aspects of multivariate statistical theory - robb muirhead (appendix)

APPENDIX

Some Matrix Theory

A I . INTRODUCTION

In this appendix we indicate the results in matrix theory that are needed in the rest of the book. Many of the results should be familiar to the reader already; the more basic of these are not proved here. Useful references for matrix theory are Mirsky (1959 , Bellman (1970), and Graybill (1969). Most of the references to the appendix earlier in the text concern results involving matrix factorizations; these are proved here.

A2. DEFINITIONS

A p X q matrix A is a rectangular array of real or complex numbers a , , , a , * , . ,. ,app, written as

so that a,, is the element in the ith row and j t h column. Often A is written as A =(u,,). We will assume throughout this appendix that the elements of a matrix are real, although many of the results stated hold also for complex matrices. If p = q A is called a square matrix of order p. If q = 1 A is a column uector, and if p = 1 A is a row vecfor. If aij=O for i = l , . . . , p , j = 1 ,... ,q, A is called a zero matrix, written A =0, and if p = q, a,, = 1 for i = 1,. . . ,p and aij = 0 for i # j then A is called the identity matrix of order p, written A = I or A = Ip. The diagonul elements of a p X p matrix A are aI I. a22,. . . , app.

512

Aspects of Multivanate Statistical Theow ROBE I. MUlRHEAD

Copyright 8 1982.2WS by John Wiley & Sons. I ~ C .

Definitions 573

The transpose of a p X q matrix A, denoted by A’, is the q X p matrix obtained by interchanging the rows and columns of A, i.e., if A=(a, , ) then A’=(a,,). If A is a square matrix of order p it is called symmetric if A = A’ and skew-symmetric if A = - A’. If A is skew-symmetric then its diagonal elements are zero.

A p X p matrix A having the form

so that all elements below the main diagonal are zero, is called upper- triangular. If all elements above the main diagonal are zero i t is called lower-triangular. Clearly, if A is upper-triangular then A’ is lower-triangular. If A has the form

so that all elements off the main diagonal are zero, i t is called diagonal, and is often written as

A = diag( a, I , . . . , app ).

The sum of two p X q matrices A and B is defined by

A + B = ( a , , + b , , ) .

If A is p X q and B is 4 X r (so that the number of columns of A is equal to the number of rows of B ) then the product of A and B is the p X r matrix defined by

The product of a matrix A by a scalar a is defined by

aA = ( aa,,).

314 Sonre Murrtw Theory

The following properties are elementary, where, if products are involved, it is assumed that these are defined:

A +( - I )A =O (AB) ’= A’A’ (A’)’ = A ( A + B) ’= A‘+ B’ A( B C ) = ( A E ) C

( A + B ) C = AC+ BC A1 = A .

A( B t- C ) = A B + AC

A p X p matrix A is called orrhogonol if AA‘= Ip and idempotent if A’ = A . If A = ( a , , ) is a p X q matrix and we write

A , , = (u , , ) , i - 1, ..., k ; j = l , ..., /

A,, “ ( a , , ) , i = l , . . . , k , j = I + 1 ,..., q

A 2 , = ( u , , ) , i = k + l , . . . , p ; j = l , , . . , /

A,, =(u, , ) , i = k t I ,..., p ; j = I + 1 ,..., q

then A can be expressed as

and is said to bepuriifioned inlo submatrices A, , , A, , , A,, and A,,, Clearly if B is a p X q matrix partitioned similarly to A as

where B , , is k XI, B I Z is k X ( q -/), B,, is ( p - k ) X / a n d B2, is ( p - k ) X ( 9 - /), then

Derermrinanrs 575

Also, if C is a q X r matrix partitioned as

where C,, is / X m , C,, is / X ( r - m ) , C,, is ( 9 - I ) X m , and C,, is ( 9 - / ) X ( r - m), then it is readily verified that

A3. DETERMINANTS

The dererminanf of a square p X p matrix A , denoted by det A or / A [ , is defined by

det A = E,,alJ,aZJ2.. . .,aPJr n

where C, denotes the summation over all p! permutations R = ( j , , . . . , J p ) of ( I , . . . , p ) and en = + 1 or - 1 according as the permutation n is even or odd. The following are elementary properties of determinants which follow readily from the definition:

(i) If every element of a row (or column) of A is zero then det A =O.

(ii) det A =det A’.

(iii) If all the elements in any row (or column) of A are multiplied by a scalar a the determinant is multiplied by a.

(iv) det(aA)=aPdet A.

(v) If B is the matrix obtained from A by interchanging any two of its rows (or columns), then det B = -det A.

(vi) If two rows (or columns) of A are identical, then det A =O.

(vii) If

b , , + c , , b , , + c , , , ..*, b , , + C l ,

aPP I* A = [ a22 9 ...,

u p 2 1 ... , I

so that every element in the first row of A is a sum of two scalars,

576 Sonie Murrix Theory

then

A similar result holds for any row (or column). Hence i f every element in ith row (or column) of A is the sum of n t e rm then det A can be written as the sum of n determinants.

(viii) If B is the matrix obtained from A by adding to the elements of its i th row (or column) a scalar multiple of the corresponding elements of another row (or column) then det B =det A.

The result given in the following theorem is extremely useful.

THEOREM A3.1. If A and B are both p X p matrices then

det(AA)=(det A)(det B )

Proo/. From the definition

where B ( k l , ..., k p ) denotes the p X p matrix whose ith row is the k,th row of B. By property (vi) d e t B ( k , ,..., k,)=O i f any two of the integers

Derermmunrs 511

k,, . . . , k, are equal, and hence

P P I'

det(AB)= 2 - . . 2 detB(k ,,..., k,) k l = l k , = I

k , # k , # . . ' Zk,

By property (v) it follows that

det B( k , ,. . . , k p ) = E,det B ,

where E, = + 1 or - 1 according as the permutation a = ( k , , . . . ,kp) of (1,. , . , p ) is even or odd. Hence

det(AB)= XE,,( fi u i k , ) .det B , r = l

= (det A )(det B ) .

A number of useful results are direct consequences of this theorem.

THEOREM A3.2. I f A , , ..., A,, are all p X p matrices then

det( A I A 2.. .A , ) = (det A , )(det A ) . , . (det A ,, ) .

This is easily proved by induction on n.

THEOREM A3.3. If A is p X p , det(AA')?O.

This follows from Theorem A3.1 and property (ii).

THEOREM A3.4. 9 X q then

If A , , is p X p , A, , is p X q, A,, is q X p , and A,, is

det [ A" ''1 = det [ A" A" ] = (det A, , )(det A 1, 22 A21 22

Proo/. It is easily shown that

det[ '' 0 A22 ]=detA2,

anu

det[ :y]=det A , , .

578 Some Mutrix Theory

Then from Theorem A3. I ,

det[ :::]=det[ I’ O A :,ldet[ ’dl :y]=(det Al,)(det A Z 2 ) .

Similarly

det[ i l l A” ] =det[ 11 det[ ,‘. ] = (dct All)(det A 2 2 ) , 21 22 2 1 A 2 1

THEOREM A3.5. If A is p X 9 and B is 9 X p then

det( Ip + A B ) =det( I9 + BA) .

Prooj We. can write

I,,+ A B A A IP [ 0 I q ] = [ ? B 1 9 ] [ B

so that

( 1 )

Similarly

det( lp + A B ) =det [ JB 4.

so that

det (Iq + BA) = det [-I.. 4- Equating ( I ) and (2) gives the desired result,

Two additional results about determinants are used often.

(ix) If T is m X m triangular (upper or lower) then det T=ny!&.

(x) If H is an orthogonal matrix then det H = t 1.

litverse of a Marnx 579

A4. MINORS AND COFACTORS

If A =(a,,) is a p X p matrix the minor of the element aIJ is the determinant of the matrix MI, obtained from A by removing the ith row andjth column. The cojucfor of a,,, denoted by ail’ is

alJ =( - I)l+’det MIJ.

I t is proved in many matrix theory texts that det A is equal to the sum of the products obtained by multiplying each element of a row (or column) by its cofactor, i.e.,

P

detA= 2 aIJcu,, ( i = l , . . . ,p) J = I

A principal minor of A is the determinant of a matrix obtained from A by removing certain rows and the same numbered columns of A. In general, if A is a p X q matrix an r-square minor of A is a determinant of an r X r matrix obtained from A by removing p - r rows and q-r columns.

A 5 . INVERSE OF A MATRIX

If A =(a,,) is p X p, with det A fO, A is calied a nunsingular matrix. In this case there is a unique matrix B such that A B = Zp. The i - j t h element of B is given by

cuJJ

det A ’ blJ = -

where aJ, is the cofactor of aJi. The matrix B is called the inoerse of A and is denoted by A - I . The following basic results hold:

(i) AA - I = A - I A = I.

(ii) ( A - I)’ = (A’) - I.

(iii) If A and Care nonsingularp X p matrices then ( A C ) - ’ = C-’A- ’ .

(iv) det(A-’)=(det A ) - ’ .

(v) If A is an orthogonal matrix, A - ’ = A‘.

580 Some Mutnx Theory

(vi) If A = diag( u , I,. . . , upp ) with a,, # 0 ( i = 1,. . . , p ) then A - = diag( a;', . . . , u;;).

(vii) If T is an rn X m upper-triangular nonsingular matrix then T-I is upper-triangular and its diagonal elements are I,; I, i = 1,. . . , in .

The following result is occasionally useful.

THEOREM A5 I . Let A and B be nonsingular p X p and y X y matrices, respectively, and let C be p X y and D be q X p. Put P = A + CBD. Then

Prooj Premultiplying the right side of ( I ) by P gives

( A + CBD) [ A - I - A - 'CB( B + BDA - ICB)-'BDA - I] =I-CB(E+ BDA-'CB)~~'EDA~'+CEDA-'

- CBDA- ICB( B + BDA - I c B ) - ' BDA - I

= I + CB [ E - I - ( I + DA - 'CB ) ( B + EDA - 'CB ) - I ] EDA -

=l+CB[B- I - B - l ( B C BDA-'CB)(B+BDA-'CO)-']BDA-'

= I ,

completing the proof.

matrix A in terms of the submatrices of A . The next theorem gives the elements of the inverse of a partitioned

THEOREM A5.2. Partition A and B as

Let A be a p X p nonsingular matrix, and let B = A-I.

whereA,, and B , , are k x k , A,, and B,, are k X ( p - k ) , A,, and B,, are ( p - k ) X k and A,, is ( p - k ) X ( p - k ) ; assume that A,, and A,, are nonsingular. Put

Inverse of a Mutrtx 581

Then

B , , =A,. ’ , , B,,=A,!,, B ,2= -A;1A1 ,A; ! ( ,

B z l = - A,?1,IA,f2.

Proof. The equation AB = I leads to the following equations:

From (6) we have BZl = - A ~ 1 A 2 1 B l l and substituting this in (4) gives A I I B , , - A , , A ~ ’ A , , B , , = I so that B , , = A;; !2 . From ( 5 ) we have B,, = - A ~ ’ A l , B 2 2 r which when substituted in (7) gives A,, B,, - A,,A,’A,,B,, = 1 so that BZ2 = A&’,,.

The determinant of a partitioned matrix is given in the following theorem.

THEOREM A5.3. Let A be partitioned as in ( I ) and let A , , . , and A, , I

be given by (3).

(a) If A,, is nonsingular then

det A =det A,,det

(b) If A , , is nonsingular then

det A =det A,,det A,,.,

Proof. To prove (a) note that if

then

502 Some Murnx Theory

(This was demonstrated in Theorem 1.2.10.) Hence

det(CAC’) =(det C)(det A)(det C’)=det A =det A , I . ,det A,, ,

where we have used Theorems A3.2 and A3.4. The proof of (6) is similar.

A6. R A N K OF A MATRIX

If A is a nonzero p X q matrix it is said to have rank r , written rank( A)= r , i f at least one of its r-square minors is different from zero while every ( r + I)-square minor (if any) is zero. If A =O it is said to have rank 0. Clearly if A is a nonsingular p X p matrix, rank(A)= p . The following properties can be readily established:

(i) rank( A)=rank( A’). (ii) If A is p X q, rank(A)smin( p , q).

(iii) If A is p X 4, B is q X r, then

rank( AB)lmin[rank( A),rank( B ) ] ,

(iv) I f A and B are p X q, then

rank( A + B)srank(A)+rank( B).

(v) I f A is P X P, B is p X (7, C is q X q, and A and C are nonsingular, then

rank( A BC) = rank( 8 ) .

(vi) If A is p X 4 and B is q X r such that AB =0, then

rank( B ) 5 q -rank(A).

A7. LATENT ROOTS A N D L A T E N T VECTORS

For a p X p matrix A the chamferistic equarion of A is given by

( 1 ) det( A - A I p ) = O .

The left side of ( I ) is a polynomial of degree p in h so that this equation has exactly p roots, called the latent roots (or characteristic roots or eigenvalues)

I-urent ROOIS md Latent Veclors 583

of A. These roots are not necessarily distinct and may be real, or complex, or both. If X i is a latent root of A then

det(A-X,l)=O

so that A - A l l is singular. Hence there is a nonzero vector x, such that ( A - A, I ) x , = 0, called a latent vector (or characteristic vector or eigenvector) of A corresponding to A,. The following three theorems summarize some very basic results about latent roots and vectors.

THEOREM A7.1. If B = CAC-', where A, B and C are all p X p, then A and B have the same latent roots.

Prooj Since

we have

det(E-AI)=detCdet(A-Al)detCL'=det(A-hl)

so that A and E have the same characteristic equation.

THEOREM A7.2. If A is a real symmetric matrix then its latent roots are all real.

Proof: Suppose that a + ifl is a complex latent root of A, and put

B = [ ( u + i p ) I - ~][(a-ip)l - A]=(*]- A > ~ +P'I .

E is real, and singular because (a + $ ) I - A is singular. Hence there is a nonzero real vector x such that B x = O and consequently

O=x'Ex =x'( al - A)'x + f12x'x

= x'( al - A)'( al - A ) x + PZX'X.

Since x'(a1- A) ' (a l - A)x>O and x'x>O we must have /3 =0, which means that no latent roots of A are complex.

THEOREM A7.3. If A is a real symmetric matrix and A, and A, are two distinct latent roots of A then the corresponding latent vectors x, and x, are orthogonal.

584 Some Mutrix Theory

Proot Since

Ax, = A , K , , Axj =A,xJr

i t follows that

x; Ax, = A,x:x,, x: Ax, = A,x:x,.

Hence ( A , - A,)x:x, ”0, so that x;x, =O. Some other properties of latent roots and vectors are now summarized.

(i) The latent roots of A and A’ are the same.

(ii) If A has latent roots A , , . , . , A p then A - k / has latent roots A , - k , . . . ,Ap - k and kA has latent roots kA, , . . ., kA,.

(iii) If A=diag(u, ,..., a , ) then al, ..., up are the latent roots of A and the vectors (l,O,. . . ,O) , (0, I , . . .,O),. . .,(O,O,. . ., 1) are associated latent vectors.

(iv) If A and R are p X p and A is nonsingular then the latent roots of A B and RA are the same.

(v) I f A, , ..., A, are the latent roots of the nonsingular matrix A then

(vi) If A is an orthogonal matrix ( A N = I ) then all its latent roots have absolute value I .

(vii) If A is symmetric i t is idempotent ( A 2 = A ) i f and only if its latent roots are 0’s and 1’s.

(viii) I f A isp X q the nonzero latent roots of AA’ and A’A are the same.

(ix) If T is triangular (upper or lower) then the latent roots of 7 are the diagonal elements.

(x) I f A has a latent root A of multiplicity r there exist r orthogonal latent vectors corresponding to A. The set of linear combinations of these vectors is called the lutent space corresponding to A. If A, and Aj are two different iatent roots their corresponding latent spaces arc orthogonal.

An expression for the characteristic polynomial p ( A ) = det( A - A Ip) can be obtained in terms of the principal minors of A. Let A,l , lz , , , , , lk be the k X k matrix formed from A by deleting all but rows and columns numbered I , , . . . , i , , and define the k th trace of A as

A - - l , ,..., Ap’are the latent rootsofA-I.

trk(A)= ~ l s , ,<12 . . .< l*~Pde t ,..., 1;

Posrrroe Definite Mutrices 585

The first trace (k = 1) is called the trace, denoted by tr(A), so that tr(A)=Zf= ,a , , . This function has the elementary properties that tr(A)= tr(A’) and if C is p X q, D is q X p then tr(CD)=tr(DC). Note also that tr,,,(A)=det(A). Using basic properties of determinants i t can be readily established that:

(xi) p(A)=det(A - XZ,)=C[=,(- A)’tr,-,(A) [tr,(A)= I]. Let A have latent roots A , , . . . , A p so that

p(A) = (-1)J’ 2 (A - hi). i = I

Expanding this product gives

where 5( A ,, . . . , A p ) denotes the j th elementary symmetric funclioti of h ,,..., A,, given by

(xii) p ( A )= Z[=,( - A )‘rp- k ( A . . ,A, 1.

r ( A , ,.. . , A p ) = 2 ~ l l h 2 . . . ~ l , *

I s i I< r l< . . . < r , S p

Equating coefficients of Ak in (xi) and (xii) shows that

(xiii) r k ( h l , . . . , A p ) = trk(A). It is worth noting that p ( h ) can also be written as

p ( X ) = ( - X)’det Adet(A-’ - A - I I )

P

k = Q =(-X)’detA z ( - A - l ) k t r p - k ( A - l )

and equating coefficients of hk here and in (xii) gives

(xiv) trk( A -‘)=det A - I trp-k( A) .

A8. POSITIVE DEFINITE MATRICES

A p X p symmetric matrix A is called positive (negative) definite if x’Ax>O (KO) for all vectors x f 0; this is commonly expressed as A >O ( A KO). It is called positive (negative) semidefinite if x ’ A x 2 O (SO) for all x Z 0 , written as A 20 ( S O ) . I t is called non-negutioe definite if A >O or A 20, i.e., if x’Ax>O for all x, and non-positive definite if A <O or A SO.

We now summarize some well-known properties about positive definite matrices.

586 Some Mufnx Theory

( i ) A is positive definite i f and only if det A l , > O for i = 1 ,..., p , where A,, , , , , , is the i X i matrix consisting of the first i rows and columns of A.

(ii) If A >O then A- '>O.

(iii) A symmetric matrix is positive definite (non-negative definite) if and only if all of its latent roots are positive (non-negative).

(iv) For any matrix B, BB'rO.

(v) If A is non-negative definite then A is nonsingular if and only if A >O.

(vi) I f A > O i s p X p a n d B i s q X p ( 9 S p ) o f r a n k r then BAB'>OiP r = q and BAB'ZO i f r C q.

(vii) I f A 10, B > O , A - B >O then B - - A - I > O and det A >det B. (viii) If A XI and B>O then det(A + B ) r d e t A +det B.

(ix) If A 1 0 and

whereA,, isa squarematrix, thenA,, >O and A , , - A,2A,1A,l >O.

A9. SOME M A T R I X FACTORIZATIONS

Before looking at matrix factorizations we recall the Gram-Schmidt orthogonalization process which enables us to construct an orthonormal basis of R"' given any other basis xI,x2, ..., xm of R". We define

Y I = X I

4 x 2 YiY,

Y;x, y;x, y3 =x3 - I Y 2 - --yt

Y2Y2 Y i Y I

yz =x2 -- --y1

......... ................. ..

and put z, =[ l/(y,'yi)'/2]yl, with i = 1,. .. ,m. Then z,,. . . ,z, form an orthonormal basis for Rm. Our first matrix factorization utilizes this process.

Some Matrix Factorizutrons 587

THEOREM A9.1. If A is a real m X m matrix with real latent roots then there exists an orthogonal matrix H such that H'AH is an upper-triangular matrix whose diagonal elements are the latent roots of A.

Let A,, ..., Am be the latent roots of A and let x t be a latent vector of A corresponding to A,. This is real since the latent roots are real. Let x 2 ,..., x, be any other vectors such that x l r x 2 ,..., x, form a basis for Rm. Using the Gram-Schmidt orthogonalization process, construct from x I , . . . , x, an orthonormal basis given as the columns of the orthogonal matrix HI, where the first column h, is proportional to x, , so that h, is also a latent vector of A corresponding to A, . Then the first column of AH, is Ah, = X,h,, and hence the first column of H i A H , ish,H;h,. Since this is the first column of A I H ; H l = A , I , , i t is (A,,O, ..., 0)'. Hence

Proot

where A , is (m - I ) X ( m - 1). Since

det(A - A I ) = ( A , -A)det(A, - A l )

and A and H ; A H , have the same latent roots, the lalent roots of A, are A 2 , ..., Am.

Now, using a construction similar to that above, find an orthogonal (m- l )X(m-1) matrix H2 whose first column is a latent vector of A, corresponding to A *. Then

where A, is ( m - 2 ) x ( m - 2 ) with latent roots h3 , . . . ,Xm.

ort hogonai matrix Repeating this procedure an additional m -3 times we now define the

and note that H A H is upper-triangular with diagonal elements equal to A , , . ..'A,,.

An immediate consequence of this theorem is given next.

588 Sonre Mutrrx Theory

THEOREM A9.2. if A is a real symmetric m X m matrix with latent roots A l , . . , , A m there exists an orthogonal rn X m matrix H such that

(2) H’AH = D = diag( A,, . . . , A m ) .

If H =[hl,. . . , h,] then Ir, is a latent vector of A corresponding to the latent root A,. Moreover, if Al, ..., An, are all distinct the representation (2) is unique up to sign changes in the first row of H.

Proof: As in the proof of Theorem A9.1 there exists an orthogonal m X m matrix H I such that

N;AN, = [: ::I- where A*, ..., A, are the latent roots of A , . Since H i A H , is symmetric i t follows that B , =O. Similarly each B, in the proof of Theorem A9.I is zero ( i = I , ..., m - I), and hence the matrix H given by ( I ) satisfies H‘AH- diag(Al, ..., A,,,). Consequently, A h , =A,h, so that 11, is a latent vector of A corresponding to the latent root A,. Now suppose that we also have Q’AQ= D for a orthogonal matrix Q. ‘Then PI)= DP with P = Q’If. If P =( p,,) i t follows that pIJA, = plJA, and, since A , # A,, p , , = O for i f J.

Since P is orthogonal i t must then have the form P = diag( * 1, -L 1 , . . . , -C I ) , and H = QP. THEOREM A9.3. If A is a non-negative definite m X m matrix then there exists a non-negntive definite m X nt matrix, written as such that A = ~ 1 / 2 ~ 1 / 2 .

Proof: Let H be an orthogonal matrix such that H’AH= D, where D=diag(A,, ..., A,) with A l , . . , , A m being the latent roots of A . Since A is non-negative definite, A , 2 0 for i = 1, . . . , m . Putting D112 = diag(A’/2,...,Alm/2), we have D t / 2 D 1 / 2 = D. Now define the matrix At’’’ by A’/* = HD1/211’. Then A ’ / , is non-negative definite and

~ 1 / 2 ~ 1 / 2 = I I D ~ / ~ H ’ ~ I D ~ / ~ I ~ ’ = H D I / ~ D V ~ ~ ~ ’ = H D H ‘ = A .

