Tests of Univariate Normality

(1)

A C T A U N I V E R S I T A T I S L 0 .0 2 I E N S 1 S FO LIA OECONOMICA 40, 19R5

W i.asiaw Wagner *

T EST S OF UNIVARIA TE NORMALITY /

l. _ _ I n t r o d u c

1

0

n

A la r g e c l a s s o f goodness o f f i t t e s t s in th e th e o r y of s t a t i s t i c a l in fo r e n c e a re t e s t s o f u n i v a r i a t e n o r m a lity . T e s ts o f n o r m a lit y e n a b le t e s t in g the goodness o f f i t o f a sampled d i s t r i b u t i o n f u n c t io n by th e norm al d i s t r i b u t i o n f u n c t io n of a g iv - en randon v a r i a b l e .

H i s t o r i c a l l y , th e t e s t s and b2 * b u i l t in the t h i r t * l e a , a re the f i r s t t e s t s f o r n o r m a lit y . O th e r t e s t s w ere d e ve loped by G e a ry ( t e s t g ) , O a v id e t a l , ( t e s t n ) , Kolm ogorov ( t e s t 0 ) , Cram er and von M ise s ( t o s t C M ), Anderson and D a r lin g ( t e s t A2 ) , S h a p ir o and W l.lk ( t e s t W ), O 'A g o s tin o ( t e s t Y ) and o t h e r s . B e s id e the above m entioned t e s t s t h e r e a re « a n y m o d if ic a t io n e w hich a r e w id e ly d le c u e s e d in th e p a p e r .

L e t a random v a r i a b l e X o f a c o n tin u o u s typ e be d i s t r i b u t e d In the **ay d e te rm in e d by a d i s t r i b u t i o n f u n c t io n F ( x ) and w it h d i s t r i b u t i o n p a ra m e te rs ji • E ( X ) and cf2 ■ D2 (X ).T h e f a c t t h a t the v a r i a b l e X i s n o rm a lly d i s t r i b u t e d w it h p a ra m e te rs and d I s denoted as X - N ( |j , d2 ) . L e t the sequence X j , . . . , xn ( { x j , d enote a sam ple c o n s is t in g o f n in d ep e n d en t o b s e r v a t io n of v a r i a b l e X, and x and a r i t h m e t i c mean, S 2 - the sum o f sq uarad d e v ia t io n s , , s2 - v a r i a n c e , and s - s ta n d a rd d e v ia t i o n from the sam p le. Sam ple { y j d e n o te s n o n - d e c re e s in g o rd e re d o b s e rv a t io n s o f

3

ample { XA}» 60 th a t y^, < . . . £ y n * F u n c t io n pn^x ^

* L e c t u r e r a t th e D epartm ent o f M a th e m a tic a l and S t a t i s t i c a l M othods, Academy o f A g r i c u l t u r e , Poznart.

(2)

13 an e m p ir ic a l d i s t r i b u t i o n f u n c t io n , v.hare x I s an a r b i t r a r y r o a l number, I . e . tho o b s e r v a t io n f u n c t io n « x , w h llo $ ( u ) nnd $ ~ * ( p ) a re th e norm al d i n t r i b u t i o n f u n c t io n and q u a n t llo o f the p -tb o r d e r of d i s t r i b u t i o n M ( 0 , 1 ) , r e s p e c t i v e l y . F o r a d e te rm in e d x the v a lu e o f F n ( x ) i s a random v a r i a b l e 0 , 1 ( n , 2 ) n I 0 0 0 $ 1 •

The o r d o r s t a t i s t i c i s a random v a r i a b l e b eing tho k - th v a r i a b l e in tho sample { y ^ } . The aomple ( y ^ j i s c a l l e d tho o r d e r s t a t i s t i c . Whon X ~ N ( y , cf2 ) than y^ I s a norm al o r d e r s t a t i s t i c , { y A J- - a sequence o f norm al o r d e r s t a t i s t i c s , and ■|ui } a sequence o f fJ(0 , 1) - o r d e r o t a t l o t l c s , where u^ » ■ ( y 4 - f O / d ). Tha o rd o r s t a t i s t i c u^ has d i s t r i b u t i o n parame- t e r s j E ( u i ) - mi# D2 ( u i ) * v ^ , C o v (u 1# u^) ■ v ^ ; 1, J -* 1 , . . . . n, then f o r N ( ¿j f d ) - o rd o r s t a t i s t i c s we have E i y j ) ■ e ♦ d u t . 0 2 ( y i ) - and C o v (y 1 , y ) - ^ ^ i j * The v a lu e s o f n i "* wi n > 1 “ 1 * •••» [ n/ 2 ] a re t a b a la r lz a d f o r v a r i o u s n s , w h ile v ^ ■ v ^ n and v ^ ■ v^^ n f o r i , J ■ 1 ,..., [ n / 2 ] ; i i J o n ly f o r n » 1, . . . , 20 ( c f . ^ 2 0 ], t a b le s 9 and 1 0 ). 2. The H yp o th es i s o f Goodness o f F i t o f th e E m p ir ic a l D i s t r ib u t io n w it h th e Normal D i s t r i b u t i o n

The s t a t i s t i c a l h y p o th e s is o f the e m p ir ic a l and norm al d i s t r i b u t i o n s i s fo rm u la te d as f o llo w s . L e t F n d en o te a c la s s o f norm al d i s t r i b u t i o n f u n c t io n and G - a c la s s o f d i s t r i b u t io n f u n c t io n o f random v a r i a b l e s h a v in g the t h i r d moment ( P

3

) o t h e r than z e ro and f i n i t e fo u r t h c e n t r a l moment (¿*

4

)# a t fl n G « 0 . L a t u

3

put a h y p o th e s is H^ : F e F^ and H^ i F e G, t h a t th e f u n c t io n F ( x ) b e lo n g s to the c l a s s o f d i s t r i b u t i o n f u n c t io n s and G. Tho hyp

0

t h

060

£> HQ o r Hj w i l l be sim p le i f the d i s t r i b u t i o n s b e lo n g in g to c la s s e s F^ o r G hava th e known d i s t r i b u t i o n p a ra m e te r s .

