The power of the Kolmogorov-Smirnov test

Item Type text; Thesis-Reproduction (electronic)

Authors Schultz, Rodney Edward, 1941-

Publisher The University of Arizona.

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THE POWER OF THE KOLMOGOROV-SMIRNOV TEST

byRodney Edward Schultz

A Thesis Submitted to the Faculty of theCOMMITTEE ON STATISTICS

In Partial Fulfillment of the Requirements For the Degree ofMASTER OF SCIENCE

In the Graduate CollegeTHE UNIVERSITY OF ARIZONA

1 9 7 2

STATEMENT BY AUTHOR

This thesis has been submitted in partial fulfill- mfiit of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.

Brief quotations from this thesis arc allowable without special permission, provided that accurate acknowl­edgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean, of the Graduate College when in his judgment the proposed use of the material is in the inter­ests of scholarship. In all other instances, however, permission must be obtained from the author.

SIGNED:

APPROVAL BY THESIS DIRECTOR This thesis has been approved on the date shown below:

ACKNOWLEDGMENTS

The author wishes to express his gratitude to th following: to Dr * John Denny for guidance and encouragement during this thesis study, and to Hughes Aircraft Company for allowing participation in their work/study fellowship program„

iii

TABLE OF CONTENTS

PageLIST OF TABLES o o o o o o o o o o o o o o o o o o o V

LIST OF ILLUSTRATIONS . • • • • \ • « • . . • • . • o . viiA. 13 S T? RA 0 T? o o © © © © © © © © © © © © © © © © © © © © "̂*GENERAL © © © © © © © © © o © © © © © © © © © © © © 1THE OR. Y O O O O O O O O O O O O O O O O O O O ^ O O O O 3

Computational Method © © © © © © © © © © © © © © 4

TYPES OF TESTS © © © © © © © © © © © © © © © © © © © 8

TEST (3 ( 1 ) O O O O O O O O O O O O O O O O O O O © © 10T ES T |3 ( 2 ) e e o o o c o o o o o o o o o o o o o o o 1 3TES T (3 ( 2A ) o o © o o o O O O © O O O O 0 . 0 O O C O © 17

T ES T |3 ( 3 ) © o o o o o o o o o o o o o o o o o o o © 1 8

COMPARISON OF TESTS © © © © © © © © © © © © © © © © 19

APPENDIX© TABLES FOR THE EXACT POWERS © © © © © © © 29

REFERENCES © © © © © © © © © © © © © © © © © © © © © 6 8

iv

LIST OF TABLES

Table Page1 o Table of R(oc 9 n) Values for n = 2 to 20 o o .» 12

2 o Table of A(oc ̂ n) and e (cc 3 n) Values forn =■ 2 to 10 O O O O O O O O O O O O O O 0. 0 15

3 o Table of A (oc 9 n) and e(oc, n) Values forn — 11 fO 20 o o o o o o o o o o o o o o 0 0 1 6

4 o Table for the Exact Powers of the Test s(3(1) Of SlZe 2 o o o o 0 0.0 o o o o o o o o 30

5 o Table for the Exact Powers of the Test{3 (I ) of Size 3 ° ° 0 ° 0 0 ° ° ° 0-6 0 0 0 0 32

60 Table for the Exact Powers of the Test(3(1) Of SlZe 4 0 0 o o o o 0 0 0 0 o o o o o 3 4

7 o Table for the Exact Powers of the Test3(1) of Size 5 ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° 36

- 80 Table for the Exact Powers of the Test3 ( I ) of Size 6 o a 0 0 0 o o o o o o o o o o 3^

. 9 o Table for the Exact Powers of the Test3(l) of Size 7 ° ° ° ° ° ° ® ° ° ° ° ° o o o 40

10 o Table for the Exact Powers of the Test3(1) of Size 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 42

1 1 o Table for the Exact Powers of the Test3(1) of Size 9 o o 0 0 0 0 0 0 0 0 0 0 0 0 0 44

12 o Table for the Exact Powers of the Test3(1) of Size 10 0 0 0 0 0 0 0 0 0 0 0 0 0 46

13 o Table for the Exact Powers of the Test3(l) of Size 11 o o o o o o o o o o o o o o 48

l4o Table for the Exact Powers of the Test3(1) of Size 12 o o o o o o o o O O O O O O 3 0

V

viLIST OF TABLES--Continued

Table Page15 ° Table for the Exact Powers of the Test

P(l) of Size 13 ° ° ° o 0 0 0 o o o o © o o 2

160 Table for the Exact Powers of the Test(3(1) O f S-XZe Ik o o o o o o 0 0 o o o o o o 5 ̂

1 ? o . Table for the Exact Powers of the Test(3(1) of Size 15 ° 0 0 0 o o o o o o o o o o

l 8 o Table for the Exact Powers of the Test(3(1). O f SlZe l6 o o o o o 0 0 o o o o o o o ^ 8

19 o Table for the Exact Powers of the Test(3(f) of Size 17 0 0 0 O O O O O O O O O O 80

20 o Table f or the Exact Powers of the Test(3(1) of Size 18 o o o o o o o o o o o o o o 6 2

21 o Table for the Exact Powers of the Test(3(1) O f SlZe 19 O O O O O O O 0 6 O 0 © O O

22 © Table for the Exact Powers of the Test.(3(1) of Size 20 0 0 0 0 0 0 0 . 0 0 0 o' 0 0 0 66

LIST OF ILLUSTRATIONS

Figure Page1 o Acceptance Region for Oc = ,05 and

n = 5 o o o o. o o o o o ° o o o O . ° ° ° ° ° 20

2 e Acceptance Region11 — 10 o e o o

for oc = o05 ando e o o o 0 21

3 • Acceptance Region n — 2 0 o o o o'

for oc = 0O 5 ando o o o o o 22

4 o Power of Test for n = 5 o ° o o o ° o ° ° ° • 23

5 o Power of Test for n = 10 o ° o o o ° ° ° ° ° . . 24

6 . Power of Test for n = to o •o o o o ° ° ° ° ° ° 25

7c Power of Test for oc = ,01 ° o o o ° ° ° ° ° « 26

8 o Power of Test for oc = ,05 o o ° ° ° ° ° ° 2?

9 • Power Of Test for OC — OCM o o o o O 0 28

vii

ABSTRACT

There are many papers that give tables and methods

for determining the level (oc) of the Kolmogorov-Smirnov

test 9 but the tables on the power (1 -type II error) are

very limited in the number of values given*. The main

purpose of this paper is to expand the tables for the power

of the Kolmogorov-Smirnov test.

The Kolmogorov-Smirnov test is a goodness of fit

test which is used when a randomly sampled population is

tested against an ja priori completely specified continuous

distribution with cumulative frequency distribution „

That is 3 for any specified value of X , the value of F^(X)

is the proportion of individuals in the population having

measurements less than or equal to X „ The cumulative step

function of a sample of n observations F^(X) is expected to

be fairly close to F^(X)» If it is not close enough, this

is evidence that the hypothetical distribution is not the

correct one and therefore some other distribution is the

true population distribution*, The paper contains a study

of the behavior of some Kolmogorov-Smirnov tests against

certain alternatives o

viii

GENERAL

In this paper we shall compare the completely specified continuous null hypothesis:

H : F = F (x) (1)o o

against a continuous alternative hypothesis:

H1 : F = G(x ), ' (2)where

G(x) = [F^(x)] 1+ ̂ (3 )

for various values of 6 0. This alternative, as given bySuzuki (1968), will be used to calculate the exact powersof three types of tests. Since 0 < F (x) < 1 , G(x) is— o —greater than or equal to F^(x) for all x . Thus we are con­cerned with one sided tests. The smaller the value of 5(delta), the closer G(x) is to F (x) and the smaller the 1 opower of the test.

We are concerned with the acceptance region:

[ Fn ( x ) < (3 [ F^ (x ) ] , for all x] ( 4 )

where P is a non-decreasing and right continuous specified function on the interval (0 ,1) . Under the alternative

2hypothesis (2 ), the probability of (3) becomes

P (F (x) < (3 [F (x)]). n n — o

THEORY

The probability integral transformation of ^ =Fx (X.) produces a random variable which is the ilb order statistic from the uniform population on the interval (0,l) and therefore U ̂ ̂ is distribution free. This property was stated and proven as below by Gibbons (1971)•

Theorem: Let the random variable X have the cumulativedistributive function . If F^ is continuous, the random variable Y produced by the transformation Y = F̂ . (X ) has the uniform probability distribution over the interval (0 ,1 ).

Proof: Since 0 < F ^ (x) < 1 for all x, we have Fy(y) =0 for y 0 and F^ (y ) = 1 for y 1 . For 0 < y < 1, defineu to be the largest number satisfying F ^ (u ) = y. ThenF (X) < y if and only if X •< u , and it follows that

Fy(y) = P(Y y) by definition= P[FX (X) < y] since Y = FX (X)= P[X < Fx 1 (y )] by taking Fx 1 of each

side with the obvious interpretation

= P[X <c u ] since y = Fx (u ) thenFx 1 (y ) = u

= Fx (u) by definition= y by definition.

Since Fy (y ) = y we have a uniform distribution over theint erval (0 ,1 ) .

The purpose of this paper is to calculate the exact power of several tests. Using the previously stated theorem we are able to determine the exact power of certain alternatives for several tests (3.

Therefore it is sufficient to give a computational method for evaluating the function:

Pn[(3] E Prob [U > p"1 (i) , j = 1, 2 . . . n] (5)

where , U^, •••, U^ is an ordered sample from theuniform distribution (0 ,l) and the inverse function (3 ̂ is uniquely determined by

P ̂ (S) = min[T >_ 0 such that (3 (T) >_ S ] ,

because of the assumption that (3 is now decreasing and right continuous.