The term A’/’ in Theorem A9.3 is called a non-negative definite square root of A. If A is positive definite A ‘ / , is positive definite and is called the positive definite square root of A.

THEOREM A9.4. If A is an m X m non-negative definite matrix of rank r then :

(i) There exists an m X r matrix B of rank r such that A = BB’.

Some Mutrix Fuctoriturtons 589

(ii) There exists an m X m nonsingular matrix C such that

A = C [ '1 C'. 0 0

Pro05 As for statement (i), let D, =diag(A,,...,A,) where Al,.,.,Ar are the nonzero latent roots of A, and let H be an m X m orthogonal matrix such that H'AH = diag( A,, . . . ,A,,O,. . , ,O) . Partition H as H =[ H I : H 2 ] , where HI is m X r and H , is m X( m - r); then

Putting DI/' =diag(h'(2,...,X'/2), we then have

where B = H I D:/' is m X r of rank r.

columns are the columns of the matrix B in (i). Then As for statement (ii), let C be an m X ni nonsingular matrix whose first r

The following theorem, from Vinograd (1950). is used often in the text.

THEOREM A9.5. Suppose that A and B are real matrices, where A is k X in and B is k X n , with m I n . Then AA'= BB' if and only if there exists an m X n matrix H with HH'= I,,, such that A H = B.

Proo/. First suppose there exists an m X n matrix H with HH'= I,?# such

Now suppose that AA'= BB'. Let C be a k X k nonsingular matrix such that A H = B. Then BB'= AHH'A'= AA'.

that

[5 #' AA'= BB'= C

(Theorem A9.4), where rank (AA')= r. Now put D = C - ' A , E = C - ' B and

590 Some Murrrx 7heoy

partition these as

where D , is r X m , U, is ( k - r ) X m , E l is r X n , and E, is ( k - r ) X n . Then

and

which imply that E l E ; = D , D ; = I , and 0, =O, E, = O , so that

Now let E2 be an ( 1 1 - r ) X n matrix such that

is an n X n orthogonal matrix, and choose an ( n - r ) X m matrix 6, and an ( n - r ) X ( n - m ) matrix b3 such that

is an n X n orthogonal matrix. Then

and

Some Morrix Fucrorrzarinns 59 I

and hence

E = [ D : 01 fi'E = [ D : O ] Q ,

where Q = D'g is n X n orthogonal. Partitioning Q as

where His m x n and P is (n - m) x n, we then have HH' = I,,, and

c - 1 ~ = E = D H ~ c - 1 ~ ~

so that B = AH, completing the proof.

The next result is an immediate consequence of Theorem A9.5.

THEOREM A9.6. Let A be an n X m real matrix of rank m ( n r m ) . Then :

(i) A can be written as A = H I B, where H , is n X m with H ; H , = In, and B is m X m positive definite.

(ii) A can be written as

where H is n X n orthogonal and B is m X m positive definite.

Proof: As for statement (i), let B be the positive definite square root of the positive definite niatrix A'A (see Theorem A9.3), so that

A'A = B 2 = B'B.

By Theorem A9.5 A can be written as A = H,B, where H , is n X m with

As for statement (ii), let H I be the matrix in (i) such that A = H , B and choose an n X ( n - m ) matrix H2 so that H = [ H , : H 2 J is n X n orthogonal. Then

H ; H , = I,.

We now turn to decompositions of positive definite matrices in terms of triangular matrices.

592 Same Murrrx I’hearv

THEOREM A9.7. If A is an m X 1 ) ~ positive definite matrix then there exists a unique m X m upper-triangular matrix T with positive diagonal elements such that A = T’T.

An induction proof can easily be constructed. The stated result holds trivially for m = 1. Suppose the result holds for positive definite matrices of size m - I. Partition the m X m matrix A as

Proof:

where A , , is ( m - I ) X ( m - I). By the induction hypothesis therc exists a unique ( m - I )X(m - I ) upper-triangular matrix T , , with positive diagonal elements such that A , , = T;,T, , . Now suppose that

where x is ( m - 1)X 1 and y E HI. For this to hold we must have x = (Til)-1a,2, and thcn

- - I y 2 = a 2 2 - x ’ x = a 2 2 - a ’ , 2 T ; ; 1 ( T ~ , ) a12==a22 -a’,2A;;1a,2.

Note that this is positive by (ix) of Section A8, and the unique satisfying this i sy =(a22 -a’,2A,Iw,2)1/2.

Y’O

THEOREM A9.8. If A is an n X m real matrix of rank m ( t i 2 m ) then A can be uniquely written as A = HIT, where H I is n X m with H ; H , = l,,, and T is m X m upper-triangular with positive diagonal elements.

Since A’A is m X m positive definite it follows from Theorem A9.7 that there exists a unique m X m upper-triangular matrix with positive diagonal elements such that A’A = T’T. By Theorem A9.5 there exists an n X m matrix H I with H;tl, = I,,, such that A = /f,T. Note that HI is unique because T is unique and rank( T ) = M.

THEOREM A9.9. I f A is an m X M positive definite matrix and B is an m X m symmetric matrix there exists an m X m nonsingular matrix L such that A = LL’ and tl= LDL’, where D=diag(dl ,..., d,,,), with d, ,..., d,,, being the latent roots of A-IB. If B is positive definite and Jlr. . . ,d, , , are all distinct, L is unique. up to sign changes in the first row of L.

Let A ’ / * be the positive definite square root of A (see Theorem A9.3). so that A = A 1 / 2 A 1 / 2 . By Theorern A9.2 there exists an m X m

Prooj:

Proof:

Some Mumx Fuctorrzurtotrs 593

orthogonal matrix H such that A - ’ / 2 B A - ‘ / 2 = HDH’, where D = diag(d,, ,..,d,,,). Putting L = A’/’H, we now have LL‘= A and B = LDL‘. Note that d ,,..., d, are the latent roots of A-IB.

Now suppose that B is positive definite and the d, are all distinct. Assume that as well as A = LL’ and B = LDL’ we also have A = MM’ and B = MDM’, where M is m X m nonsingular. Then (M- ‘L)( M - ‘L)’=

M - ‘L is orthogonal and QD = DQ. If Q =(9, , ) we then have q,,d, = q,,d, so that q,, =O for i # j . Since Q is orthogonal it must then have the form Q =diag(? I , 2 1,. . ., 2 l), and L = MQ.

THEOREM A9.10. If A is an m X n real matrix (m 5 n ) there exist an m X m orthogonal matrix H and an n X n orthogonal matrix Q such that

M- ‘LLfM- I / = M- ‘ A M - It = M- ‘MM’M’ - I - - 1, so that the matrix Q =

where d, 2 0 for i = 1,. . . , m and d:,. . .,d:, are the latent roots of A X .

Let H be an orthogonal m X m matrix such that AA’= H’D2N, where D 2 =diag(d: ,..., d i ) , with 6,220 for i = 1 ,..., m because AA‘ is non-negative definite. Let D =diag(d,, . . . ,dm) with d, 20 for i = 1,. . . ,111;

then AA’=(H‘D)(H’D)’, and by Theorem A9.5 there exists an m X n matrix Qt with Q,Q; = 1, such that A = H’DQ,. Choose an ( n - m ) X n matrix Q2 so that the n X n matrix

Proo/.

Q = [ Q! Q2

is orthogonal; we now have

A = H’DQ, = H ’ [ D : O ] Q

so that HAQ’=[D:O], and the proof is complete, The final result given here is not a factorization theorem but gives a

representation for a proper orthogonal matrix H (i.e., det H = I ) in terms of a skew-symmetric matrix. The result is used in Theorem 9.5.2.

THEOREM A9. I 1. If H is a proper m X m orthogonal matrix (det H = 1) then there exists an m X m skew-symmetric U such that

1 1 2! H =exp(U) zz I + I/ + - U * + 3 U 3 + . . - .

594 Some Murrix Theory

Proo/. Suppose H is a proper orthogonal matrix of odd size, say, m =2k f 1; it can then be expressed as

H = Q ’

cos8, -sine,

sin 8 , cos 8, 0

cos -sin 8,

sin 8, cos 8, 0

0

0 0 0 0 I . .

0

0 0 0

0

cos 0, -sin 8, 0 sin 8, cos 8, 0 0 0 1

Q,

where Q is rn X m orthogonal and -. n < 8, I n , with i = 1 , . . . , k (see, e.g., Bellman, 1970,p.65). (I f m = 2 k , the last row and column are deleted.) Putting

o - 8 , 0 8, 0 0 0

o -e2 0 82 0 0

o -ek o ek o o

0

0 0 0 0 ’ * . 0 0 0

we then have 0 = exp( Q ’ H Q ) = exp( U ), where U = Q’HQ is skew-symmetric.

Tab

k9.

xz

adju

stm

ents

to W

iks

like

lihoo

d ra

tio s

tatis

tic W

: fa

ctor

c fo

r up

pet

perc

entil

es o

f - N

log W (s

ee S

ectio

n 10

.53)

‘ m=

3

r=3

r

=4

n

0.

100

0.05

0 0.

025

0.01

0 0.

005

0.10

0 00

50

0.02

5 0.

010

0.00

5

I 1.

322

1.35

9 1.

394

M 1.

437

1.46

8 2

1.12

7 1.

140

1.15

3 3

1.07

1 1.

077

1.08

4 4

1.04

5 1.

049

1.05

3 5

1.03

2 1.

035

1.03

7

6 1.

023

1.02

6 1.

028

7 1.

018

1.02

0 1.

021

8 1.

014

1.01

6 1.

017

9 I0

12

1.01

3 1.

014

10

1.01

0 1.

011

1.01

1

I2

1.00

7 1.

008

1.00

8 14

1.

005

1.00

6 1.

006

16

1.00

4 1.

005

1.00

5 18

1.

003

1.00

4 1.

004

20

1.00

3 1.

003

1003

24

1.00

2 10

02

1.00

2 30

1.

001

1001

1.

001

40

1001

1.

001

1.00

1 60

1.o0

0 1.

o00

1.ooo

12

0 1.O

Oo

1.o0

0 1.O

Oo

Q:

1.oo

o 1.O

Oo

1.oo

o

x:,,

14.6

837

16.9

190

19.0

228

1.16

8 1.

179

1.09

2 1.

098

I058

1.06

2 1.

041

1.04

3

I030

1.

032

1.02

3 1.

025

1.01

8 1.

019

1.01

5 1.

016

1.01

2 1.

013

1009

1.

009

1007

1.

007

I005

10

06

1004

1.

005

1004

10

04

1.00

2 1.

003

1.00

2 I0

02

1.00

1 1.

001

1.OOo

lo

00

1.O

Oo

lo0

0

1.ooo

Io

oo

21 6660

23.5

894

1.37

9 1.

422

1.46

3 1.

514

1.15

9 1.

174

1.18

8 1.

207

1.09

1 1.

099

1.10

7 1.

116

1.06

0 10

65

1.07

0 1.

076

1.04

3 10

46

1050

1.

054

1.03

2 10

35

1.03

7 I0

40

1.02

5 I0

27

1.02

9 I0

31

1.02

0 I 0

22

1.02

3 1.

025

1017

10

18

1.01

9 10

21

1.01

4 10

15

1.01

6 10

17

I010

1.

011

I012

I0

12

I008

1.00

8 10

09

1010

1.

006

1.00

7 1.

007

1.00

7 1.

005

1.W5

IOM

, 1.

006

1.00

4 1.

004

1.00

5 1.

005

1003

I0

03

1.00

3 10

04

1002

I0

02

1002

1.

002

1.00

1 10

01

1.00

1 10

01

loo0

10

01

1.00

1 1.

001

lo00

lo

00

1.o0

0 lo

00

lo00

lo

00

1.oo

o 1.o

00

18.5

494

21 0

261

23.3

367

26.2

170

1.55

0 1.

220

1.12

3 I .

080

I .05

7

1.04

2 I .

033

I .026

I .

022

1.01

8

1.01

3 1.

010

I .00

8 I .

006

I .005

I .004

I .

002

I .oO

l 1.

001

I .Ooo

I .ooo

28.2

995

~~

“Her

e. M

=num

ber

of v

aria

bles

; r =

hypo

thes

is d

egre

es of

fre

edom

: n - p

=er

ror

degr

ees o

f fr

eedo

m:

M=

n-p

-m+

l:a

nd

le

vel f

or -

[ n - p - i

(m - r

+ 1)]

1o

g~

leve

l fo

r xz

on m

r de

gree

s of

free

dom

C

=

Sour

ce:

Ada

pted

from

Sch

atzo

ff (1

966a

). P

ih an

d G

upta

(l%

9).

Lee

(19

72).

and

Dav

is (

1979

). w

ith

the

kin

d permission o

f th

e B

iom

etri

ka Trustees.

~ ~

~ ~~

~~

~

m =3 -

r=5

r=6

Q

0.100

0.050

0.02

5 0.010

0.005

0.100

0.050

0.025

0.01

0 0.005

1.482

1.535

1.586

1.649

1.694

1 1.433

2 1.191

3 1.113

4

1.076

5 1.055

6 1.042

7 1.033

8 1.027

9 1.022

10

1.019

12

1.014

14

1.010

16 1.008

18

1.007

20

1.00

6

24

1.00

4 30

1.003

40

1.002

60

1.001

120

1.ooo

00

1.OOo

1.481-

I .208

1.122

1.082

1.059

1.045

1.035

1.029

1.024

I .020

1.015

1.01 1

1.009

1.007

I .006

1.004

1.003

1.002

1.001

1 .O

oo

1 .Ooo

1.527~

1.224

1.131

1.08

7 1.063

1.048

1.038

1.030

1.025

1.021

1.015

1.012

1.009

1.008

1.00

6

I .005

1 .003

1.002

1.001

1 .Ooo

1 .Ooo

1.584

1.245

1.142

1.094

1.068

1.05 1

1.040

1.033

1.027

1.023

1.017

1.013

1.010

I .008

1.007

1 .m5

1.003

1.002

1.001

I .Ooo

1 .O

oo

1.625

1.260

1.15

0 1 .w

1.071

1.054

1.042

1 .OM

1.028

1.024

1.017

1.013

1.01 1

1.

009

1.007

1.005

1.003

1.002

1.001

1 .Ooo

1 .ooo

xf,,

22.3071

24.9958

27.4884 30.5779

32.8013

1.222

1.135

1.092

1.068

1.05

2 1.041

1 .OM

1.028

1.024

1.018

1.014

1.01 1

1.009

1.007

I .005

1.00

4 1.002

1.001

1 .Ooo

I .O

Oo

25.9894

1.241

1.145

1.099

1.072

1.05

6 I .0

44

1.036

1.030

1.02

5

1.019

1.014

1.012

1.009

1.008

1.006

1.004

1.002

1.001

1 .Ooo

1 .Ooo

28.8693

1.259

1.155

1.10

5 I .077

1.059

1.047

1.038

1.032

1.027

1.020

1.015

1.012

1.010

1.008

1.006

1.004

1.002

1.001

1 .Ooo

1 .O

oo

3 1.5

264

1.282

1.167

1.1 13

I .082

1.063

1.050

1.041

1.034

1.028

1.021

1.016

1.013

1.01 1

1.009

1.006

1.004

I .002

1.001

1 .m

1 .Ooo

34.8053

1.298

1.176

1.119

1.086

1.066

1.052

1.042

1.035

1.030

1.022

1.017

1.014

1.01 I

1.009

I .007

1.004

I .003

1.001

I .Ooo

1 .O

oo

37.1564

r=7

r=8

0.100

0.050

0.025

0.01

0 0.005

0.10

0 0.

050

0.025

0.010

0.005

\

1.585

1.64

0 1.708

1.758

1.572

1.632

1.690

1 1.529

2 1.251

3 1.156

4 1.

109

5 1.081

6

1.063

7 1.050

8 1.041

9 1.034

10

1.029

12

1.022

14

1.017

16

1.014

18

1.011

20

1.00

9

24

1.007

30

1.005

40

1.003

60

1.00

1 120

1.ooo

00

1.oo

o

1.272

1.168

1.116

1.086

1.067

1.053

I .044

1.037

1.03 1

1.023

1.018

1.014

1.012

1.010

I .007

I .005

I .003

I .001

1 .O

oo

1 .Ooo

1.292

1.178

1.123

1.091

1.070

I .056

1.04

6 1.038

1.033

1.024

1.019

1.015

1.012

1.010

1.008

1.005

1.003

1.001

1 .Ooo

I .Ooo

1.317

1.192

1.132

1.097

I .075

1.06

0 1.049

1.041

1.035

I .026

1.020

1.016

1.013

1.01 I

1.00

8 1.005

1.003

1.001

I .Ooo

1 .O

oo

1.335

1.202

1.138

1.102

1.078

1.062

1.05 1

1.043

1.036

1.027

1.021

1.017

1.014

1.01 1

I .00

8 I .0

06

I .003

1.002

1 .Ooo

I .O

oo

xs,,,

29.6151

32.6706

35.4789

38.9322

41.4011

1.280

1.177

1.125

I .094

1.073

1.059

1.049

1.041

I .035

1.026

1.02 1

1.017

1.014

1.01 1

I .00

8 1.

006

I .003

1.002

1 .Ooo

1 .O

oo

1.302

1.190

1.133

1.100

1.078

I .063

1.052

1.043

1.037

1.028

1.022

1.018

1.014

1.012

1.009

1.00

6 I .0

04

1.002

I .Ooo

1 .O

oo

1.324

1.201

1.141

1.105

1.082

1.066

1.054

1.04

6 1.039

1.029

1.02

3 1.018

1.015

1.013

1.009

1.00

6 !.0

04

1.002

I .Ooo

1 .O

oo

I .763

1.350

1.216

1.150

1.1 12

1.087

1.070

1.058

1.048

1.04

1

1.031

1.024

1.019

1.016

1.013

1.010

1.007

I .004

1.002

I .Ooo

1 .O

oo

1.816

1.370

1.227

1.157

1.117

I .09

1 1.073

I .060

1.050

1.043

1.032

1.025

1.020

1.017

1.014

1.01

1.007

I .00

4 1.002

1.001

1 .Ooo

33.1963

36.4151

39.3641

42.9798

45.5585

rn =

3 -

r=9

r=

10

0.

050

0.02

5 0.

010

0.00

5 0.

100

0.050

0.02

5 0.

010

0.00

5

1 1.

612

2 1.

307

3 1.

198

4 1.

141

5 1.

107

6 1.

084

7 1.

068

8 1.

057

9 1.

048

10

1.04

1

12

1.03

1 14

1.0

25

16

1.02

0 18

1.

016

20

1.01

4

24

1.01

0 30

1.

007

40

1.00

4 60

1.00

2 12

0 1.

001

M

1.Ooo

1.67

6 1.3

3 1

1.21

1 1.

150

1.11

3

1.08

9 1.

072

1.06

0 1.

05 1

1.

043

1.03

3 1.

026

1.02

1 1.

017

1.01

4

1.01

1 1.

007

1.00

4 1.

002

1.00

1

1 .Ooo

~ ~~

1.73

7 1.

354

1.22

4 1.

158

1.1 1

9

1.09

4 1.

076

1.06

3 1.

053

1.04

5

1 .OM

1.02

7 1

.02

1.

01 8

1.

015

1.01

1

1.00

8 1 .m

1.

002

1.00

1

1 .OOo

1.81

4 1.

382

1.24

0 1.

169

1.12

7

1.09

9 1.

080

1.06

7 1.

056

1.04

8

1.03

6 1.

028

1.02

3 1.

019

1.01

6

1.01

2 1.

008

1.00

5 1.

002

1.00

1

1 .Ooo

1.87

1 1.

403

1.25

1 1.

1’76

1.13

2

1.10

3 1.

084

1.06

9 1.

058

1.05

0

1.03

8 1.

030

1.02

4 1.

020

1.01

6

1.01

2 1.

008

1.00

5 1.

002

1.00

1

1 .Ooo

xlZm

36.7

412

40.1

133

43.1

945

46.9

629

49.6

449

1.65

0 1.

333

1.21

8 1.

157

1.12

0

1.09

5 1.

078

1.06

5 1.

055

1.04

7

1.03

6 1.

029

1 .OD

1.

019

1.01

6

1.01

2 1.

008

1.00

5 1.

002

1.00

1

1 .O

oo

1.71

6 1.

359

1.23

2 1.

167

1.12

7

1.10

1 1.

082

I .06

8 1.

058

1 .om

1.

038

1.03

0 1.

024

1.02

0 1.

017

1.01

2 1.

009

1.00

5 1.

002

1.00

1

1 .Ooo

1.78

1 1.

383

1.24

5 1.

175

1.13

3

1.10

6 1.

086

1.07

2 1.

061

1.05

2

I .040

1.03

1

1.02

5 1.

021

1.01

8

1.01

3 1.

009

1.00

5 1.

003

1.00

1

1 .Ooo

1.86

2 1.

413

1.26

2 1.

187

1.14

1

1.1 1

2 1.

091

1.07

5 1.

064

1.05

5

1.04

2 1.

033

1.02

7 1.

022

1.01

9

1.01

4 1.

009

1.00

6 1.

003

1.00

1

1 .Ooo

1.92

1 1.

435

1.27

4 1.

195

1.14

7

1.1 16

1.09

4 1.

078

1.06

6 1.

057

1.04

3 1.

034

1.02

8 1.

023

1.01

9

1.01

4 1.

010

I .006

1.00

3 1.

001

1 .Ooo

40.2

560

43.7

730

46.9

792

50.8

922

53.6

720

r=

ll

r=12

0.100

0.050

0.025

0.010

0.005

0.100

0.050

0.025

0.010

0.005

I 1.685

2 1.358

3 1.237

4 1.173

5 1.133

6 1.106

7 1.087

8 1.073

9 1.062

10

1.054

12

1.041

14

1.033

16 1.027

18

1.022

20

1.019

24

1.014

30

1.009

40

1.006

60

1.003

120

1.001

00

1.OOo

x:, 43.745

1.754

1.821

1.385

1.410

1.252

1.266

1.183

1.192

1.140

1.147

1.112

1.117

1.092

1.096

1.077

1.080

1.065

1.068

1.056

1.059

1.043

1.045

1.034

1.036

1.028

1.029

1.023

1.024

1.020

1.020

1.014

1.015

1.010

1.010

1.00

6 1.

