 a

be

eca ifhae acce

ea

g

d f

ga

f he

h

g a d e.g. he

e e

c edib e ha he

i

babi i

he

e ec i

e ca

ia

c

h

he i , i

a d i hi be i e

fi d

he i i

di ide he e

e e

be ed b e a i

he ec igi a a

d a

g e ide ce c

di i

a.H

e ed a adic i g i .

e e , i b h ca e ,

e

e i gi e b α

f e ec i g H 0 he i

a

ei i

a

he fi

e. S ch a

e. If i i h

a da d de ia i e ha e he f

a

a he

) i he i g a

)b a i ce di g he

ea d dd

e h d ca be

ge e

(

ed e

be ed b e a i

he h

, he a a ee (e.g.

e a e :

e

h

d

X

Le

e e ec

f ig ifica ce.

Le

i

a

ha he e a e

Re ec i

i i ,a d

a hi g iifica ce e e ).

he ec d, e ea

ha 0 i

ea

ge ei ea

f he

a ea d

ig ifica

Th diffe .

Y

0.30023

1.27768

0.244257

1.276474

1.19835

1.733133

2.18359

0.23418

1.095023

1.0867

e a i e he he b h a e c ef ai i h he a e ai ea ( hich h d be he ca e a he c f e he a e ai

), i.e. m X

= mY . We i

e he

a i ic

89

X −Y

= U

n1n 2

nS +n S2 2 1 1

We ca c a e he a

ia e

+

n1 + n2 ( n1

2 2

− n2

2)

a i ic : 1

Mea Va ia ce

2

0.010762

0.0822094

1.888100953

1.866891821

5

5

Ob e a i

The ea i a i di

ib i

a

ea i a i i h he

e ec i

f he U

a i i ic0.08244548. Thec i ica a e f

i h 8 deg ee fUc h

f f eed

ai ed i

he i

f

he i e a [ 2.306, 2.306] i

a ed ab

he S de

= 2.306. Th ,

he ab e i tα

di

e. The e a e, he ef e,

ic

i e

g

d f

i

.

De e di g

he c

(Q = Q0)) ca ha e

e , he a e a i e h e he f f

i gf

he i ( he

h

he i H

0:

:

H1 : (Q ≠ Q0 ) , H1 : (Q > Q0 ) , H1 : (Q < Q0 ) . The fi e

90

i a

ca e i ca ed a ided e . The

ided a e a i e a d he c he

caaee ca ed igh

e

di g a a e ic

ided a d ef

ided

a e ai e , e F

e a

f

e a

ig ifica

a ic a di ec i

he a e

e, he ea i g

. Thi h f

b) O e ided (di ec i i

f

ide ed ab

e a d

e

e

ee e

ided e

a f

.

a e he

he e :

ided, f

diffe

e c

e, e a i i g he ea i g

i gh

a)

ec i e . The e a

b. Thi h

he i i

f

e a d ided, beca

he e ia e e

ii

e did a

ea

he diffe e ce. a ), f

e a

he i i

e, ie

e ea

ig ifica

he di ec i

e ha

f diffee,e hc

i i

e e

ided. The

e

f

e ided e

i g

a i ic a e i

a ed be

:

H1 : (Q > Q0 ) ( )

1−α

α

0

Q0

91

H1 : (Q < − Q0 ) ( )

1−α

α

Q0

Re

i g

e hi ha he

he e a

ec

ha he a

e

he a e age i he h he i m X

ea i a i di

0

f he U

ib i

ce ed i h ahe c

e f ai

a

e

a d ,

ai

i h aea

e e e ed b he a e .S ef ae he a e a i e h he i m X < mY. The

= mY agai

a i ic i= 0.08244548. The c i ica a e f

i h 8 deg ee

f f eed

(

e ided e ) f

tα = 1.8595 . S , he ef e, he e i

ea

T fi i h, he da a a e

he e : a c

ih

a

a

ed i

e . We

a ea

ie he F S edec

F=

i h(

1

We ge :

92

1=4,

2

1=4) deg ee

e ha a e g ea e

f eed

S12 S22 .

di

e ec he

e :

ai

he S de he ab e i h

he i .

f a ia ce

1

2

Mea

0.010762

0.0822094

1.888100953

1.866891821

Ob e a i

5

5

Df

4

4

Va ia ce

F

1.011360665

P(F<=f)

e ai

F C i ica

0.495763859

e ai

We ee ha he a ca

e ec he h ai

f

i

f he e

ai

f a iia ea ce

he i

f he e

ai

f he a ia ce he i

hich he e a

i 1.01136 a d i

E a

6.388232909

ch

e c

a e ha

e. The ea i a i

f he F

a i ic

he c i ica a e 6.3882.

e.