The h y p o th e s is HQ a g a in 3 t H^ i s v e r i f i e d u s in g one o f the t e s t s f o r n o r m a lit y . G e n e r a l l y , th e y a re d iv id e d , a c c o r d in g to th a s t r u c t u r e o f t a s t s t a t i s t i c s In t o

(3)

1 ) P e a r s o n 's X t e a t s fo r goodness o f f i t ,

2 ) t e s t s based on the com parison o f e m p ir ic a l and nortnnj. d i s t r i b u t i o n f u n c t io n s ,

3) t e s t s u sin g sample momonte,

A ) t e a t s baaed on o r d e r e t d t i s t i c s .

P e a r s o n 's X to a t w i l l not be d is c u s s e d h e r o , s in c e i t i s w e ll- k n o w n . I t i s used m a in ly f o r la r g o s a m p le s, e s p e c i a l l y v.hen the o b s e r v a t io n s a r c in tho form o f grouped d a ta .

Next we s h a l l c o n s id e r the t e s t a w it h in each group and f i n a l l y a g e n e r a l d is c u s s io n w i l l be p r e s e n te d .

3. T a t t e Based on th e Com parloon o f Emp i r i c a l and Normal D i s t r i b u t i o n F u n c t io n s

These t e s t s ore based on tho d is t a n c e o f d i s t r i b u t i o n fu n c t io n s F ( x ) e F „ _N ond F ( x ) _n b e in g tho e s t im a t o r s F ( x ) and from tho sample under the h y p o th o e is [ l 3 ] * Under tho s im p le hypo- t h e s i s F ( x ) w i l l bo r e p la c e d by F ( x ) . L o t F n ( y 1) d eno te tho v a lu e o f d i s t r i b u t i o n f u n c t io n ^ ( x ) in p o in t y 1 and z^ * ® ( u t >, whoro ( y i - p ) / <*, ~ y ) / a , ( y t ‘ * 0 / 3 . i Y t - y ) / d, d - known ( a ) ¿j, d - unknown ( b ) - known, d - unknown ( c ) - unknown, d - knoKin ( d ) n

a t b2 . ^ ( y j - f j ) 2/ n . Tha case ( a ) r e f e r s to the c l a s s F^ o f 1-1

norm al d i s t r i b u t i o n s d e te rm in e d c o m p le te ly (a 3 im p le h y p o t h e s is ) , w h ile o t h e r c a s e s do n ot d e te rm in e the d i s t r i b u t i o n f u n c t io n F(x ) w h o lly (a complex h y p o t h e s is ) . The f o llo w in g t a s t s b elong to the above m entioned g ro u p :

- Kolm ogorov-Sm irnov

sup | F n ( x ) - F ( x ) | «

(4)

KS -1 <, -1 < n JL n “ 1 Kolmogorov D+ ■ max ( l / n - z . ) , 0” - max ( * . 1 < 1 < n 1 1 < 1 < n D - max ( 0 * , o " ) , - K u ip e r V m D* ♦ D*\ *

- Cram er-von M ises

n IV2 - l / ( l 2 n ) ♦ ^ ( z l - ( 2 1 - l ) / 2 n ) 2 1-1 n n - ^ 2 z 2 - ( l / n ) ^ ( Z i - l i Z j ♦ n/3 , 1-1 1-1

W0 -

Y ]

C*1 - (2 1-l)/2 n )2

1-1

n

Y '-

( z t - l/ ( n + l) ) 2 ,

W11

1-1

n W21 - ¿ T ( * i - ( 2 1 - l ) / ( 2 ( n « - l ) ) ) 2 ,

1-1

- Wateon n U2 • W2 - n (2 - 0 . 5 ) 2, * • J 2 * i / n ’ 1 4 - ( l - l ) / n ) /

(5)

A n d e rs o n - D a rlin g n

1-1

n A g j • -n - n / ( n + l ) 2 ^■ -n - [(2 1 - 1 ) In ♦ 1-1 ♦ (21+1) In ( 1-zn_ i « . i ) ] - [ (2 n + l) In z n - In ( l - z n ) ] » n A12 " “ l/ ( n + l) 2 _ j 1 [ ln z x * “ zn - i+ l^ 1-1

I n th e above to o t s t a t i s t i c s z^ assumes the v a lu e s a c c o rd in g to th e c a s e ( a ) - ( d ) . The b e s t known t e s t s f o r th e s im p le hypo t h e s i s Hq a re D, wZ ond A2 . The above m entioned t e s t e o re g i ven in t h e i r summation form though o r i g i n a l l y th e y were p re s o n t- ®d in an i n t e g r a l fo rm , w hich we s h a l l m ention l a t e r . The t e s t s b e lo n g in g to t h i s group hove known d i s t r i b u t i o n s in case (a ). F o r 3ome t e s t s (D , V , u , V/2 , A2 ) m o d if ic a t io n s a re g iv e n f o r c a s e s ( b ) - ( d ) [ 2 5 ] , whose c r i t i c a l v o lu e s do n ot depend on th e somple s iz e but o n ly on tho s i g n i f i c a n c e l e v e l a . F o r the KS t e s t in cose a the c r i t i c a l v a lu e was g iv e n , among o t h e r s , by F 1 s z

Uo]

( t a b l e V I I I ) and f o r ( b ) by

L 1 1 1 1

e f o r s

[ 16 ].

The t e s t s o r i g i n a t i n g from Crom er-von M iso s and Andorson- “ D a r lln g t e s t s w ere g e n e ra to d from g e n e r a l I n t e g r a l fo rm s , re s p e c t i v e l y 1 W - n 0 and 1 0 whore S 1 i s a c e r t a i n f u n c t io n o f 1 and n.

(6)

The s t a t i s t i c s IVj j , W21* A?1 8nd Al 2 v',e r*1 o b ta in e d r e p la c in g i/ n by i / ( n + l ) (t h e f 1r s t ln d a x ) o r ( i ♦ 0 . 5 ) / ( n + 1 ) ( t h e second in d e x ). G r o e n and H o g a z y [1 3 ] i n t r o duced t ^ s t s f o r n o r m a lit y o f the above s t a t i s t i c s , and p roved t h e i r predom inance as f a r as t h e i r power was c o n c e rn e d , o v e r the

2 2

t e s t s W and A g iv e n in a summation form .