006

1.003

1.003

1.001

1.001

1.m

1.OOo

47.400 50.725

1.907

1.969

1.442

1.466

1.284

1.297

1.204

1.213

1.156

1.162

1.124

1.128

1.101

1.105

1.084

1.087

1.072

1.074

1.062

1.06

4

1.047

1-049

1.037

1.039

1.030

1.031

1.025

1.026

1.021

1.022

1.016

1.016

1.011

1.011

1.007

1.007

1.003

1.003

1.001

1.001

1.OOo

1.OOo

54.776 57.648

1.718

1.382

1.256

1.188

1.146

1.117

1.097

1.08 1

1 .069

1.06

0

1.04

6 I .037

1.030

1.025

I .02 1

1.016

1.011

1.007

1.003

1.001

1 .OOo

1.791

1.410

1.272

1.199

1.154

1.123

1.101

1.085

1.073

1.063

,048

,039

.032

.026

.022

.017

1.01 1

1.007

1.003

1 .001

1 .OOo

1.86

0 1.437

1.287

1.209

1.161

1.129

1.106

I .089

1.076

1.066

1.050

1.04

0 1.033

1.027

1.023

1.017

1.012

I .007

1.004

1.001

1 .ooo

1.949

1.410

1.30

6 1.22 1

1.170

1.136

1.11

1 1.093

1 .080

1.069

1.053

1.042

1.034

1.029

1.024

1.018

1.012

I .008

1.004

1.001

1 .Ooo

2.013

1.495

1.319

I .230

1.176

1.141

1.115

I .097

1.082

1.07 1

1.05

4 I .043

1.035

1.029

1.025

1.019

1.013

1.008

1.00

4 1.001

1 .Ooo

47.2122

50.9985

54.4373

58.6192

61.5812

1 1.

750

1.82

4 1.

8%

1.98

8 2.

055

2 1.

405

1.43

4 1.

462

1.49

7 1.

522

3 1.

274

1.29

1 1.

306

1.32

6 1.

340

4 1.

203

1.21

4 1.

225

1.23

8 1.

247

5 1.

158

1.16

7 1.

174

1.18

4 1.

191

6 1.

128

1.13

4 1,

140

1.14

8 1.

153

7 1.

106

1.11

1 1.

116

1.12

2 1.

126

8 1.

089

1.09

4 1.

098

1.10

2 1.

106

9 1.

076

1.08

0 1.

083

1.08

8 1.

090

10

1.06

6 1.

069

1.07

2 1.

076

1.07

8

I2

1.05

2 1.

054

1.05

6 1.

059

1.06

1 14

1.

041

1.04

3 1.

045

1.04

7 1.

048

16

1.03

4 1.

035

1.03

7 1.

038

1.04

0 18

1.

028

1.02

9 1.

031

1.03

2 1.

033

20

1.02

4 1.

025

1.02

6 1.

027

1.02

8

24

1.01

8 1.

019

1.01

9 1.

020

1.02

1 30

1.

012

1.01

3 1.

013

1.01

4 1.

014

40

1.00

8 1.

008

1.00

8 1.

009

1.00

9 60

1.00

4 1.

004

1.00

4 1.

004

1.00

4 12

0 1.

001

1.00

1 1.

001

1.00

1 1.

001

x

1.O

oo

1.OOo

1.O

Oo

1.OOo

1.O

Oo

x;",

50.6

60

54.5

72

58.1

20

62.4

28

65.4

76

1.42

7 1.

292

1.21

7 1.

171

1.13

8 1.

115

1.09

7 1 .ow

1.07

3

1.05

7 1.

046

1.03

7 1.

03 1

1.02

7

1.02

0 1.

014

1.00

9 1.

004

1.00

1

1 .M

)o

54.0

902

r=1

3

r=l4

a

0.10

0 0.

050

0.02

5 0.

010

0.00

5 0.

100

0.050

0.02

5 0.

010

0.W5

2.02

6 2.

095

1.78

0 1.

857

1.93

1 1.

458

1.30

9 1.

229

1.17

9

1.14

5 1.

121

1.10

2 1.

088

1.07

6

1.05

9 1.

048

1.03

9 1.

033

1.02

8

1.02

1 1.

015

1.00

9 1.

004

1 .00

1

1 .OOo

58.1

240

1.48

6 1.

326

1.24

0 1.

188

1.15

2 1.

126

1.10

6 1.

091

1.07

9

1.06

1 1.

049

1.04

1 1.

034

1.02

9

1.02

2 1.

015

1.00

9 1.

005

1.00

1

1 .Ooo

6 1.

7768

1.52

3 1.

346

1.25

4 1.

198

1.15

9 1.

132

1.11

1 1.

095

1.08

2

1.06

4 1.

052

I .04

2 1.

035

1.03

0

1.02

3 1.

016

1.01

0 1.

005

I .00

1

1 .Ooo

66.2

062

1.54

9 1.

361

1.26

4 1.

205

1.16

5 1.

136

1.1

15

1.09

9 1 .OM

1.

066

1.05

3 1.

044

1.03

6 1.

03 1

1.02

3 1.

016

1.01

0 1.

005

1.00

1

1 .Ooo

69.3

360

r=I5

r=

16

a

0.10

0 0.

050

0.05

0 0.

010

0.00

5 0.

100

0.05

0 0.

025

0.01

0 0.

005

I 1.

808

1.88

7 1.

964

2.06

1 2.

133

1.33

5 1.

995

2.095

2

1.44

9 3

1.30

9 4

1.23

2 5

1.183

6 1.

149

7 1.

124

8 1.

105

9 1.

091

10

1.07

9

12

1.06

2 14

1.

050

16

1.04

1 18

1.

035

20

1.03

0

24

1.02

2 30

1.

016

40

1.01

0 60

1.00

5 12

0 1.

001

00

1.OOo

x;, 57

.505

1.48

0 I .3

27

1.24

4 1.

192

1.15

6 1.

130

1.1 1

0 1.

095

1.08

3

1.06

5 I .0

52

1.04

3 I .0

36

1.03

1

I .023

1.

016

1.01

0 1.

005

1.00

1

I .Ooo

6 1.6

56

1.51

0 1.

344

1.25

6 I .z

oo

1.16

3 1.

135

1.11

5 I .099

I .08

6

1.06

7 1.

054

1.04

5 1.

037

1.03

2

1.02

4 1.

017

1.01

0 1.

005

1.00

1

1 .Ooo

65.4

10

1.54

7 1.

365

1.27

0 1.

211

1.17

1 1.

142

1.12

0 1.

103

1.09

0

I .07

0 I .0

56

I .04

7 1.

039

1.03

3

1.02

5 1.

017

1.01

1

I .00

5 1.

001

1 .Ooo

69.9

57

1.57

5 1.3

81

I .280

1.

218

1.17

7 1.

147

1.12

4 1.

107

1.093

1.07

2 1.

058

1.04

8 I .040

1.03

4

I .026

1.

018

1.01

I 1.

005

1.00

2

1 .OOo

73

.166

1.46

9 1.

325

I .245

1.1

95

1.15

9 1.1

33

1.1 1

4 1.

098

1.08

5

1.06

7 1.

054

1.04

5 1.

038

1.03

2

I .025

1.01

7 1.

01 I

1.00

5 1.

002

1 .Ooo

1.916

1.

501

1.34

4 I .2

58

1.20

4

1.16

7 1.

139

1.11

9 1.

102

I .089

1.07

0 1.

057

1.04

7 1.

039

1.03

4

1.02

6 1.

018

1.01

1 1.

006

1.00

2

I .Ooo

1.532

I .3

62

1.27

1 1.

213

1.17

4 1.

145

1.123

1.

106

1.09

2

1.01

3 1.0

59

1.049

1.

041

1.035

1.026

1.0

18

1.01

1

1.00

6 1.0

02

I .Ooo

1.57

1 1.

384

1.28

5 1.

224

1.18

2 1.

152

1.12

9 1.

111

1.09

7

1.07

6 1.

061

1.05

1 I .0

43

1.03

6

I .027

1.

019

1.01

2 I .0

06

1.00

2

1 .Ooo

2.1%

1.

599

I .400

I .2

%

1.23

2

1.18

8 1.

157

1.13

3 1.1

15

1 .ow

1.07

8 1.

063

I .05

2 I .044

1.03

7

1.02

8 1.

020

1.01

2 1.

006

1.00

2

1 .Ooo

60.9066

65.1

708

69.0

226

73.6

826

76.%

88

Tab

le 9

(C

ontin

ued)

-

m=

3 -

1x

17

r=

18

a

0.10

0 0.

050

0.02

5 0.

010

0.00

5 0.

100

0.05

0 0.

025

0.01

0 0.

005

1 1.

861

1.94

4 2.

025

2.12

7 2.

203

1.88

6 1.

971

2.05

3 2.

158

2.23

5 2

1.48

9 1.

522

1.55

4 1.

594

1.62

3 1.

508

1.54

2 1.

575

1.61

6 1.

646

3 1.

341

1.36

1 1.

379

1.40

2 1.

419

1.35

7 1.

377

1.3%

1.

420

1.43

7 4

1.25

9 1.

273

1.28

5 1.

300

1.31

2 1.

272

1.28

6 1.

299

1.31

5 1.

327

5 1.

206

1.21

6 1.2

25

1.23

7 1.

245

1.21

8 1.

228

1.23

8 1.

249

1.25

8

6 1.

169

1.17

7 1.

184

1.19

3 1.

200

1.17

9 1.

188

1.19

5 1.

204

1.21

1 7

1.14

2 1.

149

1.15

4 1.

162

1.16

7 1.

151

1.15

8 1.

164

1.17

1 1.

177

8 1.

122

1.12

7 1.

132

1.13

8 1.

142

1.12

9 1.

135

1.14

0 1.

146

1.15

1 9

1.10

5 1.

110

1.11

4 1.

119

1.12

3 1.

112

1.11

7 1.

121

1.12

7 1.

130

10

1.09

2 1.

096

1.10

0 1.

104

1.10

7 1.

099

1.10

3 1.

107

1.i

ll

1.11

4

12

1.07

3 1.

076

1.07

9 1.

082

1.08

4 1.

078

1.08

1 1.

084

1.08

7 1.

090

14

1.05

9 1.

061

1.06

4 1.

066

1.06

8 1.

064

1.06

6 16

1.

049

1.05

1 1.

053

1.05

5 1.

056

1.05

3 1.

055

18

1.04

1 1.

043

1.04

4 1.

046

1.04

7 1.

045

1.04

6 20

1.

035

1.03

7 1.

038

1.04

0 1.

041

1.03

8 1.

040

24

1.02

7 1.

028

1.02

9 1.

030

1.03

1 1.

029

1.03

0 30

1.

019

1.02

0 1.

020

1.02

1 1.

022

1.02

1 1.

021

.068

1.

071

.057

1.

059

.048

1.

050

.MI

1.04

3

.031

1.

032

.022

1.02

3

.073

.0

6 1

.05 1

.0

44

.033

.0

23

40

1.01

2 1.

012

1.01

3 1.

013

1.01

3 1.

013

1.01

3 1.

014

1.01

4 1.

015

60

1.00

6 1.

006

1.00

6 1.

006

1.00

7 1.

006

1.00

7 1.

007

1.00

7 1.

007

120

1.00

2 1.

002

1.00

2 1.

002

1.00

2 1.

002

1.00

2 1.

002

1.00

2 1.

002

00

1.Ooo

1.O

Oo

1.Ooo

1.O

Oo

1.Ooo

1.O

Oo

1.OOo

1.m

1.O

Oo

l.m

x:, 64

.295

68

.669

72

.616

77

.386

80

.747

67

.672

8 72

.153

2 76

.192

0 81

.068

8 84

.501

9

r=19

r =20

0.100

0.05

0 0.025

0.010

0.005

0.100

0.050

0.025

0.010

0.005

1 1.909

2 1.526

3 1.372

4 1.285

5 1.229

6 1.189

7 1.

160

8 1.137

9 1.119

10

1.10

5

12

1.084

14

1.068

16 1.057

18

1.048

20

1.041

24

1.032

30

1.022

40

1.014

60

1.007

120

1.002

M

1.OOo

xf,

71.040

1.9%

1.56 1

1.393

1.300

I .240

1.198

1.167

1.143

1.124

1.109

I .087

I .07 1

1.059

1.050

1.043

1.033

I .023

1.015

1.007

1.002

I .ooo

75.624

2.080

1.595

1.412

1.313

1.250

1.205

1.173

1.148

1.129

1.1 13

1.090

1.073

1.061

1.052

1.04

4

1.034

1.024

1.015

1.008

1.002

1 .Ooo

79.752

2.188

1.637

1.437

1.330

I .262

1.215

1.181

1.155

1.134

1.118

1.093

1.076

1.063

1.054

1.04

6

1.035

1 .O25

1.016

1.008

1.002

I .Ooo

84.733

2.261

1.668

1.454

1.341

1.271

1.222

1.186

1.159

1.138

1.121

1 .O%

1.078

1.065

1.055

1.047

1.036

1.025

1.016

1.008

1.002

I .Ooo

88.236

1.932

1.54

4 1.387

1.298

1.240

1.199

1.168

1.145

1.127

1.112

1.089

1.073

1.061

I .052

1.04

4

1.034

1.024

1.015

I .008

1.002

1 .Ooo

2.021

1.580

1 .a8

1.313

1.251

1.208

1.176

1.151

1.132

1.1 I6

I .092

I .075

1.063

1.053

1.04

6

1.035

I .025

1.016

1.008

1.002

I .Ooo

2.106

1.614

1.428

1.327

1.26 1

1.216

1.182

1.157

1.136

1.120

I .095

1.078

I .065

1.055

1.048

1.036

1.026

1.016

1.008

1.002

1 .O

oo

2.216

1.657

1.453

I .344

1.274

1.226

1.190

1.163

1.142

1.125

1.099

1.08 I

1.067

1.057

I .049

1.038

1.027

1.017

1.009

1.002

I .ooo

2.297

1.689

1.472

1.356

1.283

1.233

1.1%

1.168

1.146

1.128

1.102

1.083

1.069

1.059

1.05

0

1.039

1.027

1.017

1.009

1.003

1 .O

oo

74.3970

79.0819

83.2976

88.3794

91.9517

Tab

le9

(Can

linrr

ed)

m =

3 -

r =

21

r=2

2

a 0.

100

0.05

0 0.

025

0.01

0 0.

005

0.10

0 0.

050

0.025

0.01

0 0.

005

1 1.

954

2.04

4 2.

131

2.24

3 2.

325

1.97

5 2.

067

2.15

6 2

1.56

1 3

1.40

1 4

1.31

0 5

1.25

0

6 1.

208

7 1.

177

8 1.

153

9 1.

133

10

1.11

8

12

1.09

4 14

1.

077

16

1.06

5 18

1.

055

20

1.04

8

24

1.03

6 30

1.

026

40

1.01

6 60

1.00

8 12

0 1.

002

30

1.OOo

xf,,

77.7

45

1.59

8 1.

423

1.32

5 1.

262

1.21

7 1.

184

1.15

9 1.

139

1.12

2

1.09

8 1.

080

1.06

7 1.

057

1.04

9

1.03

8 I .0

27

1.01

7 1.

009

1.00

2

i .OOo

82.5

29

1.63

3 1.

677

1.44

4 1.

470

1.34

0 1.

357

1.273

1.

286

1.22

6 1.

236

1.19

1 1.

200

1.16

5 1.

172

1.14

4 1.

150

1.12

7 1.

132

1.10

1 1.

105

1.08

3 1.

086

1.06

9 1.

072

1.05

9 1.

061

1.05

1 1.

053

1.03

9 1.

040

1.02

8 1.

029

1.01

8 1.

018

1.00

9 1.

009

1.00

3 1.

003

1.ooo

1.O

Oo

86.8

30

92.0

10

1.70

9 1.

488

1.37

0 1.

295

1.24

3 1.2

05

1.17

6 1.

154

1.13

5

1.10

8 1.

088

1.07

4 1.

063

1.05

4

1.04

1 1.

029

1.01

9 1.

009

1.00

3

1 .OOo

95.6

49

1.578

1.

415

1.32

2 1.

261

1.21

8 1.

185

1.16

0 1.

142

1.12

4

1.09

9 1.

082

1.06

9 1.

059

1.05

1

1.03

9 1.

028

1.01

8 1.

009

1.00

3

1 .Ooo

1.61

6 1.

651

1.43

8 1.

459

1.33

8 1.

353

1.27

3 1.

284

1.22

7 1.

236

1.19

3 1.

200

1.16

7 1.

173

1.14

7 1.

151

1.12

9 1.

133

1.10

3 1.

106

1.08

5 1.

087

1.07

1 1.

073

1.06

1 1.

063

1.05

2 1.

054

1.04

0 1.

041

1.02

9 1.

030

1.01

8 1.

019

1.00

9 1.

010

1.00

3 1.

003

1.Ooo

1.O

oo

2.26

9 2.

353

1.6%

1.

729

1.48

5 1.

514

1.37

1 1.

384

1.29

7 1.

307

1.24

6 1.

254

1.20

9 1.

213

1.18

0 1.

183

1.15

7 1.

161

1.13

9 1.

141

1.11

0 1.

115

1.09

1 1.

093

1.07

6 1.

078

1.06

5 1.

065

1.05

6 1.

057

1.04

3 1.

044

1.03

1 1.

031

1.02

0 1.

020

1.01

0 1.

010

1.00

3 1.

0oO

1.Ooo

1.O

oo

81.0

855

85.9

649

90.3

489

95.6

257

99.3

36

m =

4 -

r =

4

r =5

0.

100

0.05

0 0.

025

0.01

0 0.

005

0.10

0 0.

050

0.02

5 0.

010

0.00

5

I 1.

405

2 1.

178

3 1.

105

4 1.

071

5 1.

051

6 1.

039

7 1.

031

8 1.

025

9 1.

020

10

1.01

7

12

1.01

3 14

1.

010

16

1.00

8 18

1.

006

20

1.00

5

24

1.00

4 30

1.

002

40

1.00

1 60

1.

001

120

1.O

oo

DTJ

1.OOo

1.45

1 1.

194

1.11

4 1.

076

1.05

5

1.04

2 1.

033

1.02

7 I .

022

1.01

8

1.01

4 1.

010

I .00

8 1.

007

I .006

I .004

1.

003

1.00

2 1.

001

1 .Ooo

1 .Ooo

1 .49

4 I .2

09

1.12

2 I .0

8 1

1.05

8

I .044

1.03

5 1.

028

I .023

1.

019

1.01

4 1.0

1 I

1.00

9 1.

007

1.00

6

I .004

I .003

1.

002

1.00

1 1 .O

oo

I .Ooo

IS50

I .2

29

1.13

2 I .0

88

1.06

3

1.04

8 1.

037

1.03

0 I .0

25

1.02

1

1.01

5 1.

012

1.00

9 1.

008

1.00

6

I .004

I .0

03

I .002

1.

001

I .Ooo

I .Ooo

1.58

9 I .2

43

1.13

9 1.

092

1.06

6

1 .OM

1.03

9 1.

032

1.02

6 1.

022

1.01

6 1.

012

1.01

0 I .008

1.00

7

I .005

1.

003

1.00

2 1.

001

1 .Ooo

I .Ooo

~5

, 23.

5418

26

.296

2 28

.845

4 31

.999

9 34

.267

2

I .43

5 1.

199

1.12

1 1.

083

1.06

1

1.04

7 1.

037

1.03

0 I .

025

1.02

1

1.01

6 1.

01 2

1.01

0 1.

008

1.00

7

I .005

I .0

03

1.00

2 1 .

001

1 .Ooo

I .O

oo

1 .48

3 1.

216

1.13

0 1.

089

I .06

5

1.05

0 1.

040

1.03

2 1.

027

1.02

3

1.01

7 1.

013

I .ow

1.00

8 1.

007

1.00

5 I .

003

1.00

2 I .

001

1 .Ooo

I .OOo

1.53

0 1.

233

1.13

9 1.

094

1.06

9

1.05

3 I .

042

1.03

4 1.

028

1.02

4

1.01

8 1.

014

1.01

1 1.

009

1.00

7

I .005

1.

004

1.00

2 1.

001

1 .Ooo

1 .m

1.58

9 1.

253

1.15

0 1.

101

I .07

4

I ,05

6 1.

044

1.03

6 1.

030

I .02

5

1.01

9 1.

014

1.01

2 1.

009

1.00

8

I .006

I .00

4 1.

002

1.00

1 1 .O

Oo

1 .OOo

I .63

2 1.

269

1.15

8 1.

106

I .077

I .05

9 1.

046

1.03

8 I .

03 I

1.02

6

1.02

0 1.

015

1.01

2 1.

010

1.00

8

1.00

6 1.

004

I .00

2 1.

001

1 .OOo

I .O

Oo

28.4

120

3 1.

4104

34

.16%

37

.566

2 29

.996

8

m =

4 -

1 =

6 r

=7

0.02

5 0.

010

0.00

5 0.

100

0.050

0.02

5 0.

010

0.00

5

1 1.

466

2 1.

222

3 1.

138

4

1.0%

5

1.07

1

6 1.

055

7 1.

044

8 1.

036

9 1.

030

10

1.02

6

12

1.01

9 14

1.

015

16

1.01

2 18

1.

010

20

1.00

8

24

1.00

6 30

1.

004

40

1.00

2 60

1.

001

120

1.OOo

30

1.Ooo

1.51

7 1.

566

1.24

0 12

57

1.14

8 1.

157

1.10

2 1.

108

1.07

6 1.

080

1.05

9 1.

062

1.04

7 1.

049

1.03

8 1.

040

1.03

2 1.

034

1.02

7 1.

029

1.02

0 1.

021

1.01

6 1.

017

1.01

3 1.

013

1.01

0 1.

011

1.00

9 1.

009

1.00

6 1.

007

1.00

4 1.

004

1.00

2 1.

003

1.00

1 1.

001

1.O

oo

1.OOo

1.OOo

1.O

oo

1.62

8 1.

279

1.16

8 1.

1 15

1 .

085

1.06

6 1.

052

1.04

3 1.

036

1.03

0

1.02

3 1.

018

1.01

4 1.

012

1.01

0

I .007

I .005

I .003

1.00

1 1 .o

oo 1 .O

oo

1.67

4 1.

295

1.17

7 1.

121

1.08

9

1.06

8 1.

055

1.04

5 1.

037

I .032

1.02

4 1.

018

1.01

5 1.

012

1.01

0

1 .00

7 1.

005

1.00

3 1.

001

1 .Ooo

1 .Ooo

x$,

33.1

963

36.4

151

39.3

641

42.9

798

45.5

585

1.49

7 1.

550

1.24

4 1.

263

1.15

5 1.

165

1.10

9 1.

116

1.08

2 1.

087

1.06

4 1.

068

1.05

2 1.

055

1.04

3 1.

045

1.03

6 1.

038

1.03

1 1.

032

1.02

3 1.

024

1.01

8 1.

019

1.01

5 1.

015

1.01

2 1.

013

1.01

0 1.

011

1.00

7 1.

008

1.00

5 1.

005

1.00

3 1.

003

1.00

1 1.

001

1.OOo

1.O

Oo

1.m

1.O

oo

1.60

1 1.

281

1.17

5 1.

122

1.09

2

1.07

1

1.05

7 1.

047

1.04

0 1.

034

1.02

6 1.

020

1.01

6 1.

013

1.01

1

1.00

8 1.0

05

1.00

3 1.