Computational MethodU . is a random variable from the uniform distribu- J

tion on the interval (0.1) . Let u . = {3 For a given’ J nU 1 U 2 Un-1Un , the conditional distribution of g— , ^— , •••, -- isn n n

that of an ordered sample from the uniform distribution on(0.1) and the density function of U is nu11 . We will ’ nevaluate the function (5 ) by letting

Prob[Uj > u j , j = 1,2,...n] = f ̂ui ’u2 ’* * * Un^ (^ )

for n = 1 ̂ f ^ ) = 1- u , and for n = 2,3,...,

" 2 ' = J lf„ - l (̂ S ’- - - ^ )un'l d u - (7)un

For example:

1 u.f2 (ui ’u2 ) = 2 J’ fi (i r )udu

U2

1 U= 2 f (1 - ) uduJ u

U22= 1 - 2u1 + 2u i u 2 - u2

f3 (ui ’u2 ’u 3 ) = 3 T f2 (i T ’i r )u2duu 3

21 2u 2u u u Q= 3 1 (1- ~ + - i r ;r),V duU ̂ 2 2

2 3= 1 - 3u]L + 6u 1u 2 - 3u 2 -

2 2 + 3u]lu^ - 6u1u2u^ + 3u2 u^.

A more direct expression was shown by Suzuki (1968)For ease of calculation of (6 ) we may proceed as follows:Let Q = 1 and for k = 1, 2, ..., leto

6

Qk = Qk (uV U2 ’*--’Uk )

(8)

Then for n = 1 , 2 ,

n nfn (ul ’U2 ’1--’Un ) = tS„ (k ) Qk (9)k =0

For example: Q t = -u

«2 = U22'iQi1=0

- U 2 2 q o - 2 U 2Q 1

= - u22 + 2u2Ul

S ' - j ' i '

- " u 33qo _ 3u32qi " 3u 3Q2o 2 2= + 3u^ u1 + 3u ^u 2 - 6u ^u 2u 1

fl(ul ) = 9kk =0

= Qo + Q 1

= 1 -i

Although it was only shown for f (u^), f u^), andf3(ul , u2 , U 3) that the more direct expressions of (8 ) and (9 ) equal the expression (?), it can be verified by an inductive proof that the two equations are equal for any

TYPES OF TESTS

The computational methods discussed and the testing situation described by (1 ) and (2 ) allow us to define the significance level (size) of the test (3 to be equal to

1 - fn [u1 , u2 , ... un ] (1 0 )

and the power of the test (1-type II error) to be equal to

1 - fn [0 (U;L) , 0 (u2 ) , ... 0 (un )] (11)where

u . = (3 "̂ (— ) = min[T > 0 such that (3(T) > — ] (12)i n — — nfor i = 1 , 2 , • • • , n and for

10(u.) = G F "*" (u . ) = u .1+6 . (13)1 0 1 1

Therefore when u for any test is determined, the level and the power of the test can be calculated. The main purpose of this paper is to calculate the exact powers for three types of tests that meet the requirements for the acceptance region (4). The powers of these tests are listed for n = 2 through 20 (see Appendix) for a = .01, .02, .03,.04, .05, .1 0 , and .2 0 ; and for 6 = 0 , .2 , .4, .6 , and 1 .0 .

The first test (3 (1 ) suggested by Suzuki (1968) is a non-linear function. Wald and Wolfowitz (1939) treated the non-linear case but their method was restrictive because of

the integral calculations e Anderson and Darling (1952)

treated the non-linear case but their method was also

restrictive due to the nature of limiting distribution of

the integral o The other tests |3 (2) , (3 (2A) , and (3(3) dis­

cussed in this paper are linear functions which appear in

Kolmogorov-Smirnov OC-statistics or ratio-type statistics

as discussed by Birnbaum and Tingey (1951) ̂ Chang Li-Chien

(1955)9 and Suzuki (1 967)° Tests (3(2) and (3 (2A) are the

classical tests discussed in most books and articles.

TEST (3(1)

Test (3 (1) is a non-linear function defined as:

(3 (1) [ T ] = (3(1)[T,R] = min[RVf,l] for fixed R > 1.

Solving equation (12) for (3 (1)u^ = min[T :> 0 such that (3 (1) [T] > — ]

= min[T 2> 0 such that min[RN/T,l] >_ —■]

2 i2= min[T 0 such that R T — — ]n

i.2= min[T > 0 such that T > — — ]n R

i2n 2R 2

since i, n, and R are :> 1

. 2By substituting u . = — ^— — into equations (8), we obtain

1 n R “

It can easily be shown that (— -— — )^ can be factored out ,n R

thus

Qk = (n 2R 2 >kk — 1 .E (k ) (k2 )k "1QT

i=0 1 . 1

where k -1 .Z ( ) (k ) Q*

i =0 1 1is independent of n and R.

10

(14)

(15)

11Equation (9) becomes

f = E O (— — -̂)k Q*n k =0 k n 2R 2 k

by letting x = -~R

f„'= k = (J2 )k Q,k -0 n

By equation (10) we know that the size of the test (level) a = 1 - f . So for a given CX and n (sample size) we would have a polynomial of size n with one unknown (x). The roots of the polynomial were calculated for x using

1 1 /2Bairstow's approximation method. By solving R(ocin) = (— ) we obtain the value R for equation (l4). The solutions for a = .01, .02, .0 3 , .04, .05, .10, and .20 and for n = 2through 20 are tabulated in Table 1. Thus, after deter­mining the value of R, the power of the test (3 (1) can be calculated using the methods discussed by letting

i2u . =1 n 2R 2

12Table 1. Table of R(Oc, n) Values for n = 2 to 20

nAlph a Levels ( Ct)

.01 .02 .03 .04 .05 . 10 . 20

2 7.1401 5.0954 4 .1968 3 .6648 3.3040 2.4195 1.80713 5.8507 4.1916 3.4667 3.0402 2.7528 2.0588 1.59314 5.0763 3-6451 3.0226 2.6583 2 .4i43 1.8351 1 .46055 4.5457 3.2689 2.7156 2.3936 2.1792 1.6797 1.36986 4.1528 2.9897 2.4872 2.1962 2 .0037 1.5646 1 . 30417 3.8470 2.7717 2.3086 2.04l6 l .8663 1.4758 1 .25488 3.6001 2.5955 2.1639 1 .9164 1.7551 1 .4054 1.21699 3.3953 2.4492 2.0437 1.8123 1.6628 l .3487 1.1871

10 3.2220 2.3252 1.9416 1.7240 1.5848 1.3023 1.163311 3.0727 2.2183 1.8537 1.6479 1.5180 1 .2642 1.144112 2.9425 2.1249 1.7768 1.5817 l .4601 1.2327 1 .128413 2.8275 2.0425 1.7089 1.5233 1.4095 l .2065 1.1153l4 2.7250 1.9689 1.6483 1.4714 l .3664 1.1845 1.104515 2.6329 1.9028 1.5939 1.4251 1.3285 1.1661 1.0952l6 2.5496 1.8430 1.5446 1.3836 1.2922 1.1505 1.087417 2.4737 1.7884 1.4998 1 .3462 l .2622 1.1372 1.080718 2.4o 42 1.7385 1.4588 1.3126 1.2360 1.1259 1.0?4819 2.3402 1.6925 1.4212 1 .2824 1.2133 1.1161 1.069720 2.2811 1.6500 1 .3865 1.2554 1.1935 1.1076 1.0652

TEST (3(2)

Test (3(2) is a linear function defined as:

(3(2) [T] = (3(2)[T,e] = min[T+e,l] for c > 0 (15)

Solving equation (12) for (3(2)

u^ = min [T > 0 such that (3(2)[T] — ]

= min [T > 0 such that min[T+e,l] — ]n

= min [T > 0 such that min[T,l-e] ^ ” e]

o, for — < c 7 n —

Birnbaum and Tingey (1951) showed that

[n(l-g)] . .P [ P ( 2) (® ; e ) ] = 1 - e Z (~)(l-G-ji)n J (e+J-) J ,j=0 j n n

where []is the Gauss notation. By inverse interpolation they found values of e such that

P n [ P ( 2 ) ( o ; e ) ] = l-oc

for various n (sample size) and oc. They also reported that the asymptotic value

14

which is derived by the asymptotic formula2

lim P [ (3( 2) (• ] = 1 - e 2en— “̂oo n n±/Z

is a little greater than the exact value of e so that the error committed by using e(cx,n) instead of e would be in the safe direction.

The asymptotic values of e(oc,n) are tabulated in Tables 2 and 3• By substituting the asymptotic value e (oc ,n) for e, the power of the test (3(2) can be calculated using the methods discussed with

I 0, for ^ < e(ot,n)u i =

— - g (oc, n ) , for — > g (oc, n ) .