4. T e s t s U sin g Sam ple Momenta

F o r th e sample o f n o b s e r v a t io n s { x ^ } we d e te rm in e a c e n t r a l moment o f th e k - th o rd e r n mfc • ( l / n ^ x , - x ) , k ■ 2, 3, . . . 1-1 S t a t i s t i c s ■ m3/yin | and b2 ■ m^/m2 a re u n b ia s e d e s t i -—

3

4 m ato rs o f p a ra m e te rs yp ^ ■ t*3/ anc* ” V4/ d

~ N C|i, or~*)» th en • 0 and ^ * 3d^, hence |/j3^ ■ 0 and ¡1^, - 3. T h is means th a t yf3^ and a re e q u a l to 0 and 3,

r e s p e c t i v e l y , f d r n o r m a lly d i s t r i b u t e d random v a r i a b l e s . T h e re fo r e d i s t r i b u t i o n s o f v a r i a b l e s f o r w h ich p a ra m e te rs (*

2

^ have v a lu e s c lo s e to ( 0 , 3 ) e re t r e a t e d as “ a lm o st n o rm a l” . F o r th e c o n s t r u c t io n o f t e s t s f o r n o r m a lit y b ein g d is c u s s e d S l u t s k y 's theorem i s used [ l O ] . I t f o llo w s from t h i s theorem th a t yiTj and b2 « r e converging- to end f$2 when n^-*-oo. The pa ra m e te rs o f d i s t r i b u t i o n o f th e s e v a r i a b l e s under HQ a re [

2

] * M ? > » 0 , E ( b 2 ) -3 (n-1 )/ ( n + l) , 0 , when n-yao, 0 2 ( y ^ ) -6 ( n - 2 )/ (n + l) ( n+3)f 6 / n , when n-+oo,

(7)

D * (b 2 ) '2 4 n ( n - 2 ) ( n - 3 ) / ( n + l) 2 (n + 3 )(n + 5 ), 2 4 /n , when n-»a>

To the d is c u s s e d group the f o llo w in g t a s t e belon g » - s ta n d a rd iz e d t h i r d sam ple momont

y ^ ; ■ rn3/ n ) ^ 2 ,

used a g a in s t th e h y p o th e s is s t a t i n g t h a t the G - c lo s s d i s t r i b u t io n s ore skew (

7

^ i 0 ) ; th e c r i t i c a l v a lu e f o r n > 25 »¿os g iv e n by P e a r s o n and H a r t l e y [1 9 ] ( t a b l e 3 4 B ) and f o r n < 25 by M u l d H o l l a n d [ l 8 ] j

- s ta n d a rd iz e d f o u r t h c e n t r a l sample moment

b2 ■ m4 / m2 ’

used a g a in s t H^ s t a t i n g t h a t s i g n i f i c a n t p o in t s c la s s d i s t r i b u t io n s e re sym m etric,* a c r i t i c a l v a lu e f o r n > 50 was g iv e n by p e a r s o n and H a r t l e y [ 19 3 J - D' A g o s t i n o-P e a r s o n [ 5 ] . F o r a g iv e n v a lu e o f the above d e te rm in e d s t a t i s t i c b^ wo e s t a b l i s h p r o b a b i l i t y P ( b2 < b2^p * " p * where b2( p , n ) i s a c r i t i c a l v a lu e o f b^ d i s t r i b u t i o n f o r g iv e n p and n (p ■ 0 .0 0 1 , 0 .0 0 2 5 , 0 .0 5 , 0.01, 0 .0 2 5 , 0 .0 5 , 0 .1 0 , 0 .2 5 , 0 .5 0 , 0 .7 5 , 0 .9 0 , 0 .9 5 , 0 .9 7 5 , 0 .9 9 , 0 .9 9 5 , 0 ,9 9 7 5 , 0 .9 9 s n • 20, 21, . . . , 200; b2 - 1 .5 4 ( 0 . 0 8 ) 7 .2 2 ). Then we d e te rm in e q u e r t l l e x (b 2 ) • $ 1 p o f the p -th o r d e r d i s t r i b u t i o n N (0 ,1 ) u s in g adoquate t a b l e s [ 2 9 ] ( t a b le 3 ) . F o r s t a t i s t i c we have th e q u a n t ile x ( y t ^ ) - S in { y ^ / A + [ ( y ^ / * ) 2 + 1 ] 1 / 2 } ’ where c o n s ta n ts 5 and 1 / X a re t a b e la r lz e d a t n • 8 (1 )5 0 , 52 (2 )1 0 0 , 1 0 5 (5 )2 5 0 , 2 6 0 (1 0 )5 0 0 , 5 2 0 (2 0 )1 0 0 0 . Wo d e te rm in e K2 t e s t based on th e s t a t i s t i c K2 - X 2

( y ^ )

♦

x2(b 2),

(8)

2

K I s the p o s s i b i l i t y o f t e s t i n g d e p a r tu r e s from n o r m a lit y caus- od by skewness and k u r t o s i3 . 3uch o t e s t l a c a l l e d th e omnibus t e a t . A t n > 200 in s t e a d o f K2 the s t a t i s t i c

K 2 - ( n / 2 4 ) [ i ( y i £ ) 2 ♦ ( b 2 - 3 ) 2 ]

i s u sed . The v a r i a b l e KZ hue an a s y m p to tic X?, d i s t r i b u t i o n .

- Bowmnn-Shenton [ l ] . Tho v a r i a b l e v l j j I s s y m m e t r ic a lly d i s t r i b u t e d , w h ile b2 i s a s y m m e t r ic a lly d i s t r i b u t e d . Tha c u rv o s used f o r each of th e se v a r i a b l e s a re ao f o l l o w s ] f o r . y ^ P e a r son s c u rv e s o f typ e V I I o r t - S t u d e n t s , f o r b2 P e a r s o n 's c u rv o s o f typ e V I o r I V . In both c a s e s Jo h n e o n *e t r a n s f o r m a t io n [1 5 ] to Sy c u rv e s g iv e s a s a t i s f a c t o r y norm al c u rv e f o r [ 3 ] ; th e s y stem o f Sy c u rv e s i t , h o w e var, l e s s s a t i s f a c t o r y f o r b2 , Su c u r v e s p r o v id e good c o n s is t e n c y f o r a t n > 8 and f o r b2 a t n >

> 25. F o r s m a ll sam ples Jo h n s o n 's 5B system I s a s u f f i c i e n t ap p r o x im a tio n f o r b2 ( e . g . fo r n • 20 I t I s P j i b g ) - 3 .0 1 9 . and P 2 <b2) « 0 . 5 4 ) . Henca wo have t e s t s t a t i s t i c s xs (V b ^ ) « S j o ln h " 1 (V b ^ / X j ) ,

xs(b2)