002

1 .Ooo

1 .ooo

1.66

7 1.

305

1.18

8 1.

130

1.09

7

1.07

6 1.

061

1.05

0 1.

042

1.03

6

I .027

I .0

2 1

1.01

7 1.

014

1.01

2

1.00

8 1.0

06

1.00

3 1.

002

1 .O

oo

1 .Ooo

1.71

5 1.

322

1.19

7 1.

136

1.10

1

1.07

9 1.

063

1.05

2 1.

044

1.03

7

I .02

8 1.

022

1.01

7 1.

014

1.01

2

1.00

9 I .0

06

1.00

4 I .0

02

1 .Ooo

I .O

oo

37.9

159

41.3

372

44.4

607

18.2

782

50.9

933

r =8

r=9

a

0.10

0 1.

050

1.025

0.01

0 0.

005

0.100

0.050

0.025

0.010

0.00

5

1.583

1.636

1 2 3 4 5 6 7 8 9 10 12

14

16 18

20

24

30

40

60

120 a0

I .528-

1.266

1.172

1.123

1.093

1.074

I .060

1.050

1.04

2 1.036

1.027

1.02 1

1.107

I .Ol4

1.012

1.009

1.006

I .003

1.002

I .Ooo

1 .O

oo

1.286

1.183

1.130

I .099

1.078

1.063

1.052

1.04

4 1.038

1.029

I .023

1.10

8 1.

015

1.013

1.009

1.00

6 1.

004

1.002

1 .Ooo

1 .Ooo

1.305

1.193

1.137

1.103

1.08 1

I .066

1.055

1.04

6 I .039

1.030

1.023

1.109

1.016

1.013

1.010

1.007

1.00

4 I .002

1 .Ooo

1 .Ooo

1 .-704

1.330

1.207

1.146

1.109

1.086

1.070

1.058

1.048

1.041

1.03 1

I .025

1.

020

1.016

1.01

4

1.010

1.007

1.00

4 I .002

1.001

1 .Ooo

1.754

~ 1.557

1.348

1.216

1.152

1.1 I4

1.089

1.072

1.06

0 1.050

1.043

.033

.026

.02 1

.017

.014

.010

1.007

1.00

4 I .002

1.001

I .Ooo

x:,

42.5847

46.1943

49.4804

53.4858

56.3281

1.288

1.189

1.137

1. I05

1.083

1.068

1.05

7 1.048

1.041

I .032

1.025

1.02

0 1.017

1.014

1.010

1.007

1.00

4 I .002

1.001

I .Ooo

1.6 I4

1.309

1.201

1.144

1.1

10

I .08

8 I .07 I

1.06

0 1.

050

.043

1.033

1.026

1.02 1

1.018

1.01

5

1.01 1

1.007

1.00

4 I .0

02

1.001

1 .OO0

I .669

1.740

1.792

1.329

1.212

1.15

2 1.115

1.09 1

1.075

1.062

1.053

1.045

1.034

1.027

I .022

1.018

1.015

1.011

1.008

1.005

1.002

1.001

I .Ooo

1.355

1.226

1.161

1.122

I .O%

1.078

1.065

1.055

I .047

1.036

1.029

I .023

1.019

1.016

1.012

1.008

I .005

I .002

1.001

1 .Ooo

1.373

1.236

1.167

1.12

7

1.10

0 1 .o

s 1

1.068

1.057

1.049

1.037

1.02

9 1.

024

1.02

0 1.017

1.012

1.008

1.005

I .002

1.001

1 .Ooo

47.2122

50.9985

54.4373

58.6192

61.5812

Tab

le9

(Con

linue

d)

m =

4

r=lO

r=

ll

n

0.10

0 0.

050

0.02

5 0.

010

0.00

5 0.

100

0.05

0 0.

025

0.01

0 0.

005

1 2 3 4 5 6 7 8 9 10

12

14

16

18

20

24

30

40

60

1 20 m

Xr”

, 2

1.58

5 1.

309

1.20

6 1.

150

1.1 1

6

1.09

3 1.

076

1.06

4 1 .OM

1.03

7

1.03

6 1.

029

I .02

3 1.

019

1.01

6

1.01

2 I .

008

1.00

5 1.

002

1.00

1

I .Ooo

5 1

.805

0

~

I .644

1.33

1 1.

218

1.15

9 1.

122

1.09

7 1.

080

1.06

7 1.

057

1.04

9

1.03

8 1.

030

1.02

4 1.

020

1.01

7

1.01

3 1.

009

1.00

5 1.

003

1.00

1

1 .ooo

-55.

7585

1.70

1 1.

352

1.23

0 1.

166

1.12

8

1.10

2 I .

083

1.07

0 1.

059

1.05

1

1.03

9 1.

03 1

1.

025

1.02

1 1.

018

1.01

3 1.

009

1.00

5 1.

003

1.00

1

1 .Ooo

59.3

41 7

1.77

4 1.

379

1.24

4 1.

176

1.13

4

1.10

7 1.

088

1.07

3 1.

062

1.05

4

1.04

1 1.

033

I .026

1.

022

1.01

9

1.01

4 1.

009

1.00

6 1.

003

1.00

1

1 .Ooo

63

.690

7

1.82

8 1.

398

1.25

5 1.

183

1.13

9

1.11

1 1.

090

1.07

6 1.

064

1.05

5

1.04

2 1.

034

1.02

7 1.

023

1.01

9

1.01

4 1.

010

1.00

6 1.

003

1.00

1

1 .Ooo

66.7

659

-

1.33

0 1.

222

1.16

4 1.

127

1.10

3 1 .OM

1.

071

1.06

1 1.

053

1.04

1 1.

033

1.02

7 1.

022

1.01

9

1.01

4 1.

010

1.00

6 1.

003

1.00

1

1 .O

oo

56.3

69

-

1.35

2 1.

235

1.17

3 1.

134

1.10

7 1.

089

1.07

5 1.

064

1.05

5

1.04

3 1.

034

1.02

8 1.

023

1.02

0

1.01

5 1.

010

1.00

6 1.

003

1.00

1

1 .O

oo

60.4

8 1

-

1.37

4 1.

247

1.18

1 1.

140

1.1 1

2 1.

092

1.07

7 1.

066

I .057

1.04

4 1.

035

1.02

9 1.

024

1.02

0

1.01

5 1.

010

1.00

6 1.

003

1.00

1

1 .Ooo

64.2

01

-

1.40

2 1.

262

1.19

1 1.

147

1.11

8 1.

097

1.08

1 1.

069

1.06

0

1 .M

1.

037

1.03

0 1.

025

1.02

1

1.01

6 1.

01 1

1.00

7 1.

003

1.00

1

1 .Ooo

68

.710

-

1.42

2 1.

214

1.19

8 1.

152

1.12

2 1.

100

1 .OM

1.07

1 1.

062

1.04

7 1.

038

1.03

1 1.

026

1.02

2

1.01

6 1.

01 1

1.

007

1.00

3 1.

001

1 .Ooo

71.8

93

r=1

2

r=1

3

u

0.10

0 0.

050

0.02

5 0.

010

0.00

5 0.

100

0.05

0 0.

025

0.01

0 0.

005

1 2 3 4 5 6 7 8 9 10

12

14

16

18

20

24

30

40

60

I 20 oc

1.63

8 1.

350

1.23

8 1.

177

1.13

9

1.1

12

1.09

3 1.

079

1.06

8 1.

059

1.04

6 1.

037

1.03

0 1.

025

I .02

1

1.01

6 1.

01 I

1.00

7 1.

003

I .00

1

I .Ooo

1.70

0 1.

373

1.25

2 1.

186

1.14

5

1.11

8 I .

097

1.08

2 1.

070

1.06

1

1.04

7 1.

038

1.03

1 I .

026

1.02

2

1.01

7 1.

01 1

1.00

7 1.

003

I .001

1 .Ooo

1.76

0 I .

3%

1.26

4 .I

95

.I52

.I22

,1

01

.085

.0

73

.063

1.83

8 I .4

24

1.28

0 1.

205

1.15

9

1.12

8 1.

106

I .089

1.

076

1.06

6

1.89

5 1.

446

I .292

1.

213

1.16

5

1.13

2 1.

109

I .092

I .

079

1.06

8

I .049

1.

039

I .032

1.

027

1.02

3

1.01

7 1.

012

I .007

1.

004

1 .00

1

1 .Ooo

.05 1

.04 1

,0

33

.028

.0

24

.018

1.

012

1.00

8 1.

004

1.00

1

I .Ooo

,053

,0

42

.034

.0

29

.024

.018

1.

013

1.00

8 1.

004

1.00

1

I .OOo

xf

,, 60

.906

6 65

.170

8 69

.022

6 73

.682

6 76

.%88

-

1.36

9 1.

254

1.19

0 1.

150

1.12

2 I.

102

1.08

6 1.

074

I .065

,050

.0

4 1

.033

,0

28

,024

,018

1.

012

1.00

8 I .0

04

1.00

1

1 .OOo

-

1.39

3 1.

268

I .20

0 1.

157

1.12

7 1.

106

1.09

0 1.

077

1.06

7

I .05

2 I .

042

1.03

5 1.

029

1.02

5

1.01

9 1.

01 3

1.00

8 1.

004

1 .00

1

1 .Ooo

-

1.41

7 1.

281

1.20

9 1.

163

1.13

2 1.

110

I .09

3 1.

080

1.07

0

1.05

4 1.

044

1.03

6 1.

030

1.02

6

1.01

9 1.

013

1.00

8 1.

004

1 .001

1 .Ooo

-

1.44

6 1.

298

I .22

0 1.

171

1.13

9 1.

115

I .09

7 1.

083

1.07

3

1.05

6 I .

045

1.03

7 I .0

3 I

I .02

7

1.02

0 1.

014

1.00

8 1.

004

I .00

1

1 .Ooo

65.4

22

69.8

32

73.8

10

78.6

16

-

I .468

1.31

0 I .

228

1.17

7

1.14

3 1.

118

1.10

0 1.

086

1.07

5

1.05

8 1.

047

I .03

8 1.

032

1.02

7

1.02

0 1.

014

1.00

9 1.

004

1.00

1

1 .Ooo

82.0

01

f

0

Tab

le9

(Con

tin

ued

) -

~~

~

m =

4 -

r=1

4

r=1

5

Q

0.10

0 0.050

0.02

5 0.

010

0.00

5 0.

100

0.050

0.02

5 0.

010

0.00

5

1 2 3 4 5 6 7 8 9 10

12

14

16

18

20

24

30

40

60

120 co

I .686

I .3

88

1.26

9 1.

203

1.16

1

1.13

1 1.

110

1.09

4 1.

08 1

1.0

7 I

1 .OH

1.

045

1.03

7 I .0

3 1

1.02

6

1.02

0 1.

014

I .009

1.

004

1.00

1

1 .Ooo

1.75

1 1.

413

1.28

4 1.

213

1.16

8

1.13

7 1.

115

1.09

7 1.

084

1.07

3

1.05

8 I .

046

I .03

8 1.

032

1.02

7

1.02

1 1.

014

1.00

9 1.

004

1.00

1

1 .ooo

1.81

4 1.

436

1.29

7 1.

222

1.17

5

1.14

2 1.

119

1.10

1 1.

087

1.07

6

1.05

9 1.

048

1.03

9 1.

033

1.02

8

1.02

1 1.

015

1.00

9 1.

005

1.00

1

1 .Ooo

1.8%

1.

467

1.31

4 1.

234

1.18

3

1.14

9 1.

124

1.10

5 1.

09 1

1.07

9

1.06

2 1.

050

1.04

1 1.

034

I .02

9

1.02

2 1.

015

1.00

9 I .0

05

1.00

1

1 .ooo

1.95

6 I .

489

1.32

7 1.

242

1.18

9

1.15

4 1.

128

1.10

9 1.

093

1.08

1

1.06

4 1.

05 1

1.

042

1.03

5 1.

030

I .02

3 1.

016

1.01

0 1.

005

1.00

1

1 .ooo

x?,

69.9

185

74.4

683

78.5

671

83.5

134

86.9

937

-

1.40

6 1.

284

1.21

6 1.

172

1.14

1 1.1

18

1.10

1 1.

087

1.07

7

1 .06

0 1.

049

1 .oQo

1.03

4 1.

029

1.02

2 1.

015

I .01

0 1.

005

1.00

1

I .ooo

74.3

97

-

1.43

2 1.

299

1.22

6 1.

179

1.14

7 1.

123

1.10

5 1.0

9 1

1.08

0

1.06

3 1.0

5 1

I .042

1.

035

1.03

0

1.02

3 1.

016

1.01

0 1.

005

1.00

1

1 .Ooo

79.0

82

-

1.45

6 1.

313

1.23

6 1.

187

1.15

3 1.

128

1.10

9 1 .o

w

1.08

2

1.06

5 1.

052

1.04

3 I .

036

1.03

1

1.02

3 1.

016

1.01

0 1.

005

1.00

1

1 .ooo

83.2

98

-

1.48

8 1.

331

1.24

8 1.

195

1.15

9 1.

133

1.1 1

3 1.

098

1.08

5

1.06

7 1.

054

1.04

5 1.

038

1.03

2

1.02

4 1.

017

1.01

1 1.

005

1.00

1

1 .Ooo

83.3

79

-

1.51

1 1.

344

1.25

6 1.

202

1.16

4 1.

137

1.1 I

6

1.10

1 1.

088

1.06

9 1.

056

1.04

6 1.

039

1.03

3

1.02

5 1.

017

1.01

1

1.00

5 1.

001

1 .ooo

91.9

52

r=16

r

= I

7 0.

100

0.05

0 0.

025

0.01

0 0.

005

0.10

0 0.

050

0.02

5 0.

010

0.00

5

1 1.

731

2 1.

423

3 1.

299

4 1.

223

5 1.

182

6 1.

150

7 1.

127

8 1.

108

9 1.

094

10

1.08

3

12

1.06

5 14

1.

053

16

1.04

4 18

1.

037

20

1.03

2

24

1.02

4 30

1.

017

40

1.01

1 60

1.00

5 12

0 1.

001

03

1.OOo

1.79

9 I .

450

1.31

4 1.

239

1.19

0

1.15

7 1.

132

1.1

I3

1.09

8 I .0

86

I .06

8 I .

055

1.04

5 1.

038

1.03

3

1.02

5 1.

018

1.01

1 I .

00s

1.00

2

1 .Ooo

I .86

4 1.

475

1.32

9 1.

249

1.19

8

1.16

3 1.

136

1.11

7 1.

101

1.08

9

1.07

0 1.

056

I .047

1.

040

1.03

4

1.02

6 1.

018

1.01

1 I .006

1.00

2

I .Ooo

I .949

I S

O7

I .347

1.

26 1

I .207

1.16

9 1.

142

1.12

1 1.

105

1.09

2

1.07

3 1.

058

I .04

9 1.

041

1.03

5

1.02

7 1.

019

1.01

2 1.

006

I .002

1 .Ooo

2.0

12

1.53

1 1.

360

1.27

0 1.

213

1.17

4 1.

146

1.12

5 1.

108

1.09

4

,074

.0

6Q

,050

,0

42

,036

,027

1.

019

1.01

2 I .006

1.00

2

1 .Ooo

2

X,, 78

.859

7 83

.675

3 88

.004

0 93

.216

8 %

.878

1

-

-

-

-

-

1.44

0 1.

468

1.49

4 1.

527

1.55

1 1.

313

1.32

9 1.

344

1.36

3 1.

377

1.24

0 1.

252

1.26

2 1.

275

1.28

4 1.

193

1.20

1 1.

209

1.21

8 1.

225

1.16

0 1.

166

1.17

2 1.

180

1.18

5 1.

135

1.14

0 1.

145

1.15

1 1.

155

1.11

6 1.

120

1.12

4 1.

129

1.13

3 1.

101

1.10

5 1.

108

1.11

2 1.

115

1.08

9

1.07

0 1.

057

1.04

8 I .

040

1.03

5

I .026

,092

1.

095

1.09

8 1.

101

,073

1.

075

1.07

8 1.

080

,059

1.

061

1.06

3 1.

065

a9

1.

051

1.05

3 1.

054

,042

1.

043

1.04

5 1.

046

.036

1.

037

1.03

8 1.

039

.027

1.

028

1.02

9 1.

030

1.01

9 1.

019

1.02

0 1.

020

1.02

1 1.

012

1.01

2 1.

012

1.01

3 1.

013

1.00

6 1.

006

1.00

6 1.

006

1.00

7 1.

002

1.00

2 1.

002

1.00

2 1.

002

1.O

oo

1.OOo

1.O

Oo

1.00

0 1.O

Oo

83.3

08

88.2

50

92.6

89

98.0

28

101.

776

Tab

le 9

(C

onfi

nw

d)

m =

4 -

r=1

8

r=1

9

0.050

0.02

5 0.

010

0.005

0.10

0 0.

050

0.02

5 0.

010

0.00

5

1 1.

7%

2 1.

457

3 1.

327

4 1.

252

5 1.

203

6 1.

169

7 1.

143

8 1.

123

9 1.

107

10

1.09

5

12

1.07

5 14

1.

061

16

1,05

1 18

1.

044

20

1.03

7

24

1.02

9 30

1.

020

40

1.01

3 60

1.

006

120

1.00

2

ffi

1.O

Oo

xs,

87.7

431

~~ 1.84

3 1.

485

I .343

1.

264

1.21

2

1.17

6 1.

149

1.12

8 1.

1 11

1.09

8

1.07

8 I .

063

1.05

3 1.

045

1.03

9

1.03

0 1.

02 1

1.01

3 1.

007

1.00

2

1 .Ooo

92.8

083

1.91

1 1.

511

1.35

9 1.

274

1.22

0

1.18

2 1.

154

1.13

2 1.

1 15

1.

101

1.08

0 1.

065

1 .OM

1.04

6 1.

040

! .03

0 1.

022

1.01

4 1.

007

1.00

2

1 .ooo

97.3

53 1

! .999

1.54

5 1.

378

1.28

7 1.

230

1.18

9 1.

160

1.13

7 1.

1 I9

1.

105

1.08

3 1.

068

1.05

6 I .

048

1.04

1

1.03

1 1.

022

1.01

4 1.

007

1.00

2

1 .Ooo

I 0

2.8 1

6

1.06

5 1.

570

1.39

2 1.

297

1.23

7

1.19

5 1.

164

1.14

1 1.

122

1.10

8

1.08

5 1.

069

I .05

8 1.

049

1.04

2

1.03

2 1.

023

1.01

4 1.

007

I .00

2

1 .Ooo

106.

648

L

1.47

3 I .

340

1.26

4 1.

214

1.17

8 1.

151

1.13

0 1.

114

1.10

1

1.08

0 1.

066

1.05

5 1.

047

1.04

0

1.03

1 1.

022

1.01

4 1.

007

1.00

2

1 .Ooo

92

.166

-

-

-

-

1.50

2 1.

529

1.56

3 1.

588

1.35

7 1.

373

1.39

3 1.

408

1.27

6 1.

287

1.30

0 1.

310

1.22

3 1.

231

1.24

1 1.

248

1.18

5 1.

191

1.19

9 1.

205

1.15

7 1.

162

1.16

9 1.

173

1.13

5 1.

140

1.14

5 1.

149

1.11

8 1.

122

1.12

6 1.

130

1.10

4 1.

107

1.11

1 1.

114

1.08

3 1.

086

1.08

9 1.

091

1.06

8 1.

070

1.07

3 1.

074

1.05

7 1.

059

1.06

1 1.

062

1.04

8 1.

050

1.05

1 1.

053

1.04

2 1.

043

1.04

4 1.

045

1.03

2 1.

033

1.03

4 1.

035

1.02

3 1.

023

1.02

4 1.

025

1.01

4 1.

015

1.01

5 1.

015

1.00

7 1.

007

1.00

8 1.

008

1.00

2 1.

002

1.00

2 1.

002

1.OOo

1.

ooo

1.Ooo

1.

Ooo

97.3

51

101.

999

107.

583

11 1.

495

r =20

r=

21

Q

0.

100

0.05

0 0.

025

0.01

0 0.

005

0.10

0 0.

050

0.02

5 0.

010

0.00

5

1 1.

812

2 1.

488

3 1.

353

4 1.

275

5 1.

224

6 1.

187

7 1.

159

8 1.

138

9 1.

121

10

1.10

7

12

1.08

6 14

1.

070

16

1.05

9 18

1.

050

20

1.04

3

24

1.03

3 30

1.

024

40

1.01

5 60

1.

008

120

1.00

2

oc

1.OOo

1.88

4 1.

518

1.37

1 1.

288

1.23

3

1.19

4 1.

165

1.14

3 1.

125

1.1

10

,088

,072

.0

6 I

,052

.0

45

.034

1.

024

1.01

6 1.

008

1.00

2

1 .OOo

1.95

4 1.

545

1.38

7 1.

299

1.24

1

1.20

1 1.

170

1.14

7 1.

129

1.11

4

1.09

1 I .0

74

1.06

2 I .053

1.04

6

1.03

5 1.

025

1.01

6 1.

008

1.00

2

1 .OOo

2.04

5 1.

580

I .408

1.31

3 1.

252

I .208

1.

177

1.15

3 1.

133

1.1

I8

1.09

4 1.

077

I .06

4 1.

055

I .047

I .03

6 1.

026

1.01

6 1.

008

I .002

1 .ooo

2.11

3 1.

606

1.42

2 1.

323

I .259

1.21

5 1.

182

1.15

7 1.

137

1.12

1

1 .O%

1.07

8 1.

066

1.05

6 1.

048

1.03

7 1.

026

1.01

7 1.

008

1.00

2

1 .Ooo

x:m

96

.578

2 10

1.87

9 10

6.62

9 11

2.32

9 11

6.32

1

-

-

-

-

-

1.50

4 1.

533

1.56

2 1.

598

1.62

4 1.

367

1.38

4 1.

401

1.42

2 1.

437

1.28

7 1.

299

1.31

1 1.

325

1.33

5 1.

234

1.24

3 1.

252

1.26

2 1.

270

1.1%

1.

203

1.21

0 1.

218

1.22

4 1.

167

1.17

3 1.

179

1.18

6 1.

190

1.14

5 1.

150

1.15

5 1.

160

1.16

4 1.

127

1.13

2 1.

136

1.14

0 1.

144

1.11

3 1.

116

1.12

0 1.

124

1.12

7

1.09

1 1.

094

1.09

6 1.

075

1.07

7 1.

079

1.06

3 1.

065

1.06

6 1.

054

1.05

5 1.

057

1.04

6 1.

048

1.04

9

1.03

6 1.

037

1.03

8

.099

1.

102

,082

1.

084

.069

1.

070

.059

1.

060

.051

1.

052

.039

1.

040

1.02

5 1.

026

1.02

7 1.

028

1.02

8 1.

016

1.01

7 1.

017

1.01

8 1.

018

1.00

8 1.

008

1.00

9 1.

009

1.00

9 1.