15Table 2 . Table of

to 10 .A (cx , n ) and g (a, n) Values for n = 2

Alpha Levels (a).01 .02 .03 .04 .05 .10 . 20

A(a, 2 ) ***** .9365 . 8728 . 8260 .7886 .6649 .5242e(a, 2 ) ***** .9889 .9363 .8971 .8654 .7587 * .6343A (a, 3) .8482 .7687 .7197 .6835 . 6546 .5582 .4478e (a, 3) .8761 .8075 .7645 .7324 .7066 .6195 .5179A (a, 4) • 7350 . 6685 .6274 • 5970 .5726 .4911 .3974e (a, 4) .7587 .6993 .6621 .6343 .6119' .5365 .4485A (a, 5) .6581 .5999 .5638 .5371 .5156 .4438 . 3610e (a, 5) . 6786 .6255 • 5922 .5674 .5473 .4799 .4012A (a, 6 ) .6015 .5491 .3166 .4925 .4731 . 4082 . 3333= (%, 6 ) .6195 .5710 .5406 0 1 7 9 .4996 .4380 . 3662A (a , 7) • 5575 .5095 .4797 .4575 .4398 . 3801 . 3111e (cx, 7) .5735 .5286 .5005 .4795 . 4626 .4055 .3391A (a , 8 ) .5220 .4775 .4498 . 4292 .4127 .3572 .2930

8 ) .5365 .4945 .4681 .4485 .4327 .3794 .3172A (a , 9) . 4926 .4509 . 4249 .4056 • 3901 .3380 .2777e (a , 9) .5058 . 4662 . 44l4 . 4229 . 4o8o .3577 .2990A(a, 10 ) .4677 . 4283 .4038 .3856 .3709 . 3217 . 2646e (a, 10 ) .4799 . 4423 .4187 . 4012 .3870 . 3393 .2837

16Table 3• Table of

to 2 0 .A(oc, n) and e(0C, n ) Values for n = 11

Alpha Levels ( Oc).01 .02 .03 .04 .05 .10 . 20

A a 11) .4463 . 4089 .3856 .3683 .3544 .3075 .2532e a ’ 1 1) .4575 .4217 .3992 .3825 .3690 .3235 .2705A a, 12) . 4276 .3919 .3697 . 3532 .3398 .2951 . 2432e a, 1 2 ) . 4380 .4037 . 3822 . 3662 .3533 .3097 .2590A a 13) . 4 m .3769 .3556 .3398 .3270 . 2841 .2343e a, 13) .4209 .3879 .3672 .3519 .3394 .2976 .2488A a 14) • 3964 .3635 .3430 .3278 .3155 .2743 . 2264G tx, 14) .4055 .3738 .3539 .3391 .3271 .2868 .2397A a 15) . 3832 .3515 .3317 .3170 .3052 . 2654 . 2192G a, 15) .3918 .3611 .3419 .3276 . 3160 .2770 .2316A Oc l6 ) .3712 . 3406 .3215 .3073 .2958 .2574 .2127G a, l 6 ) .3794 .3496 • 3310 .3172 . 3060 . 2682 .2243A a 17) .3603 .3306 . 3121 . 2984 .2873 .2500 . 2067G a. 17) . 3680 .3392 . 3211 .3077 . 2968 . 2602 .2176A a, 18) .3503 .3215 .3036 . 2902 .2794 . 2432 .2012G a. 1 8) • 3577 .3296 .3121 . 2990 . 2885 .2529 . 2114A a 19) .3411 .3131 .2957 .2827 .2722 .2370 .1961G a, 19) . 3481 .3209 . 3038 .2910 .2808 .2462 .2058A Oc 20) .3326 .3054 .2884 .2757 .2655 .2313 .1914G a. 20) . 3393 .3127 . 2961 .2837 .2737 .2399 . 2006

TEST (3(2A)

Miller (1956) added a correction factor to the asymptotic value of e(oc,n) which results in a better approximation. The new approximation of e is given by

.11143. The values of A(oc,n) are tabulated in Tables 2 and 3 • By listing the values of A (oc 1n) and e(oc,n) together it can easily be noted that:

1 • A(ot,n) is always less than e(oc,n).

2. The larger the n (sample size), the smaller thedifference is between A (oc ,n) and e(ot, n) .

3* The larger the Oc. the larger the difference isbetween A (OC, n ) and e(oc, n) .

By substituting A(oc ,n) for e, the power of the test (3 (2A) can be calculated using the methods discussed with

3/2M (17)

n

with M = .O9037(log10oc)3>/2 + .01515 (log10oc)2 - .08467 oc

0, for — < A(oc,n)

ni A (oc ,n) , for ~ > A(oc,n) . .

17

TEST (3(3)

Test (3(3) is a linear function defined as:

P(3)[T] = (3(3)[T,C] = min[CT,l] for C > 1 (l8 )

Solving equation (12) for (3(3):

u . = niin[T > 0 such that (3 ( 3) [T ] > —_L — — 11

= minf T > 0 such that minfCT 1 ] > —— u ’ -1 — n

= min[T > 0 such that min[T,i-] >

= ct *

Chang Li-Chien (1955) proved that for any n and a the coefficient C(a,n) which gives the test with size OC is given directly by

c(a,n) = i (19)

It should be noted that C(oc,n) is independent of n (sample size) and that for all concerned values of OC (i.e.,0 <_&<.!), C(ot,n) > 1 by substituting C (a , n ) for C, the power of the test (3(3) can be calculated using the methods discussed with

Oci

18

COMPARISON OF TESTS

Three types of graphical comparisons are made inFigures 1 through 9• The first comparison (Figures 1-3)plots the minimum value of T (which is the acceptancelimit) with the corresponding value of ~ for n = 5 , 1 0 , and20. If F (x ) is less than the minimum T value, the null n 1hypothesis is accepted. The second comparison (Figures 4-6) plots the exact power of the tests with the corres­ponding value of oc for n = 5, 10, and 20 . The third comparison (Figures 7-9) plots the exact power of the tests with the corresponding values of n (sample size) for oc =.0 1 , .0 5 , and .2 0 .

The power of the test (3(2) is in all cases less than the power of the test (3 (2A ) . This illustrates that the correction factor suggested by Miller (1956) to improve the asymptotic value of e(oc,n) does give a better approxima­tion of g . Test (3 (1 ) is superior in all cases to test (3(3)* Test (3 (1 ) is uniformly better than test (3 (2A ) except in the case of small oc and large n (sample size).

19

3 | H*

20

1.0(3(3)P d i P(2A)

.4 6 .8l 2 3 5 .7 .9 1.0Minimum T Value

Figure 1. Acceptance Region for a = .05 and n = 5

•H | ti

21

1.0P(2)-//VVP(2A)

Minimum T ValueFigure 2. Acceptance Region for oc = .05 and n = 10

3 | H*

22

1.0

(3( 2A)(3(2)

4 6 81 2 • 3 5 7 9 1.0Minimum T Value

Figure 3• Acceptance Region for a = .05 and n = 20

Power

of Te

st23

1

6 4 -

(3(1)

(3(2A)(3(2)

.02 .04 .06 .08 .10 .12 .14 .16 .18 .20oc (Level)

Figure 4. Power of Test for n = 5

Power

of Te

st24

1 .

9--

(3(1)

(3( 2A)5 — (3(2)

.02 .04 .06 .08 .10 .12 .14 .16 .18 .20oc (Level)

Figure 5• Power of Test for n = 10

Power

of Te

st25

1 .

(3 ( 2A ){3(1)

9(2)

6- —

.02 .04 .06 .08 .10 .12 .14 16 18 20a (Level)

Figure 6. Power of Test for n = 20

Power

of Te

st26

1.0

(3(1)

(3( 2A)(3(2)

14 16 184 6 8 12 202 10n (Sample Size)

Figure 7 • Power of Test for OC = .01

Power

of Te

st27

1 .

7— —

(3(1)

(3 ( 2A)(3(2)

184 6 14 168 2010 122n (Sample Size)

Figure 8. Power of Test for a = .05

Power

of Te

st28

1.0

9—

(3(1)

(3(2A)(3(2)

4 6 8 16 182 10 12 20n (Sample Size)

Figure 9• Power of Test for a = .20

APPENDIX

TABLES FOR THE EXACT POWERS

29

Table 4. Table for the Exact Powers of the Test |3 (I ) ofSize 2.

Level (a) 6 |3(1) 9(2) f3(2A) 9(3)

b H .0 .0100 .0000 .0038 .0100. 2 .0243 .0000 .0096 .0241.4 .0457 .0000 .0186 .0451• 6 .0732 .0000 .0306 .0720.8 .1052 .0000 .0451 .1032

1.0 .1401 .0000 .0615 .1373

b to .0 .0200 .0001 .0158 .0200. 2 .0434 .0005 .0315 .0429.4 .0750 .0016 .0516 .0737. 6 .1126 .0036 .0747 .1102

' .8 .1536 .0067 .0997 .15021.0 .1963 .0111 .1255 .1917

.03 .0 .0300 . 004l .0279 .0300. 2 .0608 .0102 .0507 .0601.4 .1000 .0196 .0776 .0981. 6 .1444 .0320 . 1068 .1412.8 .1912 .0469 .1370 . 1866

1.0 .2383 .0637 .1671 .2325.04 .0 .o4oo .0106 .0395 . 0400

. 2 .0772 .0226 .0677 .0762

.4 .1225 .0389 .0995 .1201

.6 .1720 .0583 .1327 .1681

.8 .2228 .0800 .1661 .21751 .0 .2729 .1029 .1988 . 2663

.05 .0 .0500 .0181 .0505 .0500.2 .0928 .0353 .0831 .0916.4 .1432 .0570 .1185 . l4o4• 6 .1968 .0815 .1547 .1924.8 .2506 .1077 .1904 .2447

1.0 .3027 .1346 .2247 .2955.10 .0 .1000 .0582 .0988 .1000

.2 .1639 .0935 .1454 .1621

.4 .2309 .1312 .1915 .2272• 6 .2961 .1691 .2354 .2908.8 .3573 . 2060 .2765 .3507

1.0 .4133 .24.13 . 3144 .4058

Table 4.--Continued31

Level (a) 6 (3(1) (3(2) (3 ( 2 A ) (3(3)

.20 .0 . 2000 .1337 .1785 . 2000.2 .2865 .1870 .2379 .2852.4 .3665 .2376 .2920 . 3642• 6 .4376 .2844 . 3406 .4346.8 .4997 .3270 .3839 .4962