X2

• S2 s ln h " ( b 2 - § / X 2 ), n > 25 . •- ' • A- ' ] ■ • e i ^2 ” ^ t 2 ♦ 2 * " t . ^ - bg • » < » where c o n s ta n ts 5.^, y 2# ¿>2 , X^ , X 2 a re d e te rm in e d u s in g the method p r e s e n t e d , among o t h e r s , by P e a r s o n and H a r t l e y [ 2 0 ] . vVe d e te rm in e the s t a t i s t i c

V2

- X2

(Vb^> 4 x |(b 2 ),

2

w hich under HQ has a p p ro x im a te ly % 2 d i s t r i b u t i o n . C r i t i c a l v a lu e s g e n e ra te d by th e H o n te - C e rlo method f o r n ■ 2 0 , 25, 50, 100,

150, 200, 300, 500, 1000 were g iv e n by B o w m a n end

S h o n t o n [ l ] . The y| t e s t l a a ls o an omnibus t e s t .

- P s a r s o n -O 'A g o e t 1 n o-B o w m a n [21 ] . L e t *

(9)

(<*') be th e lov»er end upper lOOotTJ-th p e r c e n t i l e o f d is -t r i b u -t i o n , and l e -t ^ ( o ' ) end ^ ( d 1) be lo w er end upper c r i t l c o l p o i n t * o f b2 d i e t r i b u t i o n . Fo u r p o ln t e w it h c o o rd in a t e * { - V 6 J ( a ' \ 2 b2 ( a ' ) } , { y ^ ( a * ) . 2b2 ( o ' ) } I { - y B i < « ') , ^ ( o ' ) } , v | Vb^ ( a 1 )»jb 2 ( a ' ) } form e r e c t e n g le . When v a r i e b l e e and e r e In d ep en d en t th en t h e i r v a lu e a d e te rm in e d from th e sample e re o u t e ld e th e r e c t a n g le w it h p r o b a b i l i t y a ' - 0 .5 ■[ 1 - ( l - a ) 1^2 } . The R t e e t d e te rm ln e e the f r e c t l o n o f p o in t s ( v £ ^ , b2 ) w hich ehould be I n s id e th e r e c t e n g le ,

5. T e s t s Seeed on O rd er S t a t l e t l c e

Under Hq o r d e r s t a t i s t i c s { V i } have e x p e cte d v e lu e e and va- r la n c e s - c o v a r la n c e s denoted by known l l n e e r f u n c t io n s of pora- m ete rs ¡j and d . T h is a llo w s us t o a p p ly tho l e a s t s q u a re s me thod to th e e s t im a t io n o f th e s e p a re m e te rs [ 1 7 ] . The b e st u n b ia sed l i n e a r e s t im a to r & o f th e p a ra m e te r d can be g e n e r a l ly w r it t e n in th e fo rm :

h

9 • 1 L dn - i * l t l ’ 1-1

where h - [ n / 2 ] , t t - y n_ 1+J - w h ile { d „ . 1+i n} a re con“ s t a n t s s a t i s f y i n g c e r t a i n c o n d lt lo n e ( e . g . t h e i r sum i s e q u a l to z e ro f o r each n ) . On th e o t h e r hand th e u n b ia sed sam ple e s t im a t o r o f th e p a ra m e te r d 2 i s e x p re s s e d by th e v a r ia n c e from the sam ple s 2 . The r a t i o o f 3 2/ s 2 w ith o u t a c o n s ta n t lo c lo s e to 1. T e s ts b u i l t on th e above r a t i o have l* f t- h a n d - e id o c r i t i c a l r e g io n s , and th e v a lu e e o f s t a t i s t i c s a re < 1. Th« f o llo w in g t e s t s e re in c lu d e d in t o th e d ls c u e s e d g ro u p : - C e • r y [ l i ]

g - Z IVi " Y l/(ns2)1/2,

1-1

(10)

128 tfle sia v Matjnar _____ _____ __________ - D u v i d-H o r t 1 a y - P e a r a o n [ & ]

u - (yn “ Vt)/®.

c r i t i c a l v a lu e s a re In th e above n ontlorted t o b ie s [1 9 ] ( t a b l e 29C); - S p l e g e l h o l t e r [ 2 6 ] T . [1/<•„»>■ . I/ O * ] 17" .

« h o re c n - ( l / 2 n ) ( n l ) 1,/™ and ib - n - 1 , and u and g a re g tv a n a b o v e ;' - S h

a

p 1 r o-W I l k [2 3 ] h

» • [ Z V l . t 'i]!!/s2'

l - l where Z - 1 . H " ° * Z a l . n • 11 1-1 i-i - S h a p 1 r a-F r a n c l a [ 2 2 ]

»' •[ Z

W

.

s2)-

1-1 where ’* ¿ ¿ 4 * • n - l4 l/ l / ^ ‘ *0 " Z #l . n * jl-x - W e l e b e r g-B I n g h a m [2 8 ] 8 - I E “ n - l . l . n ' l ] 2' « * 8 2 ’' 1-1 where / * i . n m* w l ( ' ' n i l f f l ) ' * " X ‘ 2t **•* " r

(11)

a n - A * i . n * i ■! - o 'A g o o t l n o [ 4 ] ° A ■ [ £ ‘ Vi - ^ y ] / ( " V ) U 2 1-1 or Y - V r r ( D A - 0 .2 8 2 0 9 5 ) / 0 . 029986 - F l l l l b r e n [ 9 ] h r • T . V t . t V ( s n s 2 ,1 / 2 '

1-1

where n « ! . „ ■ * ' 1 C 1 . n ) ' * « ■

1-1

V n - ( 0 - 5 ) 1 / n , « 1>n 1 " n .n -n •

( 1

- 0 .3 1 7 5 )/ (n + 0 .3 6 5 ) , 1 ■ 2.