002

1.00

2 1.

003

1.00

3 1.

003

1.OOo

1.O

Oo

1.m

1.O

Oo

1.OOo

100.

980

106.

395

I1 1.

242

117.

057

121.

126

a’

W

m =

4 -

r=2

2

a

0.10

0 0.050

0.02

5 0.

010

0.00

5

1 1.

848

1.92

2 2

1.51

8 3

1.37

9 4

1.29

8 5

1.24

3

6 1.

204

7 1.

175

8 1.

152

9 1.

134

10

1.11

9

12

1.09

5 14

1.

079

16

1.06

6 18

1.

057

20

1.04

9

24

1.03

8 30

1.

027

40

1.01

7 60

1.

009

120

1.00

3

CQ

1.Ooo

xt,

105.

372

I .54

9 1.

397

1.31

0 1.

253

1.21

2 1.

181

1.15

7 1.

138

1.12

3

1.09

8 1.

08 1

1.06

8 1.

058

1.05

1

1.03

9 1.

028

1.01

8 1.

009

1.00

3

1 .Ooo

10.8

98

1.99

4 I .5

77

1.41

4 I .3

22

1.26

2

1.21

9 1.

187

1.16

2 1.

142

1.12

6

1.10

1 1.

083

1.07

0 1.

060

I .052

1.04

0 1.

029

1.01

8 I .009

1.00

3

1 .Ooo

115.

841

2.08

8 2.

158

1.61

4 1.

436

1.33

7 1.

273

1.22

8 1.

194

1.16

8 1.

147

1.13

0

1.10

4 1.

086

1.07

2 I .

062

1.05

3

1.04

1 1.030

1.01

9 1.

010

1.00

3

1 .O

oo

12 1.

767

1.64

1 1.

45 1

1.

347

1.28

1

1.23

4 1.

199

1.17

2 1.

151

1.13

4

1.10

7 1.

088

1.07

4 1.

063

I .05

5

1.04

2 1.

030

1.01

9 1.

010

1.00

3

1 .Ooo

25.9

13

m =

5 -

r=5

r=6

0.050

0.025

0.010

0.005

0.100

0.050

0.025

0.010

0.005

I 1.448

2 1.212

3 1.132

4 1.092

5 1.

068

6

1.053

7 (1.042

8 1.035

9 1.029

10

1.025

12

1.018

14

1.014

16

1.011

18

1.009

20

1.008

24

1.00

6 30

1.004

40

1.002

60

1.00

1 120

1.OOo

33

1.OOo

xs,

34.3816

1.4%

I .230

1.141

1.098

1.072

1.056

I .045

1.037

1.03 1

I .026

1.020

1.01

5 1.012

1.010

1.008

1.00

6 I .0

04

1.002

1.001

1 .Ooo

I .O

oo

37.6525

I .544

I .246

1.150

1.103

1.076

1.059

I .047

I .039

1.032

I .027

1.020

1.016

1.013

1.010

1.009

I .006

I .004

1.002

1.00

1 I .O

oo

1 .Ooo

40.6

465

1.604

1.267

1.161

1.1 10

1.

081

1.063

I .OM

1.041

1.034

I .029

1.022

1.01

7 1.013

1.01 I

1.009

I .007

1.00

4 1.003

1.001

1 .Ooo

1 .O

oo

44.3141

1.64

9 I .283

1.169

1.1 I6

1.085

1.06

5 1.052

1.043

1.035

1.03

0

I .022

1.017

1.014

1.01 1

1.00

9

I .007

I .oo5

1.003

I .00

1 I .O

oo

1 .OOo

46.9279

I .465

1.22

8 1.

144

1.102

1.077

1.060

1.048

I .040

I .034

1.029

1.022

1.017

I .014

1.01 1

1.009

1.007

I .005

1.003

1.001

1 .Ooo

1 .O

oo

40.2560

1.514

I .245

1.15

4 1.

108

1.08 1

I .063

I .05 I

1.042

1.035

1.030

1.023

1.014

1.012

1.010

1.007

1.005

1.003

1.001

1 .Ooo

I .O

oo

43.7730

1.018

1.563

1.262

1.163

1.1 14

1.08

5

I .066

1.053

I .044

1.037

1.03 1

1.024

1.019

1.01

5 1.012

1.010

1.007

I .005

I .003

1.001

1 .OOo

1 .OOo

46.9792

1.625

I .2M

1.175

1.121

1 .om

1.070

1.056

1.04

6 I .039

1.033

I .025

1.019

1.016

1.013

1.01 1

1.00

8 1.

005

I .003

1.001

1 .Ooo

I .O

oo

50.8922

1.67 1

1.300

1.183

1.127

1.09

4

I .073

I .059

1.048

1.04

0 1.034

1.026

I .020

1.01

6 1.013

1.01 1

I .008

1.005

1.003

I .002

I .Ooo

1 .Ooo

53.6720

Tab

le 9

( C

on

tin

Ud

)

m=

5 -

r=7

r=8

0.05

0 0.025

0.010

0.005

0.100

0.05

0 0.025

0.010

0.005

1 1.484

2 1.244

3 1.158

4 1.113

5 1.086

6 1.068

7 1.055

8 1.

046

9 1.038

10

1.033

12

1.025

14

1.020

16

1.016

18

1.013

20

1.011

24

1.008

30

1.005

40

1.003

60

1.002

120

1.OOo

03

1.Ooo

xf,

16.0588

1.535

1.262

1.168

1.119

1.090

1.07 1

1.058

1.048

1.040

1.035

1.026

1.02 1

1.017

1.014

1.012

1.008

1.006

1.003

1.002

I .Ooo

1 .Ooo

49.8019

1.584

1.280

1.177

1.125

1.095

1.074

1.060

1.05

0 1 .a

42

1.036

1.027

1.02 1

1.017

1.014

1.012

1.009

1.006

i .004

1.002

I .Ooo

1 .Ooo

53.2033

1.648

1.302

1.189

1.133

1.100

1.078

1.063

1.052

1.044

1.038

1.029

1.022

1.018

1.015

1.013

1.009

1-0

6

1.004

1.002

1 .Ooo

1 .O

oo

57.3421

1.695

1.319

1.198

1.139

1.104

1.081

1.066

1.05

4 1 -046

1.039

1.030

1.023

1.019

1.015

1.013

1.010

I .006

1.00

4 1.002

1 .Ooo

1 .O

oo

60.2748

1 SO

5 1.261

1.171

1.124

1.095

1.076

1.062

1.052

1.044

1.038

1.029

I .023

1.018

1.015

1.013

1.009

1.006

1.004

1.002

1.001

1 .Ooo

5 1 .SO50

1.556

1.280

1.182

1.131

1.100

1.079

1.065

1.054

1.046

1.039

1.030

1.024

1.019

1.016

1.013

1.010

1.007

1 .w

1.002

1.001

1 .Ooo

55.7585

i .007

1.298

1.192

1.137

1.105

1.083

1.068

1.056

1 .a48

1.041

1.03 1

I .025

1.020

1.017

1.014

1.010

1.007

1.00

4 I .002

1.001

1 .Ooo

59.34 17

1.672

1.321

1.204

1.145

1.110

1.087

1.071

1.059

1.050

I .043

1.033

1.026

1.021

1.017

1.01

5

1.01 1

1.007

1.004

1.002

1.001

1 .Ooo

63.6907

1.721

1.338

1.213

1.151

1.1 14

1.09

0 1.073

1.061

I .052

1.044

1.034

1.027

1.022

1.018

1.015

1.01 1

1.008

1.005

1.002

1.001

1 .Ooo

66.7659

r=9

r=

lO

0.05

0 0.

025

0.01

0 0.

005

0.10

0 0.

050

0.02

5 0.

010

0.00

5

I 1.

526

2 1.

278

3 1.

185

4 1.

136

5 1.

105

6 1.

084

7 1.

069

8 1.

058

9 1.

049

10

1.04

3

12

1.03

3 14

1.

026

16

1.02

1 18

1.

018

20

1.01

5

24

1.01

1 30

1.

008

40

1.00

5 60

1.00

2 I2

0 1.

001

w

1.OOo

I .578

I .2

98

1.1%

,1

43

.I 10

,088

.0

72

.060

.05 1

.044

1.03

4 I .0

27

1.02

2 1.

018

1.01

5

1.01

1 1.

008

1.00

5 1.

002

1.00

1

1 .O

oo

1.63

0 1.

316

1.20

6 1.

150

1.11

5

1.09

2 I .0

75

1.06

3 1.

053

I .046

1.03

5 I .

028

1.02

3 1.

109

1.01

6

1.01

2 1.

008

1 .00

5 I .

002

1.00

1

I .Ooo

1.69

7 1.

340

1.21

9 1.

158

1.12

1

1 .O%

1.07

9 1.

066

1.05

6 1.

048

1.03

7 1.

029

I .02

4 1.

020

1.01

7

1.01

2 I .

008

1.00

5 1 .0

02

1.00

1

I .OOo

1.74

6 1.

358

I .22

9 1.

164

1.12

5

1 .ow

1.

08 1

1.06

8 1.

058

1.05

0

I .03

8 1.

030

1.02

4 1.

020

1.01

7

1.01

3 1.

009

1 .00

5 1.

003

1.00

1

1 .O

oo

x:,,,

57.5

053

61.6

562

65.4

102

69.9

568

73.1

661

1.54

7 I .2

95

1.19

9 1.

147

1.11

5

1.09

2 1.

076

I .064

1.05

5 1.

048

1.03

7 1.

029

1.02

4 1.

020

1.01

7

1.01

3 1.

009

1.00

5 1.

003

1.00

1

I .Ooo

1.60

0 1.

315

1.21

1 15

5 ,1

20

,097

,0

80

,067

.0

57

'050

1.03

8 I .0

3 1

1.02

5 1.

021

1.01

8

1.01

3 1.

009

I .005

1.

003

1.00

1

I .Ooo

I .653

1.

334

I .22

1 1.

162

1.12

5

1.10

1 I .

083

I .070

1.

059

1.05

1

1.04

0 1.

032

1.02

6 1.

022

1.01

8

1.01

4 1.

009

1.00

6 1.

003

1.00

1

1 .Ooo

1.72

1 I ,

359

1.23

5 1.

171

1.131

1.105

I .087

1.

073

1.06

2 1.

054

1.04

1 1.

033

1.02

7 1.

022

1.01

9

1.01

4 1.

010

1.00

6 1.003

1 .#2

I .Ooo

I .77

2 1.

377

1.24

4 1.

177

1.13

6

1.10

9 I .

089

1.07

5 1.

064

I .05

5

1.04

3 1.

034

1.02

8 1.

023

1.01

9

1.01

4 1.

010

I .006

1.00

3 I .0

02

I .O

Oo

63.1

671

67.5

048

71.4

202

76.1

539

79.4

900

~ ~~

~~

m=

5 -

r=ll

?=I2

n

0.100

0.050

0.025

0.01

0 0.005

0.100

0.050

0.025

0.010

0.005

1-

2

1.312

3 1.213

4 1.159

5 1.125

6 1.101

7 1.084

8 1.071

9 1.061

10

1.053

12

1.041

14

1.033

16

1.027

18

1.023

20

1.019

24

1.014

30

1.01

0 40

1.00

6 60

1.003

I20

1.001

m

1.OOO

xf,

68.796

-

1.333

1.225

1.167

1.130

1.105

1.087

1.074

1.063

1.055

1.043

1 .OM

1.028

1.023

1.02

0

1.015

1.010

1.006

1.003

1.001

1 .ooo

73.3 1

1

I

1.352

1.236

1.174

1.136

1.110

1.09 1

1.077

1.06

6 1.057

I .044

1.035

1.029

I .024

I .021

1.01

5 1.01

1 1.006

1.003

1.001

I .ooo

77.380

-

1.378

1.250

1.183

1.142

1.115

1.095

1.080

1.068

1.059

1.04

6 1.037

1.030

1.025

1.02

1

1.016

1.01 1

1.007

1.003

1.001

1 .ooo

82.292

-

1.3%

1.260

1.190

1.147

1.1

18

1.098

1.082

1.07

0 1.061

1 .a7

1.038

.03 1

.026

.022

.016

.01 I

.007

.003

1.001

1 .OO

o

85.749

1.587

1.329

I .227

1.171

1.135

1.110

I .092

1.078

1.067

I .058

1.04

6 1.037

1.030

1.025

1.022

1.016

1.011

1.00

7 1.003

1.001

1 .Ooo

74.3970

1.643

1.350

1.239

1.179

1.141

1.1 14

1.095

1.08 1

1.070

1.061

1.047

1.038

1.03 1

1.026

1.022

1.017

1.012

1.007

1.003

1.001

1 .O

oo

79.08 19

I .697

1.370

1.25 1

1.186

1.146

1.119

1.099

1.084

1.072

1.063

1.049

1.039

1.032

1.027

1.023

1.017

1.012

1.007

1.004

1.001

1 .Ooo

83.2977

1.768

1.3%

1.265

1.1%

1.153

1.124

1.103

1.087

1.075

1.065

1.05 1

1.041

1.033

1.028

1.024

1.018

1.012

1.00

8 1.

004

I .001

I .ooo

88.3794

1.821

1.415

1.275

1.203

1.158

1.128

1.106

1.089

1.077

1.067

I .052

1.042

1.034

1.029

1.024

1.018

1.013

1.008

1.004

1.001

I .Ooo

9 1.95 1

7

r=1

3

r=1

4

a

0.10

0 0.

050

0.02

5 0.

010

0.00

5 0.

100

0.05

0 0.0

25

0.01

0 0.

005

1-

2

1.34

5 3

1.24

1 4

1.18

2 5

1.14

5

6 1.

118

7 1.

099

8 1.

084

9 1.

073

10

1.06

4

12

1.05

0 14

1.

040

16

1.03

3 I8

1.

028

20

1.02

4

24

1.01

8 30

1.

013

40

1.00

8 60

1.00

4 12

0 1.

001

m

1.OOo

-

1.36

7 1.

253

1.19

1 1.

151

1.12

3 1.

103

1.08

8 1.

076

1.06

6

1.05

2 1.

042

1.03

5 1.

029

1.02

5

1.01

9 1.

013

I .00

8 1.

004

1.00

1

1 .Ooo

-

I .387

1.2

65

1.19

9 1.

157

1.12

8 1.

107

1.09

1 1.

078

1.06

8

I .05

4 I .0

43

1.03

6 1.

030

1.02

6

1.01

9 1.

013

1.00

8 1.

004

I .oOl

1 .Ooo

-

1.41

4 1.

280

1.208

1.

164

1.13

3 1.

111

1.09

4 1.

08 I

1.07

I

I .05

5 1.

045

1.03

7 I .0

3 1

I .026

1.02

0 1.

014

1.00

9 1.

004

1.00

1

1 .Ooo

-

1.43

3 1.

290

1.21

5 1.

169

1.13

7 1.1

14

I .097

1.

083

I .073

1.05

7 1.

046

1.03

8 1.

032

I .027

1.02

0 1.

014

1.00

9 1.

004

1.00

1

1 .OOo

xf;,

79.9

73

84.8

21

89.1

77

94,4

22

98.1

05

1.626

1.

361

1.25

4 1.

194

1.155

1.127

1.

107

1.09

1 1.

079

1.06

9

1.05

5 1.

044

1.037

I .0

3 1

I .026

1.02

0 1.0

14

1.00

9 1.

004

1.00

1

I .Ooo

1.68

3 1.3

83

1.26

7 1.

203

1.16

1

1.13

2 1.1

11

1.095

1.

082

1.07

2

1.05

7 1.

046

1.03

8 1.

032

1.027

1.02

1 1.

014

I .009

1.

004

1.00

1

1 .Ooo

1.740

I .404

1.27

9 1.

21 1

1.

167

1.13

7 1.

1 I5

I .098

I .0

85

1.07

4

1.05

8 1.

047

I .039

1.

033

I .028

1.02

1 1.0

15

1.00

9 1.0

05

1.00

1

1 .Ooo

1.81

3 1.

431

1.29

4 1.

22 1

1.

174

1.14

3 1.

1 I9

1.

102

1.08

8 1.

077

1.06

0 1.0

49

1.04

0 1.

034

1.02

9

1.02

2 1.

015

1.01

0 1.

005

1.00

1

1 .Ooo

1.86

7 I .4

5 1

1.30

5 1.

228

1.18

0

1.14

7 1.

123

1.10

4 I .o

m

1.07

9

1.06

2 1.

050

1.04

1 1.

035

1.03

0

1.02

2 1.

016

1.010

1.

005

1.00

1

1 .OOo

85.5

270

90.5

312

95.0

232

100.

4252

10

4.21

49

m

N

0

Tab

le9

(Cm

rin

ucd

)

m =

5 -

r=1

5

r=1

6

a

0.10

0 0.050

0.02

5 0.

010

0.00

5 0.

0100

0.050

0.01

0 0.

005

1-

2

1.37

7 3

1.26

7 4

1.20

5 5

1.16

4

6 1.

136

7 1.

115

8 1.

098

9 1.

085

10

1.07

5

12

1.05

9 14

1.

048

16

1.04

0 18

1.

034

20

1.02

9

24

1.02

2 30

1.

015

40

1.01

0 60

1.

005

120

1.00

1

00

1.oo

o

x:,

91.0

61

-

1.39

9 1.

281

1.21

4 1.

171

1.14

1 1.

119

1.10

2 1.

088

1.07

8

1.06

1 1.

050

1.04

1 1.

035

1 .OM

1.02

3 1.

016

1.01

0 I .005

1.00

1

1 .O

oo

96.2

17

-

1.42

1 1.

293

1.22

3 1.

177

1.14

6 1.

123

1.10

5 1.

091

1.08

0

1.06

3 1.

051

I .04

3 1.

036

1.03

1

1.02

3 1.

016

1.01

0 1.

005

I .ool

1 .OOo

100.

839

-

1.44

9 1.

309

1.23

3 1.

185

1.15

2 1.

127

1.10

9 1 .o

w

1.08

3

1.06

5 1.

053

1.04

4 I .0

37

1.03

2

1.02

4 1.

017

1.01

1 I .

005

1.00

1

1 .m

106.

393

-

1.46

9 1.

320

1.24

0 1.

190

1.15

6 1.

131

1.1 1

2 1.

097

I .08

5

1.06

7 1 .OM

1.04

5 1.

038

1.03

3

1.02

5 1.

017

1.01

1

1.00

5 I .

002

1 .Ooo

1 10.

286

1.66

3 1.

392

1.28

0 1.

216

1.17

4

1.14

4 1.

122

1.10

5 1.

091

1 .os 1

1.06

4 1.

052

1.04

3 1.

037

1.03

2

I .024

1.

017

1.01

1 1.

005

1.00

2

1 .Ooo

1.72

2 1.

415

1.29

4 I .

226

1.18

1

1.15

0 .1

27

.I09

.0

95

.083

.066

.ow

.045

1.

038

1.03

3

1.025

1.

018

1.01

1 1.

005

1 .m

2

1 .Ooo

1.78

0 1.

437

1.30

7 1.

234

1.18

8

1.15

5 1.

131

1.1

12

1.09

8 1.

086

1.06

8 1.

055

1.04

6 1.

039

1.03

3

I .02

5 1.

01s

1.01

1 1.

006

1.00

2

1 .Ooo

1.85

5 1.

465

1.32

3 1.

245

1.19

5

1.16

1 .1

36

.116

.I01

.089

.070

.0

57

.048

I .040

1.03

5

I .02

6 1.

01 9

1.01

2 ! .006

1.00

2

1 .OOo

1.91

1 1.

486

1.33

4 I .

253

1.20

1

1.16

5 1.

139

1.11

9 1.

104

1.09

1

1.07

2 1.

059

I .04

9 1.

041

1.03

5

1.02

7 1.

019

1.01

2 1.

006

1.00

2

1 .ooo

96.5

782

101.

8795

106

.628

6 11

2.32

88 1

16.3

21 1

r=1

7

r=1

8

0.05

0 0.

025

0.01

0 0.

005

0.10

0 0.

050

0.02

5 0.

010

0.00

5

-

I 2 1.

407

3 1.

293

4 1.

227

5 1.

184

6 1.

153

7 1.

130

8 1.

112

9 1.

098

10

1.08

6

12

14

16

18

20

24

.069

,0

56

,047

.0

40

.034

,026

30

1.

019

40

1.01

2 GO

1.

006

120

1.00

2

00

1.OOo

-

1.43

1 I .3

07

1.23

7 1.

191

1.15

9 1.

134

1.1 1

6 1.

101

I .089

.07 1

,0

58

.048

.04 I

.0

35

.027

1.

019

1.01

2 1.

006

1.00

2

1 .ooo

-

1.45

3 1.

320

1.24

6 1.

198

1.16

4 1.

139

1.11

9 1.

104

I .092

,073

.060

,050

.0

42

.036

.028

I .0

20

1.01

2 1.

006

1.00

2

1 .OOo

- 1.48

2 1.

336

1.25

7 I .2

06

1.17

0 1.

144

1.12

4 1.

I08

1.09

5

1.07

5 1.

062

1.05

1 1.

044

1.03

7

1.02

9 1.

020

1.01

3 1.

006

I .002

I .Ooo

-

1 SO

3 1.

348

1.26

5 1.

212

1.17

5 1.

147

1.12

7 1.

110

1.09

7

.077

.0

63

,052

,044

.038

.029

1.

02 1

1.01

3 1.

006

1.00

2

1 .Ooo

7

XL

10

2.07

9 10

7.52

2 11

2.39

3 11

8.23

6 12

2.32

5

1.69

8 1.

421

1.30

5 1.

238

1.19

3

1.16

1 1.

137

1.1 1

9 1.

104

1.09

2

I .073

I .060

1.05

0 I .0

43

1.03

7

1.02

8 1.

020

1.01

3 1.

006

1.00

2

1 .Ooo

1.75

8 1.

445

1.32

0 1.

248

1.20

1

1.16

7 1.

142

1.12

3 I.

I07

1.09

5

I .076

1.

062

1.05

2 1.

044

1.03

8

I .029

I .0

2 I

1.01

3 1.

007

I .002

I .Ooo

1.81

8 I .468

1.33

3 1.

257

I .20

8

1.17

3 1.

147

I. I2

6 1.

110

1.09

8

1.07

8 1.

064

I .053

1.

045

1.03

9

1.03

0 1.

02 1

1.01

3 1.

007

1.00

2

I .Ooo

1.89

5 1.

498

1.35

0 1.

268

1.21

6

1.17

9 1.

152

1.13

1 1.

114

1.10

1

1.08

0 I .0

66

1.05

5 1.

047

1.04

0

1.03

1 1.

022

1.01

4 I .0

07

1.00

2

1 .Ooo

1.95

3 1.

519

1.36

2 1.

277

I .22

2

1.18

4 1.

156

1.13

4 1.

117

1.10

3

1.08

2 I .

067

1.05

6 1.

048

1.04

1

1.03

1 I .0

22

1.01

4 I .

007

1.00

2

I .OOo

10

7.56

50 1

13.1

453

118.