A fac

d ce

hi cha ac e i ic h e if

ee e ai f

he

ai

. The

ie e

ea

ha he

ib i

a de da d ia i

i ha f

d be 4. f

d ci

, ae ia

a da d de ia i f

he ech ica e

We he ef e h

 (m,σ ) di

d c A. Cha ac e i ic X ha a

a e f σ. Tech ica e

T

ab e:

ie e

he i e H 0 (σ

We ca c a e he ea i a i

hi

e a

a ei

a e

ih

= 20

= 4.4. D e

d ci

?

= 4) .

f he

a i ic Z ,

hich ha a χ 2 di

ib i

ih

σ2 1 deg ee

f f eed

:

93

Z 20 ⋅ (4.4) 2 = ≅ 24.2 2 σ 42 Le

e α=0.01. We

a

f

he a e f

P

 Z



>z

 σ 2

0

hich a i fie :

0

= 0.01



2

F

he χ di

he ab e f

0

ib i

,

ef

d ha f

19 deg ee f eed f

= 36.19.

Th

,a

i g ha he h

he i H0 i

e, he

 Z 24.2  i >  2 σ 

P

babi i

g ea e ha α=0.01. C e

c

i

: he e i

ie e

α = 0.01

f

30.1435, a d i

E a

ea

e ec he h

he i

a e a i fied.N e ha e e a cha ge i

α = 0.05 d e

affec hi

e

ha he ech ica he e e

f ig ifica e c

: he c i ica a e i

=

i g ea e ha 24.2.

fa f

agi e

h

a e ei he g

f a iece bei g f a ed i iece

e ef

f a ed i e We

f

.A a

d

be defec i e. The

i 10%. I

hi dec a a i

a d he h

he i

d

f a ed. The

e f 30 iece h

dec a e

a

babi i

a e a d4

ha he

H 0 ( p = 0,1) . The a e a i e i igh ided, i.e.

e:

,

X =

30

∑X k =1

If he h

he i i

e, he :

P ( X = r=) We b e e = 4 . The

i

ea i ic?

( >0.1).

94

0

e.

Piece

De

he

 30   r⋅ ⋅  0,1r 0, 930−r

a e i ig e b :

k

.

f

3  30  ≥ 4) = 1 − P ( X < 4) = 1 − ∑   ⋅ 0.1 ⋅ 0.9 30− = 1 − 11.134 ⋅ 0.9 27 = 0.3527 =0  r  r

P( X

r

r

C

c

ie

i

: he e i

ea

A he e d f hiec he

ai

f

e, e

he i

a

e a i ea e a

d ced

a a e ic e , i.e. a e di

e ec he h

ha he

i

f

ed

e

ha a e f a ed i 10%.

ib i

e

d c ,

ed

fc

. S che

a ibi i

f he e

a e ca ed

i ica

a a e ic

ia ce.

We a e e a i i g he ae a e f

ib i

e

hich ca be ega ded a a

e a i e he c

i h a he e ica di

e

e f he e ie

a

ai

d ci

f

a

fac ed c

e

i e a d c a ied if a g

Le g h f e ie

6

1

C a ified

DDDDDD

Z

1

2

8 DDDDDDDD

. 50

cce i e i e

d (D) defec i e (Z):

1

2

Z

DD

4

5

1 Z

13 DDDDDDDDDDDDD

ee e N

be

f

3

6

7

e ie

2 ZZ

13

1

DDDDDDDDDDDDD

8

Z

9

hich defec

ae

e i e

ha

50

11

11

D 10

H he i H0: g d a d bad c e The a e a i e h he i H1: he ech

1

ae gica

d ced i a a d ce ead

a . b e e ce i

Acce a . a ce f he a e a i e

95

h

he i

ea

ha he e

e ab e he ide ifica i F

he da a a i ic ha

a be a

f a

e e ed he e, he f

i e

i gf

e he

ea e

> 20ha a a

he e: U

he

g

he

a i d cf i

ce ,

hich

a

be

f

.

ca ed hee e ,ie h

e e

:

Z= hich f

e a ic e d i

U − µU , σU

i ae

be e fie ,

a di be

1

ib i

.

f bad ee e

,

2

he

de e e Mea

a e: µU

S a da d de ia i I

ea

e: = 50,

=

2n1n 2

n

σU = 1=

6,

+1 , 2(nn 1

2n 1 2n− ) 2 n n 2 (n − 1)

2=

44, U = 11 :

 2n1n 2

U − Z=

2(nn 1

+ 1

 n

 = −0,39

n2n 1 2n− ) n ( n − 1) 2

2

Zα =0,05 = − 1,645 C

c

i

d ci

96

(a ef ided e ).

: he e i he e a e

ea

e ec he h e a ic cha ge i

he i

fh

he i e ih

ge ei d fe

f .