3

. . . . . « - I* l , n C r i t i c a l v a lu e s f o r th e above m entioned t a s t e a r e g iv e n by t h e i r a u t h o r s , end f o r th e W t e s t a ls o by D o a a r t s k l [ 7 ] ( t a b l e 1 1 ). 6. G e n e ra l O ls c u s s lo n o f T e a ts f o r N o r m a lit y We re v ie w e d v a r i o u s t e s t s f o r n o r m a lit y . They a re d iv id e d In t o t h r e e b a s ic g ro u p s a c c o r d in g to t h e i r s t r u c t u r e . Many o f th e se t e s t e hove th e p r o p e r t ie s o f th e omnibus t e o t . The l a t t e r

(12)

can be a d v is a b le when some e p r i o r i In f o r m a tio n on th e d e p a rtu r e from n o r m a lit y l a g iv o n . From t h l e group we d is c u s s e d th e t e 6 t s K2 , Y2 , T , W, W ', IY, D . and r . A l l o f them have been g iv e n In the l a a t 15 y e a r s . So f a r the omnibus t e s t hoe not been con s t r u c t e d in the group o f t e s t s based on th e com porieon o f n o r mal and e m p i r ic a l d i s t r i b u t i o n f u n c t io n s . I n t h i s group o f t o s t a th e Cram er-von M isee and A n d e rs o n - D « rlln g t e s t s w it h mo d i f i c a t i o n s sh o u ld be m a in ly u sed .

Many n o r m a lit y t e s t s r e v e a l s i m i l a r p r o p e r t ie s . S t a t i s t i c « on w hich th e t e s t a V b ^ , b2# u , g . W, r a re b a sed , a re in v a r ia n c e due to the s h i f t o f s c a le and lo c a t i o n . T h e r e f o r e , th e y a re s u i t a b l e f o r t e s t in g com plex Hq h y p o t h e s is . The t e s t s 0 , KS, U2 , V , A2 have c o m p le te ly d e te rm in e d d i s t r i b u t i o n s under HQ and a re s u i t a b l e f o r t e s t in g s im p le h yp o th e se s o f n o r m a lit y .

F o r the m a jo r it y o f t e s t s th e d e n s it y f u n c t io n o f t h e i r d i s t r i b u t i o n s t a t i s t i c s hove n o t been found y e t . The c r i t i c a l v a lu e s r e q u ir e d f o r them have been g e n e ra te d by mean« o f th e Mon- t a - C a r lo method. The t e s t s IV, W, and r have le f t - h a n d - s i ded c r i t i c a l r e g io n s . Some t e s t s s t a t i s t i c s need some c o n s ta n ts f o r each n ( e . g . t e s t s W, W*, ft, r ) .

The problem o f power o f th e n o r m a lit y t e s t s la r e l a t i v e l y w e ll known and has been d is c u s s e d among o t h e r s by S h a p i r o e t a l . [ 2 4 ] , S t e p h e n s [ 2 5 ] , G 1 o r g 1 and C 1 n c i [ l 2 ] . G r e e n and H e g a * y [1 3 ] and P e a r s o n e t . a l [2 1 ], H ow ever, th e r e a r e few g e n e r a l r e s u l t s w hich a re co m p lete and a p p l i c a b l e , as I t I s th e ca- so in th e th e o r y o f p a r a m e t r ic t e s t s . M a th e m a tic a l d i f f i c u l t i e s co n n ecte d w it h d e te r m in in g th e power o f t e s t a a re u s u e l- l y v e r y b ig . I t i s a ls o d i f f i c u l t to d e te rm in e p r o c t l c e l l y the G c la s s d i s t r i b u t i o n « .

I t i s known t h a t f o r th e norm al d i s t r i b u t i o n ■ O and p2* * 3. S i m i l a r l y , o th e r d i s t r i b u t i o n « can be c h a r a c t e r iz e d by g iv in g e p a i r o f v a lu e s ( v f ^ , ßj>). H ence, from the G c la s s th e d i s t r i b u t i o n s from th e b elow group s e r e chosen a c c o r d in g to the v a lu e s o f »nd (i^ . B e lo w we s h a l l p r e s e n t p a r t i c u la r group« g iv in g f o r each o f them th e typ e o f d i s t r i b u t i o n end some d i s t r i b u t i o n s b e lo n g in g to th e s s g ro u p s :

(13)

9 2

w it h lo n g t a i l s - X , lo g - n o rm a l, n o n - c e n tra l X , exponent t i a l , W o ib u l l* s , P a r e t o 's :

Group 2 : I v P 'j | > 0 . 3 , (*2 < 3 .0 : asym m etric d i e t r i b u t l o n a w it h s h o rt t a i l s - b e t a , S

0

Joh nson a;

Group 3» | V ^ T | < 0 . 3 , ft2 > 4 .5 ; sym m etric d i s t r i b u t i o n s w it h long t a i l s - d o u b le X 2 , u n ifo rm , C auchy s , L o p la c o s , T u k e y 's , S y Jo h n s o n 's , l o g i s t i c :

Group 4 : | V ^ | < 0 . 3 , (32 < 2 .5 : sym m etric d i s t r i b u t i o n s w it h o h o rt t a i l s - b e t a , d o u b le X 2 , Sg Joh n so n a , Tukey a :

Group 5

1

| V ^ l < ° . 3 , 2 .5 < (3

2

^ 4 . 5 ; alm o st norm al d i s t r i b u t i o n s - t - S t u d e n t 'a w ith 10 d e g re e s o f freedom , S

0

Jo h n s o n 's w it h p a ra m e te rs y •

0

and S • 3, l o g i s t i c , W o ib u ll s a t k •

2

.

We in c lu d e the d i s t r i b u t i o n s In t o p a r t i c u l a r groups u s in g the known v a lu e s o f snd |32. They d i f f e r s i g n i f i c a n t l y w it h in th e same d i s t r i b u t i o n d e te rm in e d a t v a r i o u s v a lu e s o f p a ra m e te rs d e te rm in in g i t . F o r in s t a n c e , f o r X

4

we have - 1.41 and (3

2

. 6 .0 0 , w h ile f o r X l Q - V ^ • 0 .8 9 and |3

2

- 4 .2 0 . Due to

t h i s the same d i s t r i b u t i o n a t the v a lu e s o f p a ra m e te rs d e te r m i ned in d i f f o r e n t w a ys, i s in c lu d e d in t o v a r i o u s g ro u p s.

T a b le 1 p r e s e n t s power o f some t e s t s f o r n o r m a lit y , e x p re sse d in p e r c e n t , f o r a - 0 ,0 5 and n -

20

, ta k in g in t o acco u n t a l t e r n a t i v e d i s t r i b u t i o n s : \ 2a , « - 1 . 2 , 4 , 10 d e g re e s o f freedom , lo g - n o rm a l L N ( ( j , r f ) w it h p a ra m e te rs y - 0 and d = 1, t - S tu d * n t s t

2

w it h two d e g re e s o f freedom , C a u c h y 's t j , b e ta B ( p , q ) w it h p a ra m e te rs p - 2 and q « 1 , u n ifo rm B ( 1 , 1 ) , and L a p la c e s.