1.0 .5534 .3657 .4225 .5496

Table 5 • Table for the Exact Powers of the Test (3(1) ofSize 3•

Level (PO 6 (3(1) 9 ( 2 ) (3(2A) (3(3)

.01 .0 .0100 .0019 .0089 .0100.2 .0261 .0054 .0196 .0258.4 .0517 .0114 .0343 .0505. 6 .0857 .0199 .0523 . 0 8 3 2.8 .1262 .0308 .0726 .1221

1.0 .1709 . 0 4 3 6 .0944 .1650.02 .0 .0200 .0071 .0186 .0200

. 2 .0467 .0163 .0362 .0458

.4 .0849 .0293 .0581 .0823• 6 .1318 .0455 .0829 .1269.8 .1840 .0642 .1094 .1766

1 . 0 .2386 .0845 .1365 . 2 2 8 7

.03 . 0 .0300 .0131 .0275 .0300. 2 .0656 .0269 .0501 .o64i.4 .1135 .0451 . 0 7 6 9 .1094. 6 .1690 .0665 .1059 .1620.8 .2285 . 0 8 9 8 .1360 . 2184

1 . 0 .2885 .1143 .1660 .2756.04 . 0 .0400 .0192 . 0 3 5 9 . 0 4 0 0

. 2 .0833 .0370 . 0 6 2 5 . 0 8 1 3

.4 .1388 . 0 5 9 3 .0929 . 1 3 3 6

.6 . 2011 .0844 .1250 .1924

.8 .2655 . 1 1 1 1 .1575 .25351 . 0 . 3289 .1384 . 1 8 9 5 . 3140

.05 . 0 .0500 .0253 .0446 .0500.2 . 1 0 0 2 .0466 .0763 • 0977.4 .1621 . 0 7 2 2 .1128 .1560• 6 .2296 .1003 . 1 5 2 0 . 2 1 9 6.8 .2977 .1295 .1925 . 2842

1 . 0 . 3 6 3 3 . 1 5 8 9 .2330 .3470

or-i . 0 . 1 0 0 0 .0592 .0959 . 1 0 0 0.2 .1764 . 0 9 9 5 .1527 . 1 7 2 2. 4 .2594 .1452 .2129 .2505• 6 .3411 .1933 .2726 . 3282.8 .4171 .2417 . 3 2 9 6 .4013

1 . 0 .4857 . 2890 .3827 .4679

33Table 5*— Continued

Level ((X) 5 P(i) (3(2) £( 2A) (3(3)

.20 .0 .2000 .1464 .1948 . 2000.2 .304? . 2184 .2765 • 3010.4 .4032 .2895 .3531 . 3 9 6 6.6 .4903 .3561 . 4220 .4817.8 .5649 .4167 . 4829 .5552

1 .0 .6277 .4711 .5360 .6175

Table 6. Table for the Exact Powers of the Test (3(1) ofSize 4.

Level (a) 6 (3(1) 9(2) (3( 2A ) 9(3)

.01 .0 .0100 .0034 . 0 0 8 6 .0100. 2 .0275 .0087 .0198 .0270.4 .0564 .0172 .0363 .0547. 6 .0959 .0286 .0577 .0922.8 .143? .0424 .0831 .1374

1.0 .1970 .0582 .1115 .1878

b to .0 .0200 .0085 .0185 .0200. 2 .0493 .0197 .0392 .o48o.4 .0929 .0362 .0674 . 0 8 9 1• 6 .1478 .0575 .1014 .1403.8 . 2096 .0828 .1391 . 1 9 8 0

1.0 .2743 .1111 . 1 7 8 8 . 2 5 8 7

.03 .0 .0300 .0148 .0286 .0300.2 .0693 .0323 .0573 .0671.4 .1242 .0569 .0940 .1181• 6 .1896 .0870 .1359 .1787.8 .2600 .1212 .1806 . 2440

1 .0 .3308 .1579 . 2260 . 3104.04 .0 .0400 .0218 .0387 .0400

.2 .0882 .0453 .0740 . 0 8 5 1

.4 .1523 .0766 .1172 .1442

. 6 .2255 .1136 .1648 .2117

.8 .3017 .1540 . 214o . 28241.0 .3762 .1959 . 2628 .3523

.05 .0 .0500 .0291 .0486 .0500.2 .1061 .0581 .0895 .1022.4 .1780 .0951 .1379 .1681. 6 .2573 .1374 . 1 8 9 8 . 2411.8 .3376 .1824 . 2422 .3157

1.0 . 4i 42 . 2280 .2933 .3879.10 .0 .1000 .0670 .0950 .1000

. 2 .1868 .1169 .1558 . 1 7 9 9.4 . 2 8 3 5 .1729 . 2203 .2687• 6 .3790 .2307 .2841 . 3 5 7 6.8 .4670 .2872 . 3446 . 4409

1.0 .5449 . 3408 . 4008 .5160

35Table 6.— Continued

Level (oc) 6 (3(1) (3(2) (3 ( 2A ) (3(3)

to o . 0 . 2 0 0 0 .1484 .1941 . 2 0 0 0. 2 • 3195 . 2 3 0 2 .2884 .3132.4 .4333 . 3 1 2 8 • 3790 .4217. 6 .5328 • 3910 . 46io . 5 1 8 0. 8 . 6 1 6 3 .4623 • 5329 .5999

1 . 0 .684? .5258 • 5950 . 6 6 8 0

Table 7. Table for the Exact Powers of the Test (3(1) ofSize 5•

Level (a) 6 |3(1) (3(2) (3(2A) (3(3)

.01 .0 .0100 .0042 .0092 .0100.2 .0286 .0115 .0228 .0280.4 .o6o4 .0241 .0436 .0582• 6 .1048 .04i8 .0706 .0999.8 .1591 .0643 .1024 .1506

1.0 .2200 .0905 .1373 .2075.02 .0 .0200 .0103 .0188 .0200

. 2 .0514 .0250 .04i6 .0498

.4 .0998 .0472 .0732 .0946

. 6 .1618 .0757 .1115 .1515

.8 . 2322 .1088 .1541 .21621 .0 .3059 .1450 .1992 . 2843

•03 .0 .0300 .0168 .0284 .0300. 2 .0724 .0378 .0596 .0696.4 .1337 .0673 .1010 .1255. 6 .2078 .1033 .1495 .1927.8 .2879 .1435 . 2021 .2657

1.0 . 3682 .1861 .2560 .3399.04 .0 ,o4oo .0237 .0383 .o4oo

. 2 .0923 .0508 .0772 .0882

.4 . i64i .0876 .1270 .1530• 6 .2473 .1314 .1835 .2279.8 .3338 .1796 . 2428 .3067

1.0 .4178 .2297 . 3020 .3845.05 .0 .0500 .0309 .0483 .0500

.2 .1112 .0642 .0943 .1059

.4 .1919 .1079 .1510 .1782• 6 . 2821 .1586 .2135 .2592.8 •3730 .2131 .2775 . 3422

1.0 .4589 .2687 . 3400 . 4221.10 .0 .1000 .0706 .0979 .1000

.2 .1959 .1297 .1698 .1861

.4 . 3049 .1983 .2485 .2837• 6 .4126 .2700 .3269 .38191 .8 .5105 . 3403 .4oo8 .47361.0 . 5 9 5 3 . 4o64 .4682 .5553

Table 7•— Continued37

Level (a) 6 P(i) 9(2) 9(2A) 9(3)

to o .0 . 2000 .1530 .1926 .2000.2 . 3322 .2435 .2953 .3231.4 .4588 .3343 .3950 .4424. 6 .5685 .4187 .4854 .5477.8 .6585 . 4940 .5642 .6359

1.0 .7300 .5598 .6314 • 7077

38

Table 8 . Table for Size 6 .

the Exact Powers of the Test (3(1) of

Level (oc) 6 (3(1) (3(2) (3(2A) 9(3)

.01 .0 .0100 .0047 .0093 .0100. 2 .0296 .0136 .0244 .0289.4 .0639 .0290 .0487 .0612• 6 .1127 .0512 .0815 .1066.8 .1730 .0794 .1209 .1623

1 .0 . 24o 8 .1126 .1649 . 2249

b to .0 .0200 .0111 .0191 .0200. 2 .0533 .0286 .0453 .0513.4 .1059 .0560 .0833 .0995.6 .1744 .0921 .1304 .1614.8 .2527 .1349 .1831 .2322

1.0 .3345 .1818 .2383 .3069.03 .0 .0300 .0182 .0290 .0300

.2 .0752 .0434 .0642 .0717

.4 .1421 .0803 .1119 .1318

. 6 .2243 .1262 .1680 . 2049

.8 .3134 .1780 .2281 .28471.0 .4021 .2324 . 2888 .3656

.04 .0 .0400 .0255 .0387 .0400.2 .0959 .0577 .0815 .0908.4 .1748 .1023 .1372 .1605. 6 .2672 .1555 .2003 .2421.8 .3632 .2133 . 2664 . 3280

1.0 .4553 .2725 .3319 .4124

.05 .0 .0500 .0330 .0484 .0500. 2 .1158 .0714 .0984 .1090.4 .2047 .1224 .1615 .1869• 6 . 3048 .1813 .2315 .2749.8 .4054 .2436 .3033 .3652