1359

124

.116

3 12

8.29

89

m

N

N

1 2 3 4 5 6 7 8 9 10

12

14

16

18

20

24

30

40

60

120 m

X5,

~~

~~~

~~

~~~

m=

5 -

r=1

9

1 =

20

a

0.10

0 0.050

0.02

5 0.

010

0.00

5 0.

100

0.05

0 0.

025

0.01

0 0.005

1.79

3 1.

853

-

1.43

6 1.

318

1.24

9 1.

203

1.17

0 1.

145

1.12

6 1.

110

1.09

7

1.07

8 1.

064

1.05

4 1.

046

1.04

0

1.03

1

1.02

2 1.

014

1.00

7 I .0

02

I .Ooo

11

3.03

8

-

1.46

0 1.

332

1.25

9 1.

210

1.17

6 1.

150

1.13

0 1.

114

1.10

1

1.08

1 1.

066

1.05

6 1.

047

1.04

1

1.03

1 1.

022

1.01

4 1.

007

1.00

2

1 .ax,

1 18.

752

-

1.48

3 1

.34

1.

268

1.21

7

1.18

1 1.

154

1.13

4 1.

117

1.10

3

1.08

3 1.

068

1.05

7 1.

049

1.04

2

1.03

2 1.

023

1.01

5 1.

007

1.00

2

1 .Ooo

12

3.85

8

-

1.51

3 1.

363

1.28

0 1.

226

1.18

8 1.

160

1.13

8 1.

121

1.10

7

1.08

5 1.

070

1.04

9 1.

050

1.04

3

1.03

3 1.

024

1.01

5 1.

008

1 .#2

1 .Ooo

129.

973

-

1.53

5 1.

375

1.28

8 1.

232

1.19

3 1.

164

1.14

1 1.

124

1.10

9

1.08

7 1.

072

1.06

0 1.

05 1

1.04

4

1.03

4 1.

024

1.01

5 I .

008

1.00

2

1 .Ooo

134.

247

i.73

i 1.

449

1.33

0 1.

259

1.21

2

1.17

8 1.

152

1.13

2 1.

116

1.10

3

1.08

3 1.

069

1.05

8 1.

049

1.04

3

1.03

3 1.

024

1.01

5 1.

008

1.00

2

1 .Ooo

1.47

4 1.

345

1.27

0 1.

220

1.18

4 1.

157

1.13

7 1.

120

1.10

6

1.08

6 1.

07 1

1.05

9 1.

05 1

1.04

4

1.03

4 1.

024

1.01

5 1.

008

1.00

2

1 .Ooo

1.49

8 1.

358

1.27

9 1.

227

1.19

0 1.

162

1.14

1 1.

123

1.10

9

1.08

8 1.

072

1.06

1 1.

052

1.04

5

1.03

5 1.

025

1.01

6 1.

00s

1.00

2

1 .OOo

1.93

3 1.

528

1.37

6 1.

291

1.23

6

1.19

7 1.

168

1.14

5 1.

127

1.1

13

1.09

1 1.

075

1.06

3 1.

053

1.04

6

1.03

6 1.

025

1.01

6 1.

008

1.00

2

1 .Ooo

~

1.99

2 1.

55 1

1.

388

1.30

0 1.

242

1.20

2 1.

172

1.14

9 1.

130

1.1

15

1.09

2 1.

076

1.06

4 1.

054

1.04

7

1.03

6 1.

026

1.01

6 1.

008

1.00

2

1 .Ooo

118.

4980

124

.342

1 12

9.56

12

135.

8067

140

.169

5

m =

6 -

r =

6 r=

7

a

0.10

0 0.

050

0.02

5 0.

010

0.00

5 0.

100

0.05

0 0.

025

0.01

0 0.

005

1 2 3 4 5 6 7 8 9 10

12

14

16

18

20

24

30

40

60

120

a,

1.47

1 1 .

237

1.15

3 1.

109

1.08

3

1.06

6 1.

053

1.04

4 1.

037

1.03

2

1.02

4 1.

019

1.01

5 1.

013

1.01

I

I .008

I .0

05

1.00

3 I .0

02

1 .Ooo

I .Ooo

1.52

0 1.

255

1.16

3 1.

116

1.08

8

1.06

9 1.

056

1.04

6 1.

039

1.03

4

1.02

5 I .0

20

1.01

6 1.

013

1.01

1

I .00

8 I .006

1.00

3 1.

002

1 .Ooo

I .Ooo

1.56

8 1.

272

1.17

2 1.

122

1.09

2

1.07

2 1.

058

I .048

1.

041

I .035

1.02

6 1.

02 1

1.01

7 1.

014

1.01

2

1.00

9 I .006

1.00

3 I .0

02

1 .Ooo

1 .OOo

1.63

1 1.

294

1.18

3 1.

129

1.09

7

1.07

6 1.

061

1.05

1 1.

043

1.03

7

1.02

8 1.

022

1.01

8 1.

014

1.01

2

1.00

9 I .0

06

1.00

4 1.

002

1 .ooo

1 .Ooo

1.67

7 1.

310

1.19

2 1.

134

1.10

1

1.07

9 1.

064

I .05

3 I .

044

I .038

1.02

9 1.

022

1.01

8 1.

015

1.01

3

1.00

9 I .006

I .00

4 1.

002

1 .Ooo

1 .Ooo

47

.212

2 50

.998

5 54

.437

3 58

.619

2 61

.581

2

1.48

1 1.

249

1.16

3 1.

118

1.09

0

1.07

2 1.

059

1.04

9 1.

042

1.03

6

1.02

7 1.

022

1.10

8 1.

014

1.01

2

1.00

9 I .0

06

1.00

4 I .0

02

I .OOo

1 .O

oo

I .53

0 1.

266

1.17

3 1.

124

I .09

5

1.07

5 1.

062

1.05

1 1.

043

1.03

7

1.02

9 1.

023

1.01

5 1.

013

1.00

9 I .006

1.00

4 I .0

02

1 .Ooo

I .O

Oo

1 .O

W

1.57

9 1.

284

1.18

2 1.

131

1.09

9

1.07

9 1.

064

1.05

3 1.

045

1.03

9

1.03

0 1.

023

1.01

9 1.

016

1.01

3

1.01

0 1.

007

I .004

1.

002

1.00

1

1 .Ooo

1.64

2 1.

306

1.19

4 1.

138

1.10

5

1.08

3 I .

067

1.05

6 I .

047

1.04

1

1.03

1 1.

024

1.02

0 1.

016

1.01

4

1.01

0 1.

007

1.00

4 1.

002

1.00

1

1 .Ooo

1.68

8 1.

322

1.20

3 1.

144

1.10

9

1.08

6 1.

070

1.05

8 1.

049

1.04

2

1.03

2 I .

025

1.02

0 1.

017

1.01

4

1.01

0 1.

007

1.00

4 1.

002

1.00

1

1 .Ooo

54

.090

2 58

.124

0 61

.776

8 66

.206

2 69

.336

0

Tab

le 9

(C

mth

ue

d)