The p ro c e d u re f o r t e s t i n g n o r m a lit y based on th e V/ t e s t has a g r e a t e r power f o r a lm o st a l l G c l a s s d i s t r i b u t i o n s than o th e r t e s t s . E s p e c i a l l y , the W t e s t i s s e n s i t i v e to asym m etry w it h a long t a i l . F o r in s t a n c e , f o r th e p o p u la t io n w it h the d i s t r i b u t io n x 'in . 8nd LN tho

v a lu ®9

° * P ower a re 29 *

S0

0nd 93&. r e s p e c t i v e l y . In the group o f t e s t s f o r n o r m a lit y based on o rd e r s t a t i s t i c s th e W t e s t sh o u ld be assumed the b e st one as f a r as power i s c o n c e rn e d , when n ^ 50. S i m i l a r power p r o p e r t i e s have the W ' and r t e s t s w hich can be p roposed f o r la r g e sam p le d o f s iz e s 50 < n ^ 100.

The t e s t s based on sam ple moments r e v e a l b ig power a t a d e f i ned typ e o f G c l a s s d i s t r i b u t i o n s : V b ^ f o r asym m etric d i s t r i b u tio n s w it h lo ng t a i l s ( X f . X * '- ^ (

0

, l ) ) a n d t e s t b

2

(14)

.' * . ■ ■ ■ _{E m p i r i c a l pow er o f} f o r te s t s a • f o r 0 .0 5 n o r e a lit y in p e r c e n t, end n *= 20 T a b 1 e 1 G c l a s s d i s t r i b u - t t io n s V f3 ^ i*2

Xz

0

w2 V U2 A2 9 _bl

b2

K2 R W w ' r Y ! 2 .8 3 1 5 ,0 94

86

₉₄ ₉₄ ₉₃ _- _- ₈₉ ₅₃ ₈₂ ₈₂ ₉₈ ₉₄ ₉₄ ₈₀ --- 1 .

2.00

_{9 .0} ₃₃ ₅₉ ₇₄ ₇₁ ₇₀ ₈₂ y 74 34 60 61 84 82 82 52 1 .4 1

6.0

₁₃ ₃₃ ₄₅ ₂₃

21

₁₅ ₁₉ ₄₉ ₂₇ ₄₀ ₃₉ ₅₀ ₂₄ _- ₂₄ --• 0 .8 9 4 .2 7 18 23 24 14 - 14 29 19 27 25 29 - - 16 -«-N (0 .1 ) 6 .1 9 1 1 3 .9 95 78

88

₈₄ ₈₅ ₉₁ ₄₉ ₈₉ ₅₈ ₈₂ ₈₁ ₉₃ ₉₄ ₉₄ ₇₇ _{_}

8

_{(1 ,1 )}

0

1.8

11

12

₁₆ ₁₇ ₁₈ ₁₇ _-

0

₂₉ ₁₆ ₁₇ ₂₃ ₄ ₄

8

14

8

_{(2 ,1 )} _{- 0 .5 7} _{2 .4}

8

_- _- ₂₃ ₁₆

12

₁₉

8

₁₃

12

11

₃₅ - mm

₆

-L a p la c e 's

0

6.0

₁₇

22

₂₆

22

₂₅ ₂₆ _- ₂₅ ₂₇ ₂₅ ₂₆ ₃₁ ₃₃ ₃₃ ₂₈ ₃₄

*1

0

- 41

86

88

87

88

98 - 89 81 79 80 89 91 92 92 94

*2

0

- - 55 - mm - - - 52 53 54 52 54 59 60 56

66

* D ir e c tio n a l t e s t • - N ot te e te d . % •

;

(15)

fo r sym m etric d i s t r i b u t i o n s w it h lo n g t a i l s ( t ^ , B ( l , tg )).A n I n t e r e s t in g power v a lu e ra n g in g from th e power o f t e s t s v rb j to b2 i s in the ca s e o f K2 and R t e s t s . H ence, a c o n c lu s io n can be drawn t h a t from th e above m entioned t e s t e th e t o s t s bA and b^ sho uld be used a t d ste rm in o d d i s t r i b u t i o n s and t e s t s K2 .and R In th e ca se when i t I s im p o s s ib le to d e te rm in e d i s t r i b u t i o n s from the G c l a s s .

The t e s t s f o r n o r m a lit y based on the m easure o f c o n s is t e n c y of e m p i r ic a l and norm al d i s t r i b u t i o n s r e v e a l s i m i l a r powers a l though th e b e s t o f them i s the A " t e s t and the w o rs t the D t e s t . T e s te V/2 , V and U2 have s i m i l a r pow ers as the A2 t o s t .

In the l i g h t o f th e above m ontlonod to o t s f o r n o r m a lit y the K2 t e s t i s much w eaker a t G c l a s s d i s t r i b u t i o n » g iv e n in the T a b le 1. T h e r e f o r e , i t sh o u ld n ot be usod in p r o c t i c e as a t e s t fo r n o r m a lit y . In s t e o d o f the t o s t in th e ca se of la r g e sam ples n > 50 w ith o u t c o n s t r u c t in g the d i s j o i n t s e r i e s , 0 A- 9 o s tln o * s Y t e s t o r W' t e s t sh o u ld be used.

As was shown by O y e r [ b ] th e t e s t s f o r n o r m a lit y w ith unknown p a ra m e te rs ^ and <S*~ have g r e a t e r powor than the t e s t s w it h unknown o'2 o n ly . B e s id e s , the power in c r e a s e s w it h the in c r e a s e o f n a t b oth unknown p a ra m e te rs £i and d .