1 .0 .4991 .3058 .3732 .4517.10 .0 .1000 .0725 .0980 .1000

.2 . 2042 .1385 .1776 .1914

.4 .3245 .2166 . 2669 .2967

. 6 .4430 .2987 .3566 .4029

.8 .5492 .3791 .4408 .50161 .0 .6391 .4541 .5167 .5885

Table 8.— Continued39

Level (a) 6 P(l) 9(2) (3 ( 2A ) 9(3)

to o . 0 . 2 0 0 0 . 1 5 8 1 .1949 . 2 0 0 0. 2 .3431 . 2 6 1 2 .3078 .3315.4 . 48io . 3 6 5 8 .4169 .4600. 6 .5991 .4625 .5142 .5728. 8 . 6 9 3 7 .5475 .5970 .66591.0 . 7 6 6 8 . 6 1 9 9 .6659 .7400

ko

Table 9• Table for the Size 7•

Exact Powers of the Test 9(1) of

Level (cx) 6 p(i) 9(2) 9(2A) 9(3).01 .0 .0100 .0052 .0094 .0100

.2 .0304 .0159 .0261 .0296

.4 .0670 .0349 .0536 . 0 6 3 9• 6 .1199 .0625 .0912 .1126.8 .1858 .0974 . 1 366 . 1 7 2 8

1 .0 • 2599 •1379 . 1 8 7 2 .2407

b to .0 .0200 .0119 .0192 .0200.2 .0549 .0318 .0478 . 0 5 2 6.4 .1114 .0637 .0908 .1038• 6 .i860 . 1 0 6 3 .1451 .1702.8 .2716 .1568 . 2064 .2466

1 .0 . 3608 . 2121 .2709 .3271•03 .0 .0300 .0191 .0291 .0300

. 2 .0776 .0475 .0679 .0735.4 .1499 .0903 .1225 .1373• 6 .2397 .1444 . 1 8 7 8 .2158.8 .3369 .2056 . 2 5 8 3 .3017

1.0 .4332 .2699 . 3294 .3885.04 .0 .o4oo .0266 .0390 .0400

. 2 .0992 .0629 .0866 .0931.4 .1847 .1149 .1504 .1672

. 6 .2857 .1777 . 2 2 3 6 .2546

.8 .3904 . 2463 .2998 • 34691.0 .4898 .3161 .3745 .4372

.05 . 0 .0500 .0343 .0489 .0500. 2 .1200 .0779 . 1042 .1117.4 .2166 .1376 .1754 .1945.6 . 3261 .2073 .2544 . 2889.8 .4354 . 2811 . 3346 . 3 8 5 6

1.0 . 5 3 5 8 .3544 .4113 .4777

oH .0 .1000 .0745 .0979 . 1000. 2 .2118 .1469 .1839 .1960.4 . 3 4 2 8 .2336 .2819 .3080• 6 .4711 • 3245 . 3 8 0 5 . 4214.8 .5842 .4125 .4726 . 5 2 6 0

1.0 .6776 .4936 .5546 .6171

Table 9•— Continued

Level (a) 6 p ( D p(2 ) p(2A) p(3)

to o . 0 . 2 0 0 0 .1609 .1964 . 2 0 0 0. 2 0525 .2742 . 3204 .3388.4 .5003 . 3902 . 4412 .4754. 6 .6254 .4968 .5479 .5946. 8 .7233 . 5 8 9 0 . 6 3 7 2 .6914

1 . 0 .7969 .6659 .7096 . 7 6 6 9

42Table 10. Table for

Size 8.the Exact Powers of the Test (3(1) of

Level (a) 5 (3(1) 9(2) P(2A) 9(3)

. 0 1 . 0 . 0 1 0 0 . 0 0 5 6 . 0 0 9 5 .0100.2 . 0 3 1 2 . 0 1 7 7 . 0 2 7 9 . 0 3 0 3.4 . 0 6 9 9 .0403 . 0 5 9 4 . 0 6 6 3• 6 . 1 2 6 5 . 0 7 3 7 . 1 0 3 1 . 1 1 8 0.8 . 1 9 7 6 .1164 . 1 5 6 1 . 1 8 2 4

1.0 . 2 7 7 8 . 1 6 5 9 .2147 . 2 5 5 1

.02 .0 .0200 . 0 1 2 5 .0193 .0200. 2 .0563 . 0 3 5 1 .0504 .0538.4 .1165 . 0 7 2 1 . 0 9 8 2 .1076.6 .1967 . 1 2 1 8 .1591 .1783.8 . 2892 . 1 8 0 3 .2279 .25971 .0 .3853 . 2 4 3 6 .2997 .3455

.03 .0 . 0 3 0 0 . 0 1 9 8 .0292 . 0 3 0 0. 2 .0798 . 0 5 1 5 .0711 .0751.4 .1572 . 1 0 0 1 . 1 3 1 6 .1423• 6 .2541 . 1 6 1 7 .2047 .2257.8 .3591 . 2 3 1 2 .2837 .3171

1.0 .4621 . 3 0 3 6 .3629 . 4092.04 .0 .0400 .0274 .0391 .o4oo

.2 .1023 . 0 6 7 6 . 0 9 0 6 . 0 9 5 1.4 .1941 . 1 2 6 1 . 1 6 1 3 . 1 7 3 2

. 6 .3033 .1974 . 2434 . 2 6 6 0

.8 . 4i6l .2749 . 3290 . 36401 .0 . 5 2 1 8 . 3532 .4121 .4594

.05 .0 . 0 5 0 0 .0353 . 0 4 9 0 . 0 5 0 0.2 .1239 . 0 8 3 3 . 1 0 9 1 .1142.4 . 2 2 8 0 .1504 . 1 8 8 3 . 2014. 6 . 3464 .2294 . 2 7 7 1 . 3015.8 .4637 .3127 . 3 6 7 0 .40 39

1.0 . 5 6 9 8 .3947 . 4 5 2 1 .5010.10 .0 .1000 . 0 7 6 3 . 0 9 8 2 .1000

.2 . 3 1 8 9 .1559 . 1 9 0 7 . 2000

.4 . 3 5 9 8 . 2 5 2 2 . 2 9 6 9 . 3 1 8 2

. 6 . 4 9 7 0 .3525 .4033 . 4 3 7 9

.8 . 6 1 5 8 . 4481 • 5015 . 5 4 7 81.0 . 7 1 1 5 .5341 . 5 8 7 6 .6422

43

Table 10.— Continued

Level (cx) 5 P(l) (3(2) P(2A) (3(3)

. 2 0 . 0 . 2 0 0 0 . 1 6 2 8 . 1 9 6 8 . 2 0 0 0. 2 .3607 . 2 8 5 1 .3303 .3453.4 .5172 .4109 . 46io .4891• 6 .6481 .5257 .5756 . 6 1 3 8. 8 .7484 .6233 .6697 .7136

1 . 0 .8218 . 7 0 3 0 . 7442 . 7 8 9 8

44Table 11 Table for

Size 9•the Exact Powers of the Test (3(1) of

Level (a) 6 (3(1) (3(2) (3 (2A ) (3(3)

.01 .0 .0100 .0059 .0096 .0100. 2 .0318 .0195 .0295 .0309.4 .0725 .0455 .0647 .0686. 6 .1327 .0843 .1144 .1231.8 .2087 .1341 .1730 .1913

1.0 .2945 .1916 . 2422 .2684.02 .0 .0200 .0130 .0194 .0200

. 2 .0576 .0380 .0531 .0548

.4 .1212 .0803 .1064 .1111• 6 . 2069 .1378 .1747 .1856.8 .3059 .2055 .2516 .2717

1.0 . 4083 .2783 .3309 .3623.03 .0 .0300 .0205 .0293 .0300

.2 .0819 .0555 .0744 .0766

.4 .1640 .1104 . l4o8 .1469

. 6 . 2678 .1802 .2215 .2348

.8 . 3801 .2583 . 3081 • 33121.0 .4893 .3384 . 3942 .4281

.04 .0 .o4oo .0282 .0391 .o4oo.2 .1051 .0722 .0944 .0969.4 .2031 .1374 .1717 .1787.6 . 3202 .2169 . 2618 . 2764.8 .4405 . 3026 .3332 .3796

1.0 .5318 . 3881 .4452 .4796

.05 .0 .0500 .0362 .0490 .0500. 2 .1276 .0885 .1134 .1163.4 .2389 .1627 .1999 .2077• 6 .3659 .2501 .2972 .3130.8 . 4906 . 34l8 .3932 .4207

1.0 .6oi4 .4309 .4871 .5220.10 .0 .1000 .0776 .0986 . 1000

. 2 .2256 . l64o .1980 .2037

.4 .3758 . 2698 .3133 .3275

. 6 .5211 . 3800 .4286 .4529

.8 .6445 . 4839 • 5331 .56731.0 .7415 .5737 .6227 .6645

45

Table 11.--Continued

Level (a) 6 (3(1) (3(2) (3( 2A) 9(3)

.20 .0 . 2000 .1645 .1969 . 2000. 2 .3677 .2951 . 3388 • 3512.4 .5319 .4298 .4784 .5015. 6 .6677 .5514 .5994 .6309.8 .7698 .6531 .6971 • 7330

1 .0 .8424 .7341 .7726 .8095

\

46

Table 1 2 . Table forSize 10.

the Exact Powers of the Test (3(1) of

Level (oc) 6 (3(1) 9(2) (3( 2A) 9(3)

.01 .0 .0100 .0061 .0100 .0100. 2 .0325 .0213 .0320 .0314.4 .0749 .0508 .0718 .0706. 6 .1385 .0954 .1284 .1278.8 .2193 .1524 .1971 .1996

1.0 .3104 .2177 .2726 . 2808.02 .0 .0200 .0134 .0200 .0200

. 2 .0589 .0408 .0568 .0558

.4 .1256 .0881 .1162 .1144

.6 .2165 .1528 .1930 .1924

.8 . 3218 .2289 .2791 . 28291.0 .4301 .3101 .3671 .3779

.03 .0 .0300 .0210 .0299 .0300. 2 .0838 .0592 .0790 .0779.4 .1705 .1203 .1526 .1511. 6 . 2809 .1986 .2419 .2431.8 . 4002 . 2860 .3371 .3443