~ ~~

~~~~

m=

6 -

r =8

r

=9

u

0.

100

0.05

0 0.

025

0.01

0 0.

005

0.10

0 0.

050

0.02

5 0.

010

0.00

5

I 1.

494

1.54

3 1.

592

1.65

6 1.

703

1.67

1 1.

719

2 1.

261

3 1.

174

4 1.

127

5 1.

098

6 1.

079

7 1.

065

8 1.

054

9 1.

046

10

1.04

0

12

1.03

1 14

1.

024

16

1.02

0 18

1.

016

20

1.01

4

24

1.01

0 30

1.

007

40

1.00

4 60

1.

002

I20

1.00

1

00

1.OOo

1.27

9 1.

297

1.18

4 1.

194

1.13

4 1.

140

1.10

3 1.

108

1.08

2 1.

086

1.06

8 1.

070

1.05

7 1.

059

1.04

8 1.

050

1.04

2 1.

043

1.03

2 1.

033

1.02

5 1.

026

1.02

1 1.

021

1.01

7 1.

018

1.01

4 1.

015

1.01

1 1.

011

1.00

7 1.

008

1.00

4 1.

005

1.00

2 1.

002

1.00

1 1.

001

1.OOo

1.O

Oo

1.31

9 I .

205

1.14

8 1.

113

1 .ow

1.

074

I .06

2 1.

052

1.04

5

1.03

5 1.

027

1.02

2 1.

108

1.01

5

1.01

1 1.

008

1.00

5 1.

002

1.00

1

1 .OOo

2

X,,,

60.9

066

65.1

708

69.0

226

73.6

826

76.%

88

1.33

6 1.

214

1.15

3 1.

1 17

1.09

3 1.

076

1.06

3 1 .OM

I .046

1.03

6 1.

028

1.02

2 1.

019

1.01

6

1.01

2 1.

008

I .00

5 1.

002

1.00

1

1 .Ooo

1.50

8 1.

275

1.18

5 1.

137

1.10

7

1.08

6 1.

07 1

1.06

0 1.

051

1.04

4

I .03

4 1.

027

1.02

2 1.

019

1.01

6

1.01

2 1.

008

1.00

5 1.

002

1.00

1

1 .Ooo

~

1.55

8 1.

607

1.29

3 1.

196

1.14

4 1.

1 12

1 .ow

1.

074

I .06

2 1.

053

1.04

6

1.03

5 1.

028

1.02

3 1.

019

1.01

6

1.01

2 1.

008

1.00

5 1.

002

1.00

1

1 .Ooo

1.31

I 1.

205

1.15

0 1.

116

1.09

3 1.

077

I .06

5 1.

055

1.04

8

1.03

6 1.

209

1.02

4 I .

020

1.01

7

1.01

3 1.

009

1.00

5 I .0

03

1.00

1

1 .OOo

1.33

3 1.

350

1.21

8 1.

227

1.15

8 1.

164

1.12

2 1.

126

1.09

8 1.

101

1.08

0 1.

083

1.06

7 1.

069

1.05

8 1.

059

1.05

0 1.

051

1.03

7 1.

037

1.03

1 1.

031

1.02

4 1.0

25

1.02

0 1.

021

1.01

7 1.

018

1.01

3 1.

013

1.00

9 1.

009

1.00

5 1.

006

1.00

3 1.

003

1.00

1 1.

002

1.ooo

1.O

Oo

67.6

728

72.1

532

76.1

921

81.0

688

84.5

016

r=lO

r=

12

0.05

0 0.

025

0.01

0 0.

005

0.10

0 0.

050

0.02

5 0.

010

0.00

5

1 1.

523

2 1.

288

3 1.

197

4 1.

147

5 1.

115

6 1.

093

7 1.

078

8 1.

066

9 1.

056

10

1.04

9

12

1.03

8 14

1.

031

16

1.02

5 18

1.

021

20

1.01

8

24

1.01

3 30

1.

009

40

1.00

6 60

1.

003

120

1.00

1

00

1.OOo

1.57

3 1.

307

1.20

8 1.

154

1.12

0

1.09

7 1.

08 I

1.06

8 I .

059

1.05

1

1.04

0 1.

032

1.02

6 1.

021

1.01

8

1.01

4 1.

010

1.00

6 1.

003

1.00

1

1 .Ooo

~

1.62

3 1.

325

1.21

8 1.

161

1.12

5

1.10

1 1 .o

w 1.

07 1

1.06

1 1.

053

1.04

1 1.

033

1.02

6 1.

022

1.01

9

1.01

4 1.

010

1.00

6 1.

003

1.00

1

1 .Ooo

1.68

7 1.

348

1.23

0 1.

169

1.13

1

1.10

6 1.

087

1.07

4 1.

063

1.05

5

1.04

2 I .0

34

1.02

7 1.

023

1.01

9

1.01

5 1.

010

1.00

6 1.

003

1.00

1

1 .Ooo

1.73

6 1.

365

I .239

1.

175

1.13

5

1.10

9 I .090

1.07

6 1.

065

I .056

1.04

3 1.

034

1.02

8 1.

023

I .020

1.01

5 1.

010

1.00

6 1.

003

1.00

1

1 .OOo

xf

, 74

.397

0 79

.081

9 83

.297

6 88

.379

4 91

.951

7

1.55

4 1.

316

1.22

1 1.

167

1.13

3

1.10

9 1.

091

1.07

8 1.

067

1.05

9

1.04

6 1.

037

1.03

1 1.

026

1.02

2

1.01

7 1.

012

1.00

7 I .0

04

1.00

1

1 .Ooo

1.60

5 1.

335

1.23

2 1.

175

1.13

8

1.11

3 1.

095

1.08

1 1.

070

1.06

1

I .048

1.03

9 1.

032

1.02

7 1.

023

1.01

7 1.

012

1.00

7 1.

004

1.00

1

1 .OOo

1.65

5 1.

354

I .242

1.

182

1.14

4

1.11

7 1.

098

1.08

3 1.

072

1.06

3

1.04

9 1.

040

1.03

3 1.

027

1.02

3

1.01

8 1.

012

1.00

8 1.

004

1.00

1

1 .Ooo

I .722

1.

378

I .255

1.

190

1.15

0

1.12

2 1.

102

1.08

6 I .0

74

1.06

5

1.05

1 1.

041

1.03

4 1.

028

1.02

4

1.01

8 1.

013

1.00

8 1.

004

1.00

1

1 .Ooo

1.77

1 1.

395

1.26

5 I.

197

1.15

4

1.12

5 1.

104

1.08

9 1.

076

1.06

7

1.05

2 I .

042

1.03

5 1.

029

1.02

5

1.01

9 1.

013

1.00

8 1.

004

1.00

1

1 .Ooo

87.7

430

92.8

083

97.3

53 I

102.

8163

106

.647

6

Tab

le9

(Con

tin

urd

) ~

~ ~~

~

m =6 -

r=14

r=15

0.050

0.025

0.010

0.005

0.100

0.050

0.025

0.010

0.005

1 1.585

2 1.343

3 1.

244

4

1.188

5 1.151

6 1.125

7 1.105

8 1.

090

9 1.078

10

1.069

12

1.055

14

1.044

16

1.037

18

1.031

20

1.027

24

1.020

30

1.014

40

1.009

60

1.00

4 120

1.001

Q:

1.OOo

1.637

1.363

1.25

6 1.1

% 1.157

1.129

1.109

1.093

1.081

1.07 1

1.056

1.04

6 1.038

1.032

I .028

1.02 1

1.015

1.00

9 1.005

I .001

1 .ooo

1.688

1.756

1.383

1.407

1.267

1.281

1.203

1.212

1.162

1.169

1.133

1.139

1.112

1.117

1.096

1.100

1.083

1.08

6 1.073

1.076

1.058

1.060

1.047

1.049

1.039

1.040

1.033

1.034

1.028

1.029

1.021

1.022

1.01

5 1.016

1.00

9 1.

010

1.005

1.005

1.001

1.001

1.OOo

1.Ooo

I .806

I .425

1.291

1.219

1.174

1.142

1.119

1.102

I .088

1.078

1.06

1 1.050

1.041

1.035

1.030

1.023

1.016

1.010

1.005

1.001

I .ooo

x;,

100.9800 1M.3948 1

1 1.2423 117.0565 121.1263

-

1.357

1.256

1.198

1.160

1.132

1.1 12

1.097

I -084

1.074

1.059

1.048

1.040

1.034

1.029

1.022

1.016

1.010

1 .005

1.001

1 .ooo

107.565

-

1.377

1.268

1.206

1.16

6

1.137

1.1 16

1.100

1.087

1.076

1.061

1.050

1.041

1.035

1.030

1.023

1.016

1.010

1.005

1.001

1 .ooo

113.145

-

1.397

1.279

1.214

1.171

1.142

1.120

1.103

1 .089

1.079

1.062

1.051

1.042

1.036

1.03 1

1 .ow

1.017

1.010

1.005

1.001

1 .OOo

118.136

-

1.422

1.293

1.223

1.178

1.147

1.124

1.106

1 -09

2 1.08 1

1.06

4 1.053

1.044

1.037

1.032

1.024

1.017

1.01 1

1.005

1.002

1 .Ooo

124.1 16

-

1.44

0 1.303

1.230

1.183

1.151

1.127

1.109

1.095

1.08

3

1.06

6 1.054

1.045

1.038

1.032

1.025

1.017

1.01

1 1.005

1 .OGt

1 .o

oo 128.299

r=1

6

r= 1

7 a

0.10

0 0.

050

0.02

5 0.

010

0.00

5 0.

100

0.05

0 0.

025

0.01

0 0.

005

1 1.

615

2 1.

370

3 1.

267

4 1.

208

5 1.

168

6 1.

140

7 1.

119

8 1.

103

9 1.

090

10

1.07

9

12

1.06

3 14

1.

052

16

1.04

3 18

1.

037

20

1.03

2

24

1.02

4 30

1.

017

40

1.01

1 60

1.

005

120

1.00

2

00

1.OOo

I .668

1.

391

1.28

0 1.

216

1.17

5

I. I4

5 1.

123

1.10

6 I .

093

1.08

2

1.06

5 1.

053

1.04

5 1.

038

1.03

3

1.02

5 1.

018

1.01

1 1.

006

1.00

2

1 .OOo

1.72

1 1.4

1 I

1.29

1 1.

224

1.18

1

1.15

0 1.

127

1.10

9 I .0

95

I .OM

I .06

7 1.

055

1.04

6 1.

039

I .033

I .025

1.

018

1.01

1 1.

006

I .002

1 .Ooo

1.78

9 1.

436

1.30

5 1.

234

1.18

8

1.15

5 1.

131

1.1

13

I .099

1.08

7

I .06

9 1.

056

1.04

7 1.

040

I .034

I .026

1.

019

1.01

2 1.

006

1.00

2

1 .Ooo

1.84

1 I .544

1.31

6 1.

241

1.19

3

1.15

9 1.

135

1.11

6 1.

101

1 .08

9

1.07

1 I .0

58

1.04

8 1.

041

1.03

5

1.02

7 1.

019

1.01

2 1.

006

1.00

2

1 .Ooo

x:,,~

114

.130

7 11

9.87

09 1

25.0

001

131.

1412

135

.433

0

-

1.38

3 1.

279

1.21

8 1.

177

1.14

8 1.

126

1.10

9 I .O

% 1.

085

1.06

8 1.

056

1.04

7 1.

040

1.03

4

1.02

6 1.

019

1.01

2 I .0

06

1.00

2

1 .Ooo

120.

679

-

I .404

1.29

1 1.

226

1.18

4

1.15

3 1.

130

1.11

3 1.

099

1.08

7

1.07

0 I .0

57

I .04

8 1.

041

1.03

5

1.02

7 1.

019

1.01

2 1.

006

I ,00

2

1 .Ooo

126.

574

-

I .424

1.

303

1.23

4 1.

190

1.15

8 1.

134

1.11

6 1.

101

1 .ow

I .072

1.

059

1.04

9 1.

042

1.03

6

1.02

8 1.

020

1.01

2 I .cQ6

1.00

2

1 .Ooo

-

1.45

0 1.

317

1.24

4 1.

197

1.16

4 1.

139

1.12

0 1.

105

1.09

2

1.07

4 1.

061

1.05

1 I .0

43

1.03

7

1.02

8 1.

020

1.01

3 I .0

06

I .002

1 .00

0

-

I .469

1.

328

1.25

1 1.

202

1.16

8 1.

142

1.12

3 1.

107

1.09

4

1.07

5 1.

062

1.05

2 1.

044

1.03

8

1.02

9 1.

02 1

1.01

3 1.

007

I .00

2

1 .Ooo

131.

838

138.

134

142.

532

Tab

le9

(Co

nti

mu

d)

rn =

€ -

r=1

8

r=1

9

a

0.10

0 0.

050

0.02

5 0.

010

0.00

5 0.

100

0.05

0 0.

025

0.01

0 0.

005

1.69

8 1.

752

1.82

2 1.

874 -

1.41

7 1.

303

1.23

7 1.

193

1.16

1 1.

138

1.1

19

1.10

5 1.

093

1.07

4 1.

061

1.05

1 1.

044

1.03

8

1.02

9 1.

02 1

1.01

3 1.

007

1.00

2

I .Ooo

1.43

8 1.

464

1.31

5 1.

329

1.24

5 1.

255

1.19

9 1.

206

1.16

6 1.

172

1.14

2 1.

146

1.12

3 1.

127

1.10

7 1.

111

1.09

5 1.

098

1.07

6 1.

079

1.06

3 1.

065

1.05

3 1.

0%

1.04

5 1-

046

1.03

9 1.

040

1.03

0 1.

031

1.02

1 1.

022

1.01

3 1.

014

1.00

7 1.

007

1.00

2 1.

002

1.OOo

1.

Ooo

1.48

3 1.

340

1.26

2 1.

212

1.17

6 1.

150

1.12

9 1.

113

1.10

0

I .08

0 1.

066

1.05

5 1.

047

1.04

1

1.03

1 1.

022

1.01

4 1.

007

1.00

2

1 .ooo

1 1.

644

2 1.

3%

3 1.

290

4 1.

228

5 1.

186

6 1.

156

7 1.

133

8 1.

116

9 1.

101

10

1.09

0

12

1.07

2 14

1.

060

16

1.05

0 18

1.

043

20

1.03

7

24

1.02

8 30

1.

020

40

1.01

3 60

1.

006

120

1.00

2

a2

1.OO

o

-

1.40

8 1.

301

1.23

7 1.

195

1.16

4 1.

140

1.12

2 1.

107

1.09

5

1.07

7 1.

063

1.05

3 1.

046

1.03

9

1.03

0 1.

022

1.01

4 1.

007

1 .oc

2

1 .Ooo

-

1.43

0 1.

314

I .246

1.

201

1.16

9 1.

145

1.12

6 1.

1 10

1.

098

1.07

9 I .

065

1 .05

5 1.

047

1.04

1

1.03

1 1.

022

1.01

4 1.

007

1.00

2

1 .ooo

-

1.45

1 1.

326

1.25

5 1.

208

1.17

4 1.

149

1.12

9 1.

113

1.10

1

1.08

1 I .

067

1.05

6 1.

048

1.04

1

1.03

2 1.

023

1.01

5 1.

007

1.00

2

1 .O

oo

-

1.47

7 1.

497

1.34

1 1.

352

1.26

5 1.

273

1.21

5 1.

221

1.18

0 1.

184

1.15

4 1.

157

1.13

3 1.

136

1.11

7 1.

119

1.10

4 1.

106

1.08

3 1.

085

1.06

9 1.

070

1.05

8 1.

059

1.04

9 1.

050

1.04

3 1.

043

1.03

3 1.

033

1.02

3 1.

024

1.01

5 1.

015

1.00

8 1.

008

1.00

2 1.

002

1.OOo

1.

OO

o

x:m

127

.21 1

1 13

3.25

69 1

38.6

506

145.

0988

149

.599

4 13

3.72

9 13

9.92

1 14

5.44

1 15

2.03

7 15

6.63

7

r=2

0

u

0.10

0 0.

050

0.02

5 0.

010

0.00

5

1 2 3 4 5 6 7 8 9 10

12

14

16

18

20

24

30

40

60

1 20

00

~

1.67

2 1.

727

1.78

1 1.

420

1.44

3 1.

464

1.31

2 1.

325

1.33

7 1.

247

1.25

6 1.

265

1.20

3 1.

210

1.21

7

1.17

1 1.

177

1.18

2 1.

147

1.15

2 1.

156

1.12

8 1.

132

1.13

6 1.

113

1.11

6 1.

119

1.10

1 1.

103

1.10

6

1.08

1 1.

084

1.08

6 1.

067

1.06

9 1.

071

1.05

7 1.

058

1.06

0 1.

049

1.05

0 1.

051

1.04

2 1.

043

1.04

4

1.03

3 1.

033

1.03

4 1.

023

1.02

4 1.

025

1.01

5 1.

015

1.01

6 1.

008

1.00

8 1.

008

1.00

2 1.

002

1.00

2

1.Ooo

1.O

oo

1.ooo

1.85

3 1.

490

1.35

3 1.

275

1.22

4

1.18

8 1.

161

1.14

0 1.

123

1.10

9

I .08

8 I .0

73

1.06

1 1.

052

1 .04

5

1.03

5 1.

025

1.01

6 1.

008

1.00

2

1 .Ooo

I .906

1.

510

1.36

4 I .

283

1.23

0

1.19

3 1.

165

I. 14

3 1.

126

1.1

I1

1 .ow

1.07

4 1.

062

1.05

3 1.

046

I .03

6 1.

026

1.01

6 1.

008

1.00

2

1 .Ooo

x:,,

140.

2326

14

6.56

74

152.

21 14

15

8.95

02

163.

6482

Tab

le9

(Co

nth

ud

)

I 1.

484

1.53

2 2

1.25

6 1.

273

3 1.

170

1.18

0 4

1.12

5 1.

131

5 1.

096

1.10

1

6 1.

077

1.08

1 7

1.06

3 1.

066

8 1.

053

1.05

5 9

1.04

5 1.

047

10

1.03

9 1.

041

12

1.03

0 1.

031

14

1.02

4 1.

025

16

1.01

9 1.

020

18

1.01

6 1.

017

20

1.01

4 1.

014

24

1.01

0 1.

010

30

1.00

7 1.

007

40

1.00

4 1.

004

60

1.00

2 1.

002

120

1.00

1 1.

001

00

1.Ooo

1.O

Oo

1.38

0 1.

643

1.29

0 1.

312

1.18

9 1.

201

1.13

7 1.

145

1.10

5 1.

111

1.08

4 1.

088

1.06

9 1.

072

1.05

8 1.

060

1.04

9 1.

051

1.04

2 1.

044

1.03

2 1.

034

1.02

6 1.

027

1.02

1 1.

022

1.01

7 1.

018

1.01

5 1.

015

1.01

1 1.

011

1.00

7 1.

008

1.00

4 1.

005

1.00

2 1.

002

1.00

1 1.

001

1.Ooo

1.O

oo

1.68

9 1.

329

1.21

0 1.

150

1.11

4

1.09

1 I .

074

1 .M

2 1.

053

1.04

5

1.03

5 1.

028

1.02

2 1.

019

1.01

6

1.01

2 I .

008

1.00

5 1.

002

1.00

1

1 .Ooo

m =7 -

r=7

r=

8 a

0.10

0 0.

050

0.02

5 0.

010

0.00

5 0.

100

0.05

0 0.

025

0.01

0 0.

005

1.49

0 1.

538

1.586

1.

648

1.69

4 1.

265

1.28

2 1.

179

1.18

9 1.

132

1.13

9 1.

103

1.10

8

1.08

3 1.

086

1.06

8 1.

071

1.05

8 1.

060

1.04

9 1.

051

1.04

3 1.

044

1.03

3 1.

034

1.02

6 1.

027

1.02

1 1.

022

1.01

8 1.

018

1.01

5 1.

016

1.01

1 1.

012

1.00

8 1.

008

1.00

5 1.

005

1.00

2 1.

002

1.00

1 1.

001

1.O

oo

1.OOo

1.29

9 1.

321

1.19

8 1.

210

1.14

5 1.

152

1.11

2 1.

117

1.09

0 1.0

94

1.07

4 1.

077

1.06

2 1.

065

1.05

3 1.

055

1.04

6 1.

048

1.03

5 1.

037

1.02

8 1.

028

1.02

3 1.

023

1.01

9 1.

020

1.01

6 1.

017

1.01

2 1.

012

1.00

8 1.

009

1.005

1.

005

1.00

2 1.

003

1.00

1 1.

001

1.Ooo

1.O

Oo

1.33

7 1.

218

1.15

8 1.

121

I .09

7 1.

080

I .06

7 I .0

57

1.04

9

1.03

8 1.

029

1.02

4 1.

020

1.01

7

1.01

3 1.

009

1.00

5 1.

003

1.00

1

1 .Ooo

xt,,,

62.0

375

66.3

386

70.2

224

74.9

195

78.2

307

69.9

185

74.4

683

78.5

672

83.5

134

86.9

938

r =

9

r=lO

01

0.100

0.050

0.025

0.01

0 0.005

0.100

0.050

0.025

0.010

0.005

I 1.499

I .547

1.594

1.656

1.703

1 so9

1.557

2 1.275

3 1.188

4 1.140

5 1.110

6 1.089

7 1.074

8 1.063

9 1.

054

10

1.047

12

1.036

14

1.029

16

1.024

18

1.020

20

1.017

24

1.013

30

1.009

40

1.005

60

1.003

120

1.001

oc

1.Ooo

1.292

1.198

1.147

1.1 15

I .093

1.077

I .065

1.056

1.048

1.038

1.030

1.025

1.02 1

1.018

1.013

1.009

1.006

1.003

1 .001

I .Ooo

1.309

1.207

1.153

1.1 19

I .096

1 .080

1.067

1.058

1.05

0

1.039

1.03 1

I .025

1.021

1.018

1.013

1.00

9 1.006

1.003

1.001

1 .Oo

o

1.33 1

1.219

1.161

1.125

1.100

I .083

1.070

1.06

0 1.052

1 .w

1.032

1.026

1.022

1.019

1.014

1.010

1.00

6 1.003

1.001

1 .Ooo

1.348

1.228

1.16

6 1.129

1.103

1.085

I .072

1.062

I .053

.04 1

,033

.027

.023

.019

.014

1.010

I .006

1.003

1 .001

1 .Ooo

1.285

1.197

1.148

1.117

I .o%

1.080

1.068

1.058

1.05 1

1.

040

1.032

1.026

1.022

1.019

1.014

1.010

1.00

6 1.003

1.001

I .O

oo

1.303

1.208

1.155

1.122

1.099

1.083

1.070

I .060

1.053

I .042

1.034

1.028

1.023

1.019

1.014

1.010

1.006

1.003

1.001

I .O

oo

1.60

4 1.320

1.217

1.162

1.127

1. I03

I .086

1.073

1.062

1.054

1.042

1.034

1.028

1.023

1.020

1.01

5 1.010

1.006

1.003

1 .ool

1 .Ooo

x:,,,

77.7454

82.5287

86.82%

92.0100

95.6493

1.666

1.342

I .229

1.169

1.132

1.107

1.089

1.075

1.065

1.056

,044

.036

.029

,024

.020

.015

1.

01 1

1.007

1.003

1.001

1 .Ooo

1.713

1.359

1.238

1.175

1.136

1.110

1.09

1 1.

077

1.066

1.058

1.045

1.036

1.029

I .024

1.021

1.01

6 1.01 1

1.007

I .003

1.001

1 .#

85.5271

90.5312 95.0231 100.4250 104.2150

1-

2

1.29

7 3

1.20

7 4

1.15

7 5

1.12

5

6 1.

102

7 1.

086

8 1.

073

9 1.

063

10

1.05

5

12

1.04

3 14

1.

035

16

1.02

9 18

1.

024

20

1.02

1

24

1.01

6 30

1.

011

40

1.00

7 60

1.00

3 I2

0 1.

001

m

1.O

oo

xz,,,

93.2

70

-

1.31

5 1.

218

1.16

4 1.

130

1.10

6

Table9

(Contin

ued

)

m=

7 -

r=ll

r=

12

a

0.10

0 0.050

0.02

5 0.

010

0.00

5 0.

100

0.05

0 0.025

0.10

0 0.

005

1.62

7 1.

690

1.73

7

,089

,0

76

,065

,0

57

.045

.0

36

.030

1.

025

1.02

1

1.01

6 1.

01 1

1.00

7 1.

003

1.00

1

1 .Ooo

98.4

84

-

1.33

2 I .

227

1.17

1 1.

135

1.1

10

1.09

2 1.

078

1.06

7 I .0

59

1.04

6 I .

037

I .03 1

1.

026

1.02

2

1.01

7 1.

012

1.00

7 1.

004

1.00

1

1 .OOo

10

3.1 5

8

-

1.354

1.23

9 1.

179

1.14

0

1.11

4 1.

095

1.08

1 1.

070

1.06

1

1.04

8 1.

038

1.03

2 1.

027

1.02

3

1.01

7 1.

012

1.00

7 1.

004

1.00

1

1 .OOo

108.

771

-

1.37

1 I .

248

1.18

4 1.

145

1.11

8 1.

098

1.08

3 1.

072

1.06

2

1.04

9 I .

039

1.03

2 1.

027

1.02

3

1.01

7 1.

012

I .008

1.

004

1.00

1

1 .Ooo

11

2.70

4

1.53

1 1.

308

1.21

7 1.

166

1.13

3

1.10

9 1.

092

1.07

9 1.

068

1.06

0

1.04

7 1.

038

1.03

2 I .

027

1.02

3

1.01

7 1.

012

I .No8

1.00

4 1.

001

1 .Ooo

1.57

9 1.

326

1.22

8 1.

173

1.13

8

1.11

3 1.

095

1.08

I I .

070

1.06

2

1.04

9 1.

039

1.03

3

1.02

4

1.01

8 1.

012

1.00

8 1.

004

1.00

1

1 .O

oo

I .02

8

I .34

4 1.

238

1.18

0 1.

143

1.11

7 1.

098

1 .OM

1.07

3 I .064

1 .ow

1.04

1 1 .OM

1.02

8 1.

024

1.01

8 1.

013

1.00

8 1.

004

1.00

1

I .Ooo

1.36

6 1 .

w)

1.18

8 1.

149

1.12

2 ,1

02

,087

,0

75

.066

.052

,0

42

,035

1.

029

1.02

5

1.01

9 1.

013

1.00

8 1.

004

1.00

1

1 .Ooo

1.38

3 1.

259

1.19

4 1.

153

1.12

5 1.

104

I .089

1.

077

1.06

7

1.05

3 1.

043

1.03

5 1.030

1.02

5

1.01

9 1.

013

1.00

9 I .0

04

1.00

1

1 .Ooo

100.

9800

106

.394

8 11

1.24

23 1

17.0

565

121.

1263

r=1

3

r=1

4

a

0.10

0 0.

050

0.02

5 0.

010

0.00

5 0.

100

0.05

0 0.

025

0.01

0 0.

005

I-

2

1.32

0 3

1.22

8 4

1.17

5 5

1.14

1

6 1.

116

7 1.

098

8 1.

084

9 1.

073

10

1.06

4

12

1.05

1 14

1.

042

16

1.03

5 18

1.

029

20

1.02

5

24

1.01

9 30

1.

013

40

1.00

8 60

1.00

4 12

0 1.

001

#)

1.OOo

-

1.33

8 1.

238

1.18

2 1.

146

1.12

0 1.

102

1.08

7 1.

076

1.06

6

1.05

3 1.

043

1.03

6 1.

030

1.02

6

1.02

0 1.

014

1.00

9 1.

004

1.00

1

1 .Ooo

-

1.35

6 I .2

48

1.18

9 1.

151

1.12

4 1.

105

1.09

0 1.

078

1.06

8

1.05

4 1.

044

1.03

6 1.

03 1

1.02

6

I .020

1.

014

I .00

9 1.

004

1 .oO

l

I .Ooo

-

1.37

8 1.

26 1

1.19

7 1.

157

1.12

9 1.

108

1.09

3 1.

080

1.07

1

1.05

6 I .

045

1.03

8 1.

032

1.02

7

1.02

1 1.

014

1.00

9 1.

004

1.00

1

1 .Ooo

-

1.39

5 1.

270

1.20

3 1.

161

1.13

2 1.

111

1.09

5 1.

082

1.07

2

I .05

7 1.

046

1.03

8 1.

032

1.02

8

1.02

1 1.

015

1.00

9 1.

005

1.00

1

1 .ooo

xf,,,

108.

661

114.

268

119.

282

125.

289

129.

491

1.55

6 1.

33 I

1.23

8 1.

184

1.14

9

1.12

3 1.

105

I .090

1.

078

I .06

9

1.05

5 1.

045

1.03

7 1.

032

I .027

1.02

1 1.

015

1.00

9 1.

005

1.00

1

1 .Ooo

I .m-

1.35

0 I .2

49

1.19

2 1.

154

1.12

8 1.

108

1.09

3 1-

08 I

I .07

I

1.05

7 1.

046

1.03

9 1.

033

1.02

8

1.02

1 1.

015

1.00

9 1.

005

1 .oOl

1 .Ooo

1.65

2 1.

368

1.25

9 1.

198

1.15

9

1.13

2 1.1

1 I

1 .o%

I .08

3 1.

073

1.05

8 1.

047

1.03

9 1.

033

1.02

9

1.02

2 1.

015

1.01

0 1.

005

1.00

1

1 .Ooo

1.71

5 1.

391

1.27

2 1.

207

1.16

5

1.13

7 1.1

15

1.09

9 I .0

86

I .076

1.06

0 1.

049

1.04

1 I .0

34

1.03

0

I .022

1.01

6 1.

010

1.00

5 1.

001

1 .Ooo

1.76

3 1 .

a8

1.

28 1

1.21

3 1.

170

1.14

0 1.

118

1.10

1 1.

088

I .077

1.06

1 1.

050

1.04

1 1.

035

1.03

0

1.02

3 1.

016

1.01

0 1.

005

1.00

1

1 .Ooo

11

6.31

53 1

22.1

077

127.

2821

133

.475

7 13

7.80

32

OI

W

W

. . . . 4 - e m -

. . . . . - - e m -

. , . . . - . I ce

liii

c

2

i

634

r=1

7

r=1

8

(I

0.10

0 0.

050

0.02

5 0.

010

0.00

5 0.

100

0.05

0 0.

025

0.01

0 0.

005

1-

2

1.36

5 3

1.268

4

1.21

1 5

1.17

3

6 1.

145

7 1.

124

8 1.

108

9 1.0

95

10

1.08

4

12

1.06

7 14

1.

056

16

1.04

7 18

1.

040

20

1.03

4

24

1.02

6 30

1.

019

40

1.01