I n s i g n i f i c a n t d i f f e r e n c e s in the c r i t i c a l v a lu e s o f t e s t s f o r n o r m a lit y based on th e com p arison o f norm al and e m p ir ic a l d

i-2 , , 2 a.2 t i

s t r i b u t l o n f u n c t io n s o c c u r when cf i s e s tim a te d by s o r a ♦ The 2 c r i t i c a l v a lu e s a re much lo w o r when both p a ra m e te rs ¡J and d ° r e to be e s t im a te d . The power o f th e n o r m a lit y t e a t in c r e a s e s s i g n i f i c a n t l y , os shown by P e a r s o n e t a l . [ 2 1 J , whon in s te a d o f th e omnibus t e s t a d i r e c t i o n a l t e s t i s u se d . The d i- • "o ctlo n a l t e s t s f o r n o r m a lit y a re used f o r d e te rm in e d G c l a s s d i- a t r ib u t lo n s . F o r in s t a n c e , D 'A g o s tln o Y t e s t can be t r e a t e d a s an omnibus t e s t f o r v a r i o u s G c l a s s d i s t r i b u t i o n s , how ever f o r some of them, when ft2 < 3 a le ft - h a n d - s id e d Y t e s t can bo u sed , w h ile fo r o t h e r s , when > 3 a r ig h t- h a n d - s id e d Y t e s t i s em ployed.

I t sh o u ld be n oted t h a t th e power o f t e s t in c r e a s e s w it h the in c r e a s e o f th e sam ple e iz e n . F o r In s t a n c e , f o r the W t e s t a t « • 0 .0 5 and d i s t r i b u t i o n s \ \ Q. l n ( O . I ) th e power l e as

(16)

n X₄

2

X 2_*10 L N ( O . l ) 10 24 11 60 20 50 29 93 30 71 35 99 40 87 48 100 50 95 56 lOO

Th at I s why when t e s t i n g the n o r m a lit y o f g iv e n v a r i a b l e a t l e a e t one sam ple o f s iz e n > 30 sh o u ld be u sed . On the o th e r h and , th e power o f t e s t d e c re e s e e w it h the d e c re a s e o f s i g n i f i c a n c e l e v e l a . F o r in s t a n c e , f o r t e s t s Y end W a t the d i s t r i b u t i o n L N (0 ,1 ) we have th e f o llo w in g p o w e r» ! X oi 0 .1 0 0 .0 5 0 .0 2 0 .01 n \ Y W Y W Y W Y W 10 51 68 42 58 34 45 28 38 20 80 95 75 92 66 86 61 81 30 93 99 90 99 86 97 82 96 40 97 100 94 lOO 92 99 90 99 50 99 100 98 100 97 100 96 100

H ence, o f g r e a t im p o rta n c e i s an a d eq u ate c h o ic e o f s i g n i f ic a n c e l e v e l cx to v e r i f y the h y p o th e s is HQ.

I t can be su g g ested t h a t in the above groups th e b e et t e s t e f o r n o r m a lit y from the p o in t o f v ie w o f t h e i r power a t ad eq uate eample s iz e s a re the f o llo w in g t e s t s i

Group 1 - W t e e t , f o r n < 50, vv' t e s t f o r 50 < n ^ 100, an a r b i t r a r y W2 , V , V2 t e s t , f o r n > lO O j Group 2 - W t e s t f o r n < 50, t e s t s K2 , R , f o r n > 50; Group 3 - r t e s t , f o r n < 50, one o f the W' o r r t e s t s , f o r 50 < n « 1 0 0 , Y t e s t , f o r n > 100;

(17)

Group 4 - bg t e s t , f o r n < 20, K2 t o s t , f o r 20 < n < 200: Group 5 - W t e s t , f o r n < 20, b0 t o s t , f o r 20 < n < 50, one o f th e W# o r r t e s t s , f o r 50 < n < 100, one o f th e K2 o r Y t e s t s , f o r n > 100. G e n e r a ll y , the p ro c e d u re o f v e r i f i c a t i o n o f Hq w it h norm al d i s t r i b u t i o n o f random v a r i a b l e X on the b e s ts o f a s im p le sam- P le tak e n from a p o p u la t io n a c c o rd in g to a c o rre s p o n d in g “ cheme o f s a m p lin g , sh o u ld be as f o llo w s . VVa d e te rm in e the v a lu es o f V b ^ and b2 from sam ple They a re the e s t im a te s

and p o. Then we choose one o f the above m entioned groups and a r e s p e c t iv e t o s t f o r n o r m a lit y a c c o rd in g to the sam ple s iz e n* I f th e ty p e of d e p e r tu r e from n o r m a lit y ( e . g . sk e w n e ss) l a known, we choose th e t e a t o f th e g r e a t e s t power w hich would °o rre s p o n d to the d e te rm in e d h y p o th e s is H j ( e . g . th e >/bj t e s t fo r asym m etric skew d i s t r i b u t i o n s ) and i f th e re i s no such an 0 P r i o r i In fo r m a tio n one o f the omnibus t e s t s I s u sed .

BIBLIO G RAPH Y

f l ] B o w m a n K. 0 . , S h e n t o n L . R . (1 9 7 5 ): Omnibus C o n to u rs f o r D e p a rtu re s from N o r m a lit y Based on V b ^ and b2 , “ B io m e t r lk a " 62, p . 243-250.

(2 ] C r a m e r H. (1 9 4 6 ), M a th e m a tic a l Methods o f S t a t i s t i c ® , P r in c e t o n U n i v e r s i t y P r e s s (w yd. p o l. Metody raatem atyczna w s t a t y s t y c e , PWN, Warszawa 1 9 5 8 ). t 3 ] D 'A g o S t 1 n o R . B . (1 9 7 0 ), T r a n s fo r m a tio n s to Norma l i t y o f th e N u ll D i s t r i b u t i o n g1# “ B io m e t r ik a " 57, p . 679-681. U ] D ' A g m t l n o R . B . (1 9 7 1 ), An Oonlbue T e s t N o r m a lit y f o r M o d erate and L a rg e S iz e S a n p le e , " B lo m e t r ik a 58, p . 341--348. y -t 5 ] O 'A g o s -t 1 n o R . B . , P e a r s o n E . S (1 9 7 3 ), T e s ta f o r D e p a rtu re f r o « N o r m a lit y , E m p ir ic a l R e s u lt s f o r th e D i s t r i b u t i o n s o f b2 end V b ^ , " B i o « e t r i k a " 60, p . 613- -622.