1 .0 .5150 .3747 . 4300 .4455.04 .0 . 0400 .0289 .0399 .o4oo

. 2 .1079 .0768 .0996 .0986

.4 .2117 .1490 .1842 .1838• 6 .3365 .2372 .2827 . 2860.8 .4639 .3317 . 3838 . 3940

1.0 .5801 .4242 .4797 .4981

.05 .0 .0500 .0369 .0498 .0500.2 .1312 .0936 .1190 .1183.4 .2496 .1752 .2131 .2135.6 .3848 .2711 . 3188 .3237.8 .5162 .3706 . 4242 .4361

1.0 .6310 .4658 .5215 .5413.10 .0 .1000 .0780 .0993 . 1000

. 2 .2317 .1713 .2057 .2071

.4 .3907 .2859 . 3304 .3359

. 6 .5432 . 4o48 .4541 .4667

.8 .6704 .5155 .5649 .58501.0 .7678 .6117 .6580 .6844

4?Table 12.— Continued

Level (a) 6 (3(1) 9(2) (3( 2A) 9(3)

. 20 .0 . 2000 . 1662 .1975 . 2000. 2 .3736 .3050 .3477 . 3566.4 .3446 .4479 .4952 .5127. 6 .6848 .5755 .6217 . 6464.8 .7881 .6799 • 7217 .7503

1 .0 .8596 .6713 .7972 . 8266

48

Table 13- Table for Size 1 1 .

the Exact Powers of the Test (3(1) of

Level (a) 6 (3(1) (3(2) (3 ( 2A ) (3(3)

.01 .0 .0100 .0063 .0100 .0100. 2 .0330 .0230 .0336 .0319.4 .0772 .0563 .0772 .0725• 6 . i44o .1070 .1398 .1321.8 .2293 .1718 .2157 .2074

1.0 .3254 . 2454 .2983 .2925.02 .0 .0200 .0137 .0200 .0200

. 2 .0600 .0435 .0592 .0567.4 .1298 .0959 .1237 .1174

. 6 . 2258 .1678 .2077 .1988

.8 .3370 .2520 . 3016 .29341.0 .4508 • 3407 .3968 .3925

• 03 .0 .0300 .0214 .0299 .0300. 2 .0856 .0627 .0821 .0791.4 .1768 .1298 .1619 .1551. 6 .2937 . 2162 .2594 .2510.8 .4196 .3119 . 3628 .3565

1.0 .5395 . 4o8i .4627 . 46l6.04 .0 .o4oo .0294 .0399 .0400

. 2 .1105 .0810 .1034 .1001

.4 . 2202 .1601 .1951 .1885

. 6 • 3525 .2570 .3018 .2950

.8 .4864 . 3600 .4105 .40741.0 . 6068 .4597 .5119 .3152

.05 .0 .0500 .0375 .0498 .0500.2 .1346 .0985 .1233 .1201.4 . 2600 .1876 . 2246 .2189. 6 . 4033 .2925 .3383 .3335.8 .5409 . 4002 .4503 .4504

1.0 .6588 .5014 .5521 .5589.10 .0 . 1000 .0796 .0994 .1000

. 2 .2374 .1782 .2117 . 2101

.4 . 4o46 . 3011 .3447 .3438

.6 • 5636 .4279 .4759 .4793.8 .6938 .5444 .5919 .60131.0 .7910 .6437 .6875 .7024

49

Table 13 «--Continued

Level (oc) 6 (3(1) (3(2) (3 (2A ) (3(3)

. 20 .0 . 2000 .1677 .1978 . 2000. 2 . 3786 . 3148 • 3557 .3615.4 .5556 . 466l .5108 .5231• 6 .6996 .5991 .6418 .6605.8 .8037 .7055 .7434 .7658

1.0 .8?4l .7863 .8183 .8416

Table l 4 . Table for the Exact Powers of the Test (3(1) ofSize 12.

Level (a) 6 (3(1) (3(2) (3( 2A) (3(3)

. 0 1 . 0 . 0 1 0 0 . 0 0 6 5 . 0 1 0 0 . 0 1 0 0. 2 .0336 .0246 .0351 .0324.4 . 0 7 9 4 . 0 6 1 6 .0827 .0743• 6 . 1 4 9 2 .1184 . 1 5 1 6 .1363.8 . 2 3 8 9 .1909 .2352 .2147

1 . 0 .3399 .2727 .3254 .3035. 0 2 . 0 . 0 2 0 0 . 0140 . 0 2 0 0 . 0 2 0 0

. 2 . 0 6 1 1 .0462 .0615 .0575

.4 .1338 . 1 0 3 8 .1313 . 1 2 1 2

. 6 .234? . 1 8 3 1 . 2224 .2048

. 8 .3516 .2753 . 3238 . 30321.0 .4706 .3713 .4253 . 4o6i

.03 . 0 . 0 3 0 0 . 0 2 1 8 .0299 . 0 3 0 0. 2 .0873 . 0 6 6 1 . 0 8 5 2 . 0 8 0 2.4 .1829 .1392 . 1 7 1 0 .1587. 6 . 3 0 6 1 12334 .2762 .2583. 8 . 4383 . 3370 .3870 .3679

1 . 0 . 5 6 2 8 .4397 .4927 .4767.04 . 0 .o4oo .0299 .0399 .0400

. 2 . 1 1 3 0 . 0 8 5 0 . 1 0 7 0 . 1 0 1 6

.4 . 2 2 8 5 . 1 7 0 8 .2055 . 1 9 2 8

. 6 . 3 6 8 1 .2759 . 3204 . 3 0 3 4

. 8 .5083 .3867 .4364 .41991.0 . 6 3 2 3 .4924 . 5 4 3 1 • 5310

.05 .0 . 0 5 0 0 .0381 .0498 . 0 5 0 0. 2 . 1 3 8 1 . 1 0 3 2 . 1 2 7 6 . 1 2 1 8.4 . 2 7 0 4 .1995 . 2 3 6 2 .2239. 6 .4214 . 3 1 2 9 .3580 . 3428. 8 .5647 .4282 .4769 .4637

1.0 .6849 .5349 . 5 8 3 0 .5752.10 . 0 . 1 0 0 0 .o8o4 .0994 . 1 0 0 0

. 2 . 2426 . 1 8 5 0 . 2 1 7 6 .2130

.4 .4173 . 3 1 6 0 • 3584 • 3511. 6 . 5 8 2 2 . 4 5 0 2 .4964 . 4 9 1 1

. 8 .7148 .5715 . 6 1 6 7 .61621.0 . 8 1 1 2 .6729 .7139 . 7 1 8 8

51

Table 14.— Continued

Level (a) 6 (3(1) (3(2) (3( 2A) (3(3)

.20 .0 . 2000 .1690 .1981 . 2000. 2 . 3828 . 3242 .3638 . 366O.4 .5651 .4839 .5263 .5327• 6 .7124 .6223 .6615 .6734.8 .8171 .7308 .7640 .7798

1 .0 . 8862 .8108 .8374 .8548

52

Table 15• Table forSize 13•

the Exact Powers of the Test (3(1) of

Level (a) 6 (3(1) 9(2) (3(2A) 9(3)

.01 .0 .0100 .0067 .0100 .0100. 2 .0340 .0262 .0365 .0328.4 .08l4 .0670 .0881 .0759.6 .1543 .1299 .1633 .1402.8 . 2480 .2100 . 2544 .2217

1 .0 • 3537 .2995 .3518 .3139.02 .0 .0200 .0143 .0200 .0200

. 2 .0621 .0488 .0639 .0582

.4 .1376 .1116 .1390 .1229

. 6 .2432 .1985 .2374 .2105

.8 .3657 .2990 . 3461 .31251.0 . 4896 . 4022 .4535 .4189

.03 .0 .0300 .0222 .0299 .0300. 2 .0890 .0695 .0882 .0813.4 .1888 .1487 .1801 .1622• 6 .3182 . 2508 .2927 .2652.8 .4565 . 3620 . 4104 .3786

1 .0 .5852 .4705 .5211 .4907.04 .0 .0400 .0303 .0399 .o4oo

.2 .1155 .0890 .1106 .1029

.4 .2367 . 1814 .2156 .1969• 6 .3835 .2945 . 3382 .3113.8 .5296 .4125 . 4608 .4316

1.0 .6565 .5232 .5719 .5458• 05 .0 .0500 .0386 .0498 .0500

. 2 . l4l4 .1077 .1316 .1234

.4 .2806 . 2111 . 2474 . 2286

. 6 .4392 .3325 . 3770 .3515

.8 .5876 .4547 • 5021 .47611.0 .7095 .5658 .6120 .5904

.10 .0 .1000 .0812 .0994 .1000. 2 .2472 .1916 . 2234 .2157.4 . 4289 .3308 .3718 .3579• 6 .5991 .4720 .5162 .5021.8 .7335 .5976 .64oo .6300

1.0 . 8289 . 7002 .7379 .7337

53

Tab1e 15 • "-Continued

Level (a) 6 (3(1) (3(2) J3( 2A) (3(3)

to o .0 . 2000 .1701 .1983 .2000.2 . 3861 .3329 .3718 .3703.4 • 5733 .5004 .5417 • 5417. 6 .7235 .6436 .6811 .6854.8 .8287 .7533 .7842 .7925

1.0 . 8966 .8321 .8561 . 8666

/

54

Table l 6 . Table for Size 14.

the Exact Powers of the Test (3(1) of

Level (a) 6 (3(1) (3(2) (3( 2A) (3(3)

.01 .0 .0100 .0068 .0100 .0100.2 .0345 .0278 .0380 .0332.4 .0834 .0725 .0936 .0775.6 .1591 .1417 .1751 .1439.8 .2569 . 2294 .2735 . 2283

1.0 .3670 .3262 .3776 .3237.02 .0 .0200 .0145 .0200 .0200

.2 .0630 .0513 .0662 .0589

.4 .1413 .1194 .1467 .1254• 6 .2516 .2137 .2525 .2158.8 .3794 . 3220 . 3686 .3213

1.0 .5079 .4319 .4818 . 4310• 03 .0 .0300 .0225 .0299 .0300

. 2 .0906 .0728 .0912 .0823

.4 .1946 .1582 .1892 .1654

. 6 • 3301 . 2682 .3094 .2718

.8 .4743 .3869 .4336 .38871.0 .6067 .5008 .5487 .5040

.04 .0 .o4oo .0307 .0399 .0400.2 .1180 .0929 . ll4i . 104l.4 .2449 .1920 .2258 . 2008. 6 . 3988 .3130 • 3557 . 3187.8 .5504 .4377 . 4843 .4427

1.0 .6796 .5528 .5989 .5597.05 .0 .0499 .0390 .0499 .0500

. 2 .1444 .1121 .1356 .1249.4 . 2902 . 2226 .2583 .2330• 6 .4560 .3518 .3953 .3597.8 .6089 . 4801 .5259 .4878

1.0 .7319 .5948 .6387 .6o46.10 .0 .1000 .0819 .0995 .1000

. 2 .2513 .1980 . 2292 . 2181

.4 .4393 .3452 .3852 . 3644

.6 .6143 .4935 .5356 .5124

.8 .7501 .6229 .6622 .64291.0 .8442 .7263 .7602 .7475

Table l 6 .— Continued55

Level (a) 6 (3(1) 9(2) (3( 2A) 9 ( 3 )

to o . 0 . 2 0 0 0 . 1 7 1 1 .1984 . 2 0 0 0. 2 . 3 8 8 8 . 3411 .3793 .3742.4 . 5 8 0 2 .5159 .5561 .5500• 6 .7332 . 6 6 3 1 . 6 9 9 1 . 6964. 8 . 8 3 8 8 .7735 . 8 0 2 5 . 8o4i

1 . 0 .9034 . 8 3 0 7 . 8 7 2 6 .8771

Table 17 • Table for the Exact Powers of the Test |3 (I ) ofSize 15•

Level (a) 6 (3(1) (3(2) (3 ( 2A ) 9(3)

.01 .0 .0100 .0069 .0100 .0100. 2 .0350 .0294 .0395 .0336.4 .0852 .0780 .0992 .0790.6 .1637 .1537 .1871 .1474.8 .2655 .2491 . 2927 .2347

1.0 .3798 • 3531 .4031 .3332.02 .0 .0200 .0147 .0200 .0200

. 2 .0639 .0537 .0685 .0596

.4 .1449 .1271 .1544 .1277

. 6 .2597 . 2288 .2674 .2209

.8 . 3928 . 3446 .3905 .32971.0 .5255 . 46o4 .5090 .4425

.03 .0 .0300 .0228 .0300- .0300. 2 .0921 .0760 .0942 .0832.4 . 2003 .1675 .1984 .1685. 6 .3419 .2853 . 3260 . 2780.8 .4917 . 4112 .4566 . 3984

1.0 .6274 .5301 .5756 .5165.04 .0 ,o4oo .0310 .0399 .0400

. 2 .1204 .0967 .1176 .1053

.4 .2531 . 2026 .2359 .2045

. 6 . 4i4i .3314 • 3731 . 3258

.8 .5707 .4625 .5073 .45311.0 .7017 .5814 .6247 .5727

.05 .0 .0497 .0394 .0499 .0500. 2 .1473 .1164 .1395 .1263.4 .2997 . 2340 .2691 .2372.6 .4723 .3708 .4131 .3674.8 .6293 .5048 .5487 .4989

1.0 .7527 .6223 . 66 36 .6178.10 .0 .1000 .0824 .0995 . 1000

. 2 .2548 . 2042 .2348 .2205.4 .4486 .3592 . 3984 . 3704

. 6 .6279 .5139 .5546 .5221

.8 .7648 .6467 .6838 .65491 .0 .8576 .7505 . 7814 .7601

57Table 17•— Continued

Level (a) 6 (3(1) (3(2) (3 ( 2A ) 9(3)

. 20 .0 . 2000 .1719 .1985 . 2000. 2 .3909 .3491 .3864 .3780.4 .5861 .5306 .5697 .5579. 6 .7417 . 68l4 .7157 .7967.8 .8476 .7918 .8188 .814?

1.0 .9130 . 8670 .8868 . 8866

58

Table 1 8 • Table forSize 16.

the Exact Powers of the Test (3(1) of

Level (a) 6 (3(1) (3(2) (3 ( 2A) 9(3)

.01 .0 .0100 .0070 .0100 .0100. 2 .0354 . 0 3 0 9 .o4io .0340.4 . 0 8 7 0 .0835 . 1048 . 0 8 0 4• 6 . 1 6 8 2 . 1 6 5 8 . 1 9 9 3 . 1 5 0 8.8 .2738 . 2 6 8 8 . 3 1 2 0 .24071.0 . 3922 .3797 . 4 2 8 5 . 3422

.02 .0 .0200 .0149 .0200 .0200. 2 .0648 .0562 . 0 7 0 8 . 0 6 0 3.4 .1484 .1349 . 1 6 2 1 . 1 3 0 0• 6 . 2 6 7 6 .2439 . 2 8 2 2 . 2 2 5 8.8 .4058 .3669 .4119 . 3376

1.0 .5426 .4880 .5349 .4534.03 .0 . 0 3 0 0 . 0 2 3 0 . 0 3 0 0 . 0 3 0 0

.2 .0936 .0791 . 0 9 7 1 .0841

.4 . 2 0 6 0 .1767 . 2 0 7 4 .1714

. 6 .3536 . 3 0 2 1 . 3424 .2839.8 . 5 0 8 8 .4346 .4791 .40751.0 .6474 • 5579 . 6 0 1 6 .5283.04 .0 .0400 . 0 3 1 3 .0399 .0400

. 2 .1229 .1004 .1211 .1064

.4 . 26l4 . 2 1 3 0 . 2460 . 2 0 8 0

.6 .4293 . 3494 . 3 9 0 4 .3326

.8 .5906 .4867 .5298 .46311.0 . 7 2 2 8 . 6 0 8 9 .6495 .5850

.05 .0 .0500 .0397 .0499 .0500. 2 .1513 . 1 2 0 6 .1435 . 1 2 7 6.4 .3109 .2453 .2799 . 2412.6 .4905 .3895 .4308 .3748.8 . 6 5 1 0 .5289 .5708 .5093

1 .0 • 7739 .6487 . 6 8 7 2 .6303.10 .0 .1000 . 0 8 3 0 .0996 .1000

. 2 . 2 5 7 8 .2101 . 2403 . 2 2 2 7

.4 . 4 5 6 8 .3727 . 4112 .3762

. 6 .64oi .5334 .5729 . 5 3 1 3.8 .7779 . 6 6 8 9 .7042 . 6 6 6 21.0 . 8 6 9 2 .7723 . 8 0 1 1 . 7 7 1 9

Table 18.— Continued59

Level (oc) 6 (3(1) (3(2) (3 ( 2A ) p(3)

to o . 0 . 2 0 0 0 . 1 7 2 8 . 1 9 8 6 . 2 0 0 0. 2 • 3925 . 3 5 6 8 • 3932 .3815.4 • 5912 .5448 . 5 8 2 6 .5653. 6 .7492 .6985 • 7311 .7164. 8 .8553 . 8 0 8 6 .8337 .8245

1 . 0 .9196 .8814 .8994 . 8 9 5 1

I 60Table 19• Table for the

Size 17.Exact Powers of the Test (3(1) of

Level (oc) 6 (3(1) (3(2) (3 ( 2A ) (3(3)

.01 .0 .0100 .0071 .0100 .0100. 2 .0358 .0325 .0425 .0343.4 .0887 .0891 .1105 . 0 8 1 8• 6 .1726 .1779 .2115 .1540.8 .2819 .2883 .3313 .2465

1.0 .4043 . 4058 .4536 .3508.02 .0 .0200 .0150 .0200 .0200

. 2 ' .0656 .0586 .0731 .0609

.4 .1518 .1427 .1698 . 1 3 2 2

.6 .2754 .2591 .2970 .2305

.8 .4185 .3891 .4330 .34531.0 / .5592 .5150 .5599 .4638

.03 .0 .0300 .0232 .0300 . 0 3 0 0. 2 .0951 .0822 .1000 .0849.4 .2117 .1859 . 2164 .1742.6 . 3 6 5 3 . 3188 . 3 5 8 5 . 2 8 9 6.8 .5257 .4575 .5007 .4163

1.0 .6668 .5843 .6261 .5395.04 .0 .o4oo . 0 3 1 6 .0399 .o4oo

. 2 . 1 2 5 3 ,io4i .1245 .1074

.4 .2697 .2232 . 2 5 6 0 .2113

. 6 .4444 .3671 . 4074 .3390

.8 .6101 .5099 .5517 .47251.0 .7428 . 6 3 4 7 .6733 . 5 9 6 6

.05 .0 .0500 .o4oo .0499 .0500.2 .1545 .1248 .1474 . 1 2 8 8.4 .3207 . 2564 .2907 .2450. 6 .5067 .4078 .4482 . 3819.8 .6701 .5521 .5923 .5193