(18)

[ 6 ] D a v i d H. A . , H a r t l e y H. 0. , P e a r a o n E . S . (1 9 5 4 ), The D i s t r i b u t i o n o f the R a t i o , in a S in g le Sam ple, of Range to S ta n d a rd D e v ia t io n , “ B io m e t r lk a " 41, p . 482-493. [ 7 ] D o m a n o k l C. (1 9 7 9 ), S t a t y s t y c z n e t e s t y nieparam e- t r y c z n e , PWE Warszawa. [ 8 ] D y e r A . R. (1 9 7 4 ), C om parisons o f T «9 t f o r N o r m a lity w ith a C a u t io n a r y N o te , " B io m e t r lk a “ 61, p . 185-189. [ 9 ] F 1 1 1 i b r o n 3. L . (1 9 7 5 ), The P r o b a b i l i t y P l o t C o r r e la t i o n C o e f f i c i e n t T e s ts f o r N o r m a lit y , "T n c h n o m e trte e “ 17, p . 111-117. [1 0 ] F 1 s z M. (1 9 6 7 ), Rachunek prawdopodobiertstwo 1 s t a t y s - tyk a m ate ia a tycz n a , PWN, Warszawa.

[1 1 ] G e a r y R. C. (1 9 4 7 ), T o s tln g f o r N o r m a lit y , “ Blome- t r l k a “ 34, p . 209-242. [1 2 ] G 1 o r g 1 G. M ., C i n c 1 S . (1 9 7 5 ), S u l l e fu n z lo - n i d i a c r lm ln a t o r ie d i a lc u n l t e s t e s u n i v a r i a t e d l n o r m a li t é , S tu d , d i Eco n . 6, p . 100-120. [1 3 ] G r e e n 0. R . , H e g a z y Y. A. S . (1 9 7 6 ), Pow er f u l M o d lfied -C D F Goodness o f F i t T e s t s , 3ASA 71, p. 204-209. [1 4 ] H e g a z y Y. A. S . , G r e e n 3. R . (1 9 7 5 ), Some New Goodness o f F i t T e s ts u s in g O rd er S t a t i s t i c s , A p p l. S t a t i s t . , 24, p. 299-308. [1 5 ] G o h n e o n 3. , K o t z S . ( l 9 7 0 ) , C o n tin u o u s U n i v a r i a t e D i s t r i b u t i o n , V o l. 2 , B o s to n , H o u g h t o n - M ifflin . [1 6 ] L i 1 l i e f o r s H. W. (1 9 6 7 ), On th e Kolm ogorov-Sm l-

rnov T e s t f o r N o r m a lit y w ith Mean and V a r ia n c e Unknown, 3ASA 62, p . 399-402.

[1 7 ] L l o y d E , H. (1 9 5 2 ), L e a s t S q u a re s E s t im a t io n o f Lo c a t io n and S c a le P a ra m e te rs u s in g O rd er S t a t i s t i c s , “ Biome- t r i k a " 39, p . 88-95.

[1 8 ] M u l d h o l l a n d H. P . (1 9 7 7 ), On the D i s t r i b u t i o n o f v lT j f o r Sam ples o f S iz e a t Most 25, w ith T a b le s , “ B io - m e t r ik a “ 64, p . 401-409.

[1 9 ] P e a r a o n E . S , , H • r * 1 • y H. 0. (190 6), B io - ro e trik a T a b le s f o r S t a t i s t i c i a n s , V o l. 1, Cam bridge U n i v e r s i t y P r« 3 6 .

(19)

[2 1 ] P e a r s o n E. S . , D 'a g o 9 t 1 n o R. B . , B o w m a n K. 0 . (1 9 7 7 ), T e s te f o r D e p a rtu re from N o rm a lity : Com parison o f P o w e rs, “ B lo m e t r lk a “ 64, p . 231-246,

[2 2 ] S h a p i r o S. S . , F r a n c i s R. S . (1 9 7 2 ), Ap p ro x im a te A n a ly s is of V a r ia n c e T e s t f o r N o r m a lit y , 3ASA 67, p . 215-216.

[2 3 ] - S h a p i r o S . S . , IV 1 1 k M. B . (1 9 6 5 ), An Ana l y s i s o f V a r ia n c s Te9 t f o r N o r m o llty (C om p lete S a m p le » ;, “ B lo m e t r lk a “ 52, p . 591-611,

[2 4 ] S h a p i r o S, S , . VV 1 1 k M. B . , C h e n H. 3. (1 9 6 8 ), A C o m p a ra tive S tu d y o f V a r ia n c s T e s t f o r N o rm a li t y , 3ASA 63, p . 1343-1372.

[2 5 ] S t e p h e n s M. A. (1 9 7 4 ), EOF S t a t i s t i c s f o r Good ness o f F i t snd Some C om p ariso n s, 3ASA 69, p . 730-737,

[2 6 ] S p l e g e l h a l t e r 0 . 3. (1 9 7 7 ), A T o st f o r Nor m a li t y A g a in s t Sym m atrio A l t s r n a t l v e s , " B lo m e t r lk o 64, p . 415-418. [2 7 ] V a n S o e S t 3. (1 9 6 7 ), Soma E x p e r im e n ta l R e s u lt s C o n c s rn in g T s s t s o f N o r m a lit y , S t a t i s t . N e o rls n d . 21, p . 91- -97. [2 8 ] W s l s b e r g S . , B i n g h a m C. (1 9 7 5 ), An Ap p ro x im a te A n a ly s is o f V a r ia n c e T e s t f o r N o n - N o rm a lity o u i- t a b l s f o r M achine C a l c u l a t i o n , “ T e c h n o m e trlc s “ 17, p . 133- -134. [2 9 ] 2 1 e 1 i rt s k 1 R. (1 9 7 2 ), T a b llc e s t o t y s t y c z n e , PWN, W arszaw a. W ie s ła w Wagner

TESTY 3EDNOWYMIAROWE3 NORMALNOŚCI

W a r t y k u l e p rsz sn to w a n s s9 t e s t y J^ n o w y m ia ro w e J

w p o d z ia le na t e s t y o p a r te na P o r ó w n a n i u d y s t r y b u a n t r o z k ł a c u e m

P lr y c z n o g o 1 norm aln eg o , _{Oroz t e s t y o p a r te} _{na s t a t y s t y k a c h p o z y c y jn y c h .}t o s t y Omówione z o s t a K_{Omówione z o sT a cy} Podstawowe w ła s n o ś c i p o s z c z e g ó ln y c h to s tó w , ^ ¡ t o w e n e zos-PMnktu w id z e n ia o d s tę p s tw a od n o rm a ln o ś c i. n r* v d a tn o ś ć do za-t a ł a ró w n ież moc p rz e d s za-ta w io n y c h za-te s za-tó w 1 1 i ocigtMn

9

twa od nor-®tosowart z p u nktu w id z e n ia lic z e b n o ś c i próby