1 . 0 .7923 .6738 • 7097 .6420.10 . 0 .1000 .0835 .0996 . 1000

. 2 .2603 .2160 .2457 . 2248

.4 . 464l . 3 8 5 8 . 4236 .3817• 6 .6509 .5520 .5903 .5400.8 .7895 .6896 .7231 .6768

1.0 .8793 .7922 . 8 1 8 9 .7827

6lTable 19•— Continued

Level (a) 6 (3(1) 9(2) (3 ( 2A) 9(3)

.20 .0 . 2000 .1735 .1986 . 2000. 2 .3937 .3643 .4000 .3849.4 • 5955 .5585 .5951 •5723. 6 .7557 .7148 .7457 .7254.8 . 8621 . 8240 .8472 .8335

1.0 .9254 .8943 .9104 .9028

62Table 20 Table for

Size l 8 .the Exact Powers of the Test (3(1) of

Level (a) 6 (3(1) 9(2) (3( 2A) 9(3)

.01 .0 .0100 .0072 .0100 .0100. 2 .0361 .0340 .0439 .0347.4 .0904 .0948 .1163 .0831• 6 .1768 .1902 .2238 .1572.8 .2897 .3079 .3505 .2521

1.0 .4159 .4313 .4780 .3591.02 .0 .0200 .0152 .0200 .0200

. 2 .0664 .0610 .0754 .0614

.4 .1550 .1505 .1775 .1342

. 6 . 2830 .2742 .3117 .2350

.8 .4310 . 4110 .4538 .35261 .0 .5752 .5413 .5841 .4738

.03 .0 .0300 .0234 .0300 .0300. 2 .0966 .0852 .1029 .0857.4 .2174 .1951 .2254 .1769. 6 .3770 .3353 .3743 .2950.8 .5424 .4798 .5217 .4246

1.0 .6855 .6095 .6493 .5302.04 .0 .0400 .0318 .0399 .0400

. 2 .1278 .1077 .1279 . io84

.4 .2782 .2334 . 2658 .2145

.6 .4595 . 3844 . 4240 .34521 .8 .6290 .5322 .5726 .4816

1.0 .7618 .6590 .6956 .6076

• 05 .0 .0500 .0403 .0499 .0500. 2 .1576 .1289 .1512 .1300.4 . 3302 .2673 . 3012 .2487. 6 .5221 .4257 .4652 .3886.8 . 6880 .5744 .6130 .5287

1.0 .8090 .6972 • 7310 .6531.10 .0 .1000 .0839 .0996 .1000

. 2 . 2623 .2217 .2509 . 2268

.4 .4703 .3987 .4357 . 3869• 6 . 6605 .5700 .6069 .5483.8 . 7 9 9 7 .7092 .7408 . 6 86 8

1.0 .8882 .8105 .8351 .7929

Table 20.--Continued63

Level (a) 6 (3(1) 9(2) (3( 2A) 9 (3)

. 20 .0 . 2000 .1742 .1987 . 2000. 2 . 3944 .3717 . 4065 . 388I.4 .5990 .5719 .6072 .5790. 6 .?6l4 .7302 .7594 .7339.8 .8681 .8382 .8596 .8418

1.0 .9304 .9058 .9203 .9098

64

Table 21. Table for the Size 19•

Exact Powers of the Test (3(1) of

Level (a) 6 (3(1) (3(2) (3( 2A) 9(3)

.01 .0 .0100 .0073 .0100 .0100.2 .0365 .0335 .0454 .0350.4 .0920 .1005 .1221 .0844• 6 .1809 . 2026 . 2362 .1602.8 .2974 .3274 . 3694 .2575

1 .0 .4273 .4565 .3018 .3670.02 .0 .0200 .0153 .0200 .0200

. 2 .0671 .0634 .0777 .0620

.4 .1583 .1383 .1853 .1362• 6 .2906 . 2892 .3263 .2393.8 .4433 .4324 .4743 .3597

1.0 .5908 .3666 .6076 .4833.03 .0 .0300 .0236 .0300 .0300

. 2 .0980 .0883 .1038 .0863

.4 .2231 . 2043 . 2344 .1795• 6 .3888 .3516 • 3901 .3003.8 .5589 • 3017 .5422 .4327

1 .0 • 7037 .6338 .6714 .5604.04 .0 .0400 .0320 .0399 .o4oo

.2 .1303 .1113 .1313 .1094

.4 . 2866 .2433 .2736 .2176

. 6 .4745 . 4oi4 . 4402 • 3511

.8 .6474 .5538 .5928 . 49021 .0 .7798 .6819 .7163 .6180

ino .0 .0500 .0406 .0499 .0500. 2 .1603 .1329 .1350 .1312.4 • 3391 .2781 .3117 .2521.6 .5367 . 4431 .4818 .3931.8 .7046 .5956 .6328 .3378

1.0 . 8242 .7190 .7509 .6636.10 .0 .1000 .0843 .0996 .1000

.2 .2639 .2274 .2561 .2287

.4 .4738 .4113 .4473 .3919

.6 . 66 90 .5874 .6229 .5362

.8 .8089 .7276 .7574 .69621 .0 . 8960 .8272 .8499 .8023

Table 21.— Continued65

Level (a) 6 (3(1) 9(2) (3( 2A) 9 ( 3 )

.20 .0 . 2000 .1748 . 1 9 8 8 . 2000.2 .3949 . 3 7 8 9 .4130 . 3 9 1 1.4 . 6 0 2 0 . 5 8 4 8 . 6 1 8 9 .5853. 6 .7664 .7449 • 7725 . 7 4 1 9.8 .8734 . 8 5 1 5 .8711 . 8 4 9 6

1 . 0 .9348 . 9 1 6 2 .9290 . 9 1 6 2

66

Table 22. Table forSize 20.

the Exact Powers of the Test (3(1) of

Level (a) 6 (3(1) (3(2) (3( 2A) 9 ( 3 )

.01 .0 .0100 .0074 .0100 .0100. 2 .0368 .0370 .0469 .0353.4 .0935 .1062 .1279 .0856.6 .1849 .2150 . 2486 .1631.8 . 3048 .3469 . 3883 .2627

1 .0 .4384 .4812 .5251 .3747.02 .0 .0200 .0155 .0200 .0200

. 2 .0679 . 0 6 5 7 .0800 .0625

.4 . l6l4 .1661 .1930 .1381

. 6 .2980 . 3041 . 3410 .2434

.8 .4554 .4533 .4943 .36651.0 .6061 .5907 .6302 . 4924

. 03 .0 .0300 .0238 .0300 .0300. 2 .0995 .0913 .1087 .0872.4 . 2289 .2134 .2433 .1819. 6 .4006 . 3 6 7 8 .4056 . 3 0 5 3.8 • 5753 .5229 .5621 . 44o4

1 .0 .7213 .6569 .6924 • 5702.04 .0 .o4oo .0322 . 0 3 9 9 .0400

. 2 .1328 .1148 .0346 .1103

.4 .2951 .2535 .2854 .2205

.6 . 4 8 9 2 . 4182 .4562 .3568

.8 .6651 .5747 .6122 .49851.0 .7965 .7037 . 7 3 6 2 .6280

.05 .0 .0500 . o4o8 .0499 .0500.2 .1632 .1369 .1587 .1322.4 .3476 . 2888 . 3220 .2555. 6 .5503 . 4602 .4979 . 4013.8 .7198 .6160 .6517 .5464

1.0 . 8 3 7 9 . 7 3 9 5 . 7 6 9 4 . 6 7 3 6.10 .0 .1000 . 0 8 4 7 .0996 .1000

. 2 .2651 .2329 .2613 .2306

.4 . 4 8 0 5 .4237 .4592 .3967

. 6 . 6766 .6042 .6383 .5637

.8 .8170 .7451 .7731 .70521.0 .9029 .8427 . 8 6 3 4 .8111

67Table 22.— Continued

Level (a) 6 (3(1) (3(2) (3 ( 2A ) 9(3)

.20 .0 . 2000 .1754 .1989 .2000. 2 .3950 .3858 .4194 • 3940.4 .6045 .5973 .6303 .5914. 6 .7709 .7589 .7849 .7495.8 .8782 .8637 .8816 ,8568

1.0 .9388 .9255 .9369 .9220

REFERENCES

Anderson 9 T „ ¥ o , and Darling , D o A o Asymptotic Theory of Certain Goodness of Fit Criteria Based on Stochastic Processes o Annals of Mathematical Statistics 23 (1952) , 193-212.

Birnbaum, Z = W « , and Tingey, F . One Sided ConfidenceContours for Probability Distribution Functions o Annals of Mathematical Statistics, 22 (1951)^592-596.

Chahg Li-Chien <> On the Ratio of an Empirical Distribution Function to the Theoretical Distribution Function» ACTA Mathematica Sinica, 5 (1955), 3^7 -3 6 8.

Gibbons, Jo Do Non Parametric Statistical Inferenceo McGraw-Hill Book Company (1971) , 22-24 o ^

Miller, L o H » Table of Percentage Points of Kolmogorov Statistics o Journal of the'American StatisticalAssociation, 51 (1956), 111-121o

Suzuki, Go On Exact Probabilities of Some Generalized Kolmogorov 1s D-Statistics® Annals of the Institute of Statistical Mathematics, 19 (196?)> 373-388.

Suzuki, Go Kolmogorov-Smirnov Tests of Fit Based on Some General Bounds* American Statistical Association Journal (1968) , 919-924 *

Wald, A o, and Wolfowitz, J * Confidence Limits forContinuous Distribution Functions o Annals of Mathematical Statistics, 10 (1939) ̂ 105-118 *

68