2 60

1.00

6 12

0 1.

002

a0

1.OOo

-

1.38

5 1.

280

,219

.1

79

.I50

.I28

.I11

.0

97

.086

1.06

9 1.

057

1.04

8 1.

041

1.03

5

1.02

7 1.

019

1.01

2 1.

006

1.00

2

1 .Ooo

-

1.40

3 I .2

9 1

1.22

6 1.

184

1.15

4 1.

131

1.11

4 1.1

00

1.08

8

1.07

I 1.

058

1.04

9 1.

042

1.03

6

I .02

8 1.

020

1.01

3 1.

006

I .00

2

1 .Ooo

-

1.42

7 1.

304

1.23

5 1.

191

1.15

9 1.

136

1.11

7 1.

103

1.09

1

1.07

3 1.

060

1.05

0 1.

043

1.03

7

1.02

8 1.

020

1.01

3

1.00

7 1 .

002

I .Ooo

-

I .445

1.31

4 1.

242

1.1%

1.163

1.

139

1.12

0 1.

105

1.09

3

I .074

1.

061

I .05

1 1.

044

1.03

8

1.02

9 1.

021

1.01

3 1.

007

1.00

2

I .Ooo

2

Xr

m

139.

149

145.

461

151.

084

157.

800

162.

481

I .60

5 1.

377

I .27

8 1.

220

1.18

1

1.15

2 1.

131

1.11

4 1.

100

1.08

9

I .072

1.

059

1.05

0 1.0

43

1.03

7

1.02

8 1.

020

1.01

3 1.

007

1.00

2

I .Ooo

1.65

4 1.

3%

I .290

1.

228

1.18

7

1.15

7 1.1

34

1.1 1

7 1.

103

1.09

1

I .074

1.

061

I .05 1

1.04

4 I .038

1.02

9 1.

02 1

1.

013

1.00

7 I .0

02

1 .Ooo

I .703

1.4

15

1.30

1 I .2

36

1.19

2

1.16

1 1.

138

1.12

0 1.

105

1.093

I .075

1.0

62

1.05

2 1 .045

I .039

1.03

0 I .0

2 I

1.01

4 1.0

07

I .002

1 .Ooo

I .768

I .4

39

1.31

5 1.

245

1.19

9

1.16

7 1.

142

1.12

4 1.

108

1 .O%

1.07

7 I .0

54

1.06

4 1.

046

1.04

0

1.03

1 1.

022

1.01

4 1.

007

I .002

I .OOo

1.81

6 1.

457

1.32

4 1.

252

1.20

4

1.17

1 1.

146

1.12

6 1.

111

1.09

8

1.07Y

1.

065

1.05

5 1.

047

1.04

0

1.03

1

1.02

2 1.

014

1.00

7 1.

002

1 .OOo

146.

7241

15

3.19

79 1

58.9

264

165.

8410

17

0.63

41

r=1

9

r=2

0

Q

0.10

0 0.

050

0.02

5 0.

010

0.00

5 0.

100

0.050

0.02

5 0.

010

0.00

5

I-

2

1.38

8 3

1.28

8 4

1.22

9 5

1.18

9

6 1.

159

7 1.

137

8 1.

119

9 1.

105

10

1.09

4

I2

1.07

6 14

1.

063

16

1.05

3 18

1.

045

20

1.03

9

24

1.03

0 30

1.

022

40

1.01

4 60

1.

007

120

1.00

2

00

1.OOo

-

1.40

8 1 .

300

I .23

7 1.

195

1.16

4 1.

141

1.12

3 1.

108

1.09

6

1.07

8 1.

065

1.05

4 I .@

I7

1 .w

1.03

1

1.02

2 1.

014

I .007

1.

002

1 .O

oo

-

1.42

7 1.

311

1.24

5 1.

201

1.16

9 1.

145

1.12

6 1.

111

1 .@

?9

1.08

0 1.

066

1.05

6 1.

048

1.04

1

I .032

1.

023

1.01

5 1.

007

1.00

2

1 .m

-

1.45

1 1.

325

1.25

4 1.

208

1.17

4 1.

149

1.13

0 1.

114

1.10

1

1.08

2 1.

068

I .057

1.

049

1.04

2

1.03

3 1.

023

1.01

5 1.

008

1.00

2

1 .Ooo

-

1.46

9 1.

335

1.26

1 1.

213

1.17

8 1.

152

1.13

2 1.

116

1.10

3

I .08

3 1.

069

1.05

8 1.

050

1.04

3

1.03

3 1.

024

1.01

5 1.

008

1.00

2

1 .Ooo

x:m

154

.283

16

0.91

5 16

6.81

6 17

3.85

4 17

8.75

5

1.62

9 1.

679

1.72

8 1.

793

1.84

3 1.

399

1.41

9 1.

438

1.46

2 1.

480

1.29

8 1.

310

1.32

1 1.

335

1.34

6 1.

238

1.24

6 1.

254

1.26

3 1.

270

1.1%

1.

203

1.20

9 1.

216

1.22

1

1.16

6 1.

172

1.17

6 1.

182

1.18

6 1.

144

1.14

8 1.

151

1.15

6 1.

159

1.12

5 1.

129

1.13

2 1.

136

1.13

9 1.

111

1.11

4 1.

117

1.12

0 1.

122

1.09

9 1.

101

1.10

4 1.

107

1.10

9

1.08

0 1.

082

1.08

4 1.

086

1.08

8 1.

067

1.06

8 1.

070

1.07

2 1.

073

1.05

6 1.

058

1.05

9 1.

060

1.06

2 1.

048

1.05

0 1.

051

1.05

2 1.

053

1.04

2 1.

043

1.04

4 1.

045

1.04

6

1.03

3 1.

033

1.03

4 1.

035

1.03

5 1.

024

1.02

4 1.

025

1.02

5 1.

026

1.01

5 1.

015

1.01

6 1.

016

1.01

6 1.

008

1.00

8 1.

008

1.00

8 1.

008

1.00

2 1.

002

1.00

2 1.

002

1.00

2

1.OOo

1.w

1.

Ooo

1.O

oo

1.OO

o

161.

8270

168

.613

0 17

4.64

78 1

81.8

403

186.

8468

m=

8

a

0.10

0 0.

050

0.02

5 0.01

0 0.

005

1 1.

491

-

r =8

2 1.2

70

3 1.1

85

4 1.1

38

5 1.1

08

6 1.0

88

7 1.0

73

8 1.

061

9 1.0

53

10

1.04

6

12

1.036

14

1.0

28

16

1.023

18

1.0

20

20

1.017

24

1.012

30

1.0

09

40

1.005

60

1.003

12

0 1.

001

cr)

1.Ooo

xR 7

8.85%

1.53

8 1.

288

1.195

1.1

44

1.113

1.09

1 1.0

76

I .064

1.05

5 1.

048

1.03

8 1.

030

1.025

1.0

2 1

1.01

7

1.013

1.

009

1.005

1.

003

1.001

1 .Ooo

83

.675

3

1.58

5 1.3

05

I .204

1.

150

1.11

7

1 .w5

1.07

8 I .0

66

1.05

1 1.0

49

1.03

9 1.

031

1.026

1.

022

1.01

8

1.01

3 1.

009

1.006

1.

003

1.00

1

1 .ooo

88.0

041

1.64

6 1.

692

1.32

6 1.

215

1.158

I.

123

1.09

9 I .

082

I .069

1.05

9 1.

05 1

1.04

0 1.0

3 1

1 .ox

I .022

1.01

8

1.01

4 1.

009

1.00

6 1.0

03

1.00

1

1 .Ooo

93.2

169

1.34

2 1.

224

1.163

1.1

26

1.10

2 1.0

84

1.07

1 I .

060

1.052

1.04

1 1.

032

1.027

1.0

23

1.018

1.01

4 1.0

10

1.00

6 1.0

03

1.00

1

1 .Ooo

96

.878

1

h

w

-a

m

w

m

r =

9 r=

lO

0.10

0 0.050

0.02

5 0.

010

0.00

5 0.

100

0.05

0 0.025

0.01

0 0.

005

1 1.

495

2 1.

277

3 1.

192

4 1.

144

5 1.

114

6 1.

093

7 1.

077

8 1.

066

9

1.05

7 10

1.

049

12

1.03

9 14

1.

031

16

1.02

6 18

1.

021

20

1.01

8

24

1.01

4 30

1.

010

40

1.00

6 60

1.00

3 12

0 1.

001

m

1.OOo

x:,,

87.7

430

1.54

1 1.

295

1.20

2 1.

151

1.1 1

9

1.09

7 1.

080

1.06

8 1.

059

1.05

1

1.04

0 1.

032

I .02

6 1.

022

1.01

9

1.01

4 1.

010

1.00

6 1.

003

1.00

1

1 .Ooo

92.8

083

1.58

7 1.

311

121

1 1.

157

1.12

3

1.10

0 1.

083

1.07

0 1.

061

1.05

3

1.04

1 1.

033

1.02

7 1.

023

1.01

9

1.01

5 1.

010

1.00

6 1.

003

1 .oOl

1 .OOo

1.64

8 1.

333

1.22

2 1.

165

1.12

9

1.10

4 1.

086

1.07

3 1.

063

1.05

5

1.04

3 1 .ON

1.02

8 1.

024

1.02

0

1.01

5 1.

010

1.00

6 1.

003

1.00

1

1 .Ooo

1.69

4 1.

349

1.23

1 1.

170

1.13

2

I. 10

7 1.

089

1.07

5 1.

064

I .O%

1.04

4 1.

035

I .029

1.

024

1.02

0

1.01

5 1.

01 1

1.00

7 1.

003

1.00

1

1 .Ooo

97.3

531

102.

8163

10

6.64

76

I SO

1 1.

286

1.20

0 1.

151

1.12

0

I .098

1.

082

1.07

0 1.

061

1.05

3

1.04

2 1.

034

1.02

8 1.

023

1.02

0

1.01

5 1.

01 1

1.00

6 1.

003

1.00

1

1 .Ooo

1.54

7 1.

303

1.20

9 1.

158

1.12

5

I. 10

2 1.

086

1.073

1.

063

1.05

5

1.04

3 1.

035

1.02

9 1.

024

1.02

1

1.10

6 1.

01 1

1.00

7 I .

003

1.00

1

1 .Ooo

1.59

3 1.

319

1.21

9 1.

164

1.13

0

1.10

6 1.

088

I .075

1.065

1.

057

1.04

4 1.

036

1.03

0 1.

025

1.02

1

1.10

6 1.

01 1

1.

007

1.00

3 1.

001

1 .Ooo

1.65

3 1.

341

1.23

0 1.

172

1.13

5

1.1

10

1.09

2 1.

078

1.06

7 1.

059

1.04

6 1.

037

1.03

0 1.

026

1.02

2

1.01

6 1.

01 I

1.00

7 1.

003

1.00

1

1 .m

1.69

8 1.

357

I .239

1.

177

1.13

9

1.11

3 1.

094

1.08

0 1.

069

1.06

0

1.04

7 1.

038

1.03

1 1.

026

1.02

2

1.01

7 1.

012

1.00

7 1.

004

.I .00

1

1 .Ooo

%.5

782

101.

8795

10

6.62

86

112.

3288

11

6.32

1 1

r=ll

r=12

0.050

0.025

0.010

0.005

0.100

0.050

0.025

0.010

0.005

1-

2

1.294

3 1.208

4 1.159

5 1.127

6 1.104

7 1.088

8 1.075

9 1.065

10 1.057

12

1.045

14

1.037

16

1.030

18

1.026

20

1.022

24

1.017

30

1.012

40

1.007

60

1.004

120

1.001

00

1.OOo

-

1.312

1.218

1.165

1.132

I. 108

1.09

1 1.078

I .067

1.059

I .046

1.038

1.03 1

I .026

1.022

1.017

1.012

I .007

1.00

4 1.001

1 .Ooo

-

1.328

1.227

1.172

1.136

1.112

I .094

1.080

1.069

1.061

1.048

1.039

1.032

1.027

1.023

1.017

1.012

1.008

1.00

4 1.001

1 .Ooo

-

1.350

1.239

1.179

I. 142

1.1 I6

1.097

1.083

I .072

1.063

I .049

I .040

1.033

I .028

1.024

1,018

1.013

1.008

1.00

4 1.001

1 .OOo

-

1.366

1.247

1.185

1.14

6

1.119

1.100

I .085

1.073

1.06

4

1.050

1.041

1.034

I .028

I .024

1.01

8 1.013

1.008

I .004

I .

001

1 .Ooo

x;,,,

105.372

110.898

I15.841

121.767

125.913

1.516

1.30

4 1.216

1.166

1.134

1.11

1 1.093

1.080

1.070

1.061

,049

.039

.033

.028

.024

.018

1.013

1.008

1.00

4 1.001

1 .O

oo

1.562

1.321

1.226

.I73

.I39

,114

.097

.083

.072

.063

1.050

1.041

1.034

1.029

1.024

1.019

1.013

I .008

1.00

4 1.001

1 .OOo

1.608

I .338

1.236

1.180

1.143

1.1 I8

I .ow

1.085

I .074

1.065

1.05 1

1 .042

1.035

I .029

1.025

1.019

1.013

1.008

1.00

4 1.

001

1 .Ooo

1.667

I .359

1.248

1.187

1.149

1.122

1.103

I .088

1.076

1.067

1.053

1.043

I .036

1.030

1.026

1.020

1.014

1.009

1.00

4 1.001

1 .Ooo

1.713

1.375

1.256

1.193

1.153

1.126

1.105

1.09

0 1.078

1.068

.OH

.0

44

.036

.03 1

.026

.020

1.01

4 1.009

1.00

4 1.001

1 .Ooo

114.1307 119.8709 125.0001 131.1412 135.4330

Tab

le 9

(C

on

fke

d)

r=1

3

r= 1

4 a

0.10

0 0.050

0.02

5 0.

010

0.00

5 0.

100

0.050

0.02

5 0.

010

0.00

5

l-

2

1.31

3 3

1.22

5 4

1.17

4 5

1.14

1

6 1.

117

7 1.

099

8 1.

085

9 1.

074

10

1.06

6

12

1.05

2 14

1.

043

16

1.03

5 18

1.

030

20

1.02

6

24

1.02

0 30

1.

014

40

1.009

60

1.00

4 12

0 1.

001

00

1.OOo

x;,

122.

858

-

1.33

1 1.

235

1.18

1 1.

146

1.12

1 1.

102

1.08

8 1.

077

1.06

7

1.05

4 1.

044

I .036

I .

03 1

1.02

7

I .020

1.

014

1.00

9 1.

004

1.00

1

I .Ooo

28.8

04

-

1.34

7 1.

245

1.18

8 1.

151

1.12

5 1.

105

1.09

0 1.

079

1.06

9

1.05

5 1.

045

1.03

7 1.

032

1.02

7

1.02

1 1.

015

1.00

9 1.

005

1 .oOl

1 .Ooo

34.1

11

~ .

-

1.36

9 1.

257

1.1%

1.

156

1.12

9 1.

109

1.09

3 1.

08 1

1.07

I

1.05

7 1.

046

1.03

8 1.

033

1.02

8

1.02

I 1.

015

1.00

9 1.

005

1.00

1

1 .Ooo

14

0.45

9

-

1.38

5 1.

266

1.20

1 1.

161

1.13

2 1.

111

1.09

6 I .

083

1.07

3

1.05

8 1.

047

1.03

9 1.

033

1.02

8

1.02

2 1.

015

1.01

0 1.

005

1.00

1

1 .Ooo

44.8

91

1.53

5 1.

323

1.23

4 1.

182

1.14

8

1.12

3 1.

105

1.09

1 1.

079

t ,07

0

1.05

6 1.

046

1.03

8 1.

032

1.02

8

1.02

1 1.

015

1.01

0 1.

005

1.00

1

1 .ooo

1.58

1 1.

626

1.34

1 1.

357

1.24

4 1.

254

1.18

9 1.

1%

1.15

3 1.

158

1.12

7 1.

131

1.10

8 1.

111

1.09

3 1.

0%

1.08

2 1.

084

1.07

2 1.

074

1.05

7 1.

059

1.04

7 1.

048

1.03

9 1.W

1.

033

1.03

4 1.

029

1.02

9

1.02

2 1.

022

1.01

6 1.

016

1.01

0 1.

010

1.00

5 1.

005

1.00

1 1.

001

1.OOo

1.O

Oo

1.68

6 1.

379

1.26

6 1.

204

1.16

4

1.13

6 1.

1 15

1.

099

1.08

6 1.

076

1.06

1 1.

050

1.04

1 1.

035

1.03

0

I .023

1.

016

1.01

0 1.

005

1.00

1

1 .Ooo

1.73

1 1.

395

1.27

5 1.

210

1.16

8

1.13

9 1.

1 18

1.

101

1.08

8 1.

078

1.06

2 1.

051

1.04

2 1.

036

1.03

1

1.02

3 1.

017

1.01

0 1.

005

1.00

1

1 .OO

o

31.5

576

137.

7015

143

.180

1 14

9.72

69 1

54.2

944

r=15

r=

16

u

0.

100

0.05

0 0.

025

0.01

0 0.

005

0.10

0 0.

050

0.02

5 0.

010

0.00

5

I-

2

1.33

3 3

1.24

3 4

1.19

0 5

1.15

5

6 1.

130

7 1.

111

8 1.

0%

9 1.

084

10

1.07

4

12

1.06

0 14

1.

049

16

1.04

1 18

1.

035

20

1.03

0

24

1.02

3 30

1.

016

40

1.01

0 60

1.00

5 12

0 1.

002

00

1.OOo

-

1.35

1 I .

253

1.19

8 1.

160

1.13

4 1.

114

1 .O

N

1.08

7 1.

076

1.06

1 1.

050

1.04

2 1.

036

1.03

1

1.02

4 1.

017

1.01

1 1.

005

I .00

2

1 .Ooo

-

1.36

8 1.

263

1.20

4 1.

165

1.13

8 1.

117

1.10

1 1.

089

1.07

8

1.06

3 I .0

52

1.04

3 1.

037

1.03

2

1.02

4 1.

017

1.01

I 1.

005

1.00

2

I .Ooo

-

1.38

9 1.

275

1.21

2 1.

171

1.14

3 1.

121

1.10

5 1.

091

I .08

1

I .06

5 1.

053

1.04

4 1.

038

1.03

2

1.02

5 1.

018

1.01

I I .

006

I .002

1 .Ooo

-

1.40

6 1.

284

1.21

8 1.

176

1.14

6 1.

124

1.10

7 I .0

93

1.08

2

1.06

6 I .054

1.04

5 1.

038

1.03

3

1.02

5 1.

018

1.01

1 I .006

1.00

2

1 .Ooo

xf

, 14

0.23

3 14

6.56

7 15

2.2l

l 15

8.95

0 16

3.64

8

1.55

5 I .3

43

1.25

2 1.

198

1.16

2

1.13

6 1.

117

1.10

1 1.

089

1.07

9

1.06

4 1.

052

1.04

4 1.

038

1.03

2

1.02

5 1.

018

1.01

1 1.

006

I .002

I .Ooo

1.60

1 1.

36 1

1.26

3 1.

206

1.16

8

1.14

1 1.

120

1.10

4 I .0

92

1.08

1

1.06

5 I .0

54

1.04

5 1.

038

1.03

3

1.02

6 1.

018

1.01

2 1.

006

I .002

1 .Ooo

I .646

1.37

8 I .2

72

1.21

2 1.

173

1.14

5 1.

123

1.10

7 1.

094

1.08

3

1.06

7 1.

055

1.04

6 1.

039

I .034

I .026

1.

019

1.01

2 I .006

I .002

1 .Ooo

I .706

1.

400

1.28

5 1.

221

1.17

9

1.14

9 1.

127

1.1 1

0 1.

097

1.08

5

1.06

9 I .

056

1.04

7 1.

040

I .035

1.02

7 1.

019

1.01

2 1.

006

I .002

I .Ooo

I .75

1 1.

416

I .29

4 ,2

21

. I83

.153

.1

30

.113

,099

.087

1.07

0 1.

057

I .04

8 1.

041

1.03

5

1.02

7 1.

019

1.01

2 1.

006

I .002

1 .Ooo

148.

8853

155

.404

7 16

1.20

87 1

68.1

332

172.

9575

m P

Tab

le9

(Conllr

ared

)

m=

8 -

r=1

7

r=1

8

(I

0.10

0 0.

050

0.02

5 0.

010

0.00

5 0.

100

0.05

0 0.

025

0.01

0 0.

005

1.57

5 1.

621

1.66

7 1.

727

1.36

3 1.

381

1.27

0 1.

281

1.21

5 1.

222

1.17

7 1.

183

1.15

0 1.

154

1.12

9 1.

133

1.11

2 1.

115

1.09

9 1.

102

1.08

8 1.

091

1.07

1 1.

073

1.05

9 1.

061

1.05

0 1.

051

1.04

3 1.

044

1.03

7 1.

038

1.02

9 1,

029

1.02

1 1.

021

1.01

3 1.

014

1.00

7 1.

007

1.00

2 1.

002

1.OOo

1.

Ooo

1.39

8 1.

291

1.22

9 1.

188

1.15

8 1.

136

1.11

8 1.

104

1.09

3

1.07

5 1.

062

I .05

2 1.

045

1.03

9

1.03

0 1 .O

22

1.01

4 I .

007

1.00

2

1 .Ooo

1.42

0 1.

304

1.23

8 1.

194

1.16

3 1.

140

1.12

2 1.

107

1.09

5

1.07

7 1.

064

1.05

4 1.

046

1.04

0

1.03

1 1.

022

1.01

4 1.

007

1.00

2

1 .Ooo

1 2 3 4 5 6 7 8 9 10

12

14

16

18

20

24

30

40

60

120

a2

-

1.35

3 1.

261

1.20

7 1.

170

1.14

3 1.

123

1.10

7 I .094

1.08

4

1.06

7 1.

056

1.04

7 1.

040

1.03

5

1.02

7 1.

019

1.01

2 1.

006

1.00

2

1 .Ooo

-

1.37

1 1.

272

1.21

4 1.

175

1.14

7 1.

126

1.11

0 1.

097

1.08

6

1.06

9 1.

057

1.04

8 1.

041

I .036

1.02

7 1.

020

1.01

3 1.

006

1.00

2

1 .Ooo

-

1.38

8 1.

282

1.22

1 1.

180

1.15

1 1.

130

1.1 1

3 1.

099

I .088

1.07

1 1.

058

1.04

9 1.

042

1.03

6

1 .OX8

1.02

0 1.

013

I .00

7 1.

002

1 .O

oo

-

-

1.41

0 -

1.29

4 1.

303

1.22

9 1.

235

1.18

7 1.

191

1.15

6 1.

160

1.13

4 1.

136

1.11

6 1.

118

1.10

2 1.

104

1.09

0 1.

092

1.07

3 1.

074

1.06

0 1.

061

1.05

0 1.

051

1.04

3 1.

044

1.03

7 1.

038

1.02

9 1.

029

1.02

1 1.

021

1.01

3 1.

013

1.00

7 1.

007

1.00

2 1.

002

1.OOo

1.

Ooo

1.77

3 1.

437

1.31

3 1.

244

1.19

9

1.15

7 1.

143

1.12

4 1.

109

1.09

7

I .07

8 1.

065

1.05

5 1.

047

1 .040

1.03

1 1.

022

1.01

4 I .

007

1.00

2

1 .Ooo

x;

, 15

7.51

8 16

4.21

6 17

0.17

5 17

7.28

0 18

2.22

6 16

6.13

18 1

73.0

041

179.

1 I37

18

6.39

30 1

91.4

585

m=

9 -

r=9

r=

lO

0.05

0 0.

025

0.01

0 0.

005

0.10

0 0.

050

0.02

5 0.

010

0.00

5

1 1.

495

2 1.

282

3 1.

197

4 1.

149

5 1.

119

6 1.

097

7 1.

081

8 1.

069

9 1.

060

10

1.05

2

12

1.04

1 14

1.

033

16

1.02

7 18

1.

023

20

1.02

0

24

1.01

5 30

1.

010

40

1.00

6 60

1.

003

120

1.00

1

m

1.w

I .54

0 1.

299

1.20

7 1.

156

1.12

3

1.10

1 I .OM

I .07

2 I .

062

1.05

4

1.04

3 I .

034

1.02

8 1.

024

1.02

0

1.01

5 1.

011

1.00

7 1.

003

1 .00

1

1 .O

oo

1.58

5 1.

315

1.21

6 1.

162

1.12

8

1.10

4 1.

087

1.07

4 I .064

1.05

6

1.04

4 1.

035

1.02

9 1.

024

1.02

1

1.01

6 1.

01 1

I .007

1.

003

1.00

1

1 .Ooo

1.64

5 1.

337

1.22

7 1.

169

1.13

3

1.10

8 I .ow

1.07

7 1.

066

I .05

8

I .045

1.

036

1.03

0 1.

025

1.02

2

1.01

6 1.

01 1

1.00

7 1.

003

1.00

1

1 .Ooo

1.69

0 I .3

53

1.23

6 1.

175

1.13

7

1.1 1

1 1.

093

1.07

9 1.

068

1.05

9

1.04

6 1.

037

1.03

I 1.

026

1.02

2

1.01

7 1.

012

I .007

1.

003

I .001

1 .Ooo

x:,

97.6

7% 1

03.0

095

107.

7834

113

.512

4 11

7.52

42

1.49

7 I .

288

I .203

1.

155

1.12

4

1.10

2 1.

086

1.07

3 I .0

64

1.05

6

1.04

4 I .0

36

1.03

0 1.

025

1.02

I

1.01

6 1.

01 1

I .007

1.

003

1.00

1

1 .Ooo

1.54

2 1.

305

1.21

3 1.

162

1.12

9

1.10

6 1.

089

1.07

6 1.

066

I .058

1.04

5 1.

037

1.03

0 1.

026

I .02

2

1.01

7 1.

012

1.00

7 1.

004

1.00

1

1 .OOO

1.58

6 1.

321

I .222

1.

168

1.13

3

1. I0

9 1.

092

1.07

8 I .0

68

I .05

9

1.04

7 1.

038

1.03

1 1.

026

I .023

1.01

7 1.

012

1.00

7 1 .w

1.

001

I .Ooo

I .645

1.

342

1.23

3 1.

175

1.13

9

1.1

I3

1.09

5 1.

08 1

1.07

0 1.

061

1.04

8 1.

039

I .032

1.

027

I .023

1.01

8 1.

012

I .008

I .m

1.

001

1 .Ooo

1.69

0 1.

357

1.24

2 1.

181

1.14

3

1.11

6 1.

097

1.08

3 1.

072

1.06

3

1.04

9 I .0

40

1.03

3 1.

028

1.02

4

1.01

8 1.

013

1.00

8 I .0

04

1.00

1

1 .Ooo

10

7.56

50 1

13.1

453

118.

1359

124

.1 1

63 1

28.2

989

QI e

Tab

te9

(Contin

ued

)

m =

9

r=

ll

r=1

2

0.05

0 0.

025

0.01

0 0.

005

0.10

0 0.

050

0.02

5 0.

010

0.00

5

1-

2

1.29

4 3

1.21

0 4

1.16

1 5

1.13

0

6 1.

107

7 1.

091

8 1.

078

9 1.

068

10

1.05

9

12

1.04

7 14

1.

038

16

1.03

2 18

1.

027

20

1.02

3

24

1.01

8 30

1.

012

40

1.00

8 60

1.

004

120

1.00

1

03

1.OOo

x;,

1 17.

407

-

1.31

1 1.

219

1.16

8 1.

134

1.11

1 1.

094

1.08

0 1.

070

1.06

1

I .04

8 1.

039

1.03

3 1.

028

1.02

4

1.01

8 1.

013

1.00

8 1.

004

1.00

1

1 .ooo

123.

225

-

1.32

7 1.

229

1.17

4 1.

139

1.1 1

4 I .0

96

1.08

3 1.

072

1.06

3

I .050

1.

040

1.03

4 1.

028

1.02

4

1.01

8 1.

013

1.00

8 1.

004

1.00

1

1 .OoO

128.

422

-

I .348

1 2

40

1.18

2 1.

144

1.11

9 1.

100

1.08

5 1.

074

I .065

1.05

I 1.

042

1.03

5 1.

029

1.02

5

1.01

9 1.

013

1.00

8 1.

004

1.00

1

1 .Ooo

134.

642

-

I .364

1.

248

1.18

7 1.

148

1.12

2 1.

102

1.08

7 1.

076

1.06

6

1.05

2 1.

043

1.03

5 1.

030

1.02

5

1.01

9 1.

014

1.00

8 1.

004

1.00

1

1 .Ooo

138.

987

I so6

1.

302

1.21

7 1.

168

1.13

6

1.11

3 1.

095

I .082

1.

072

1.06

3

1.05

0 1.

041

1.03

4 1.

029

1.02

5

1.01

9 1.

013

1.00

8 I .0

04

1.00

1

I .Ooo

.550

.3

19

.227

.1

75

.I41

.I16

1.

099

I .08

5 1.

074

1.06

5

1.05

2 1.

042

1.03

5 1.

030

1.02

6

1.01

9 1.

014

1.00

9 1.

004

1.00

1

1 .Ooo

1.594

I .

335

1.23

6 1.

181

1.14

5

1.12

0 1.

101

1.08

7 1.

076

1.06

7

I .053

1 .@

I3

1.03

6 1.

030

1.02

6

1.02

0 1.

014

1.00

9 1.

004

1.00

1

1 .OOo

1.45

2 1.

355

1.24

7 1.

188

1.15

1

1.12

4 1.

105

I .ow

1.

078

1.06

9

1.05

5 1.

044

1.03

7 1.

03 1

1.02

7

1.02

0 1.

014

1.00

9 1.

005

1.00

1

1 .Ooo

~

1.6%

1.

37 1

1.

256

1.19

4 1.

155

1.12

7 1.

107

1.09

2 1.

080

1.07

0

I .056

1.

045

1.03

8 1.

032

1.02

7

1.02

1

1.01

5 1.

009

1 .OM

1.00

1

1 .Ooo

127.

21 1 I

13

3.25

69 1

38.6

506

145.

0988

149

.599

4

r=1

3

r=1

4

(1

0.10

0 0.

050

0.025

0.

010

0.00

5 0

.10

0.

050

0.02

5 0.

010

0.00

5

1-

2

1.31

0 3

1.22

4 4

1.17

5 5

1.14

2

6 1.

118

7 1.

101

8 1.

087

9 1.

076

10

1.06

7

12

14

16

18

20

24

,054

.w

,0

37

.03

I .0

27

,020

30

1.

015

40

1.00

9 60

1.00

5 12

0 1.

001

w

1.O

oo

-

I .326

1.

234

1.18

1 I.

I47

1.12

2 1.

104

I .089

1.

078

1.06

9

.055

,045

.038

.0

32

.028

.02 1

1.

015

1.00

9 1.

005

1.00

1

1 .Ooo

-

1.34

3 1.

243

1.18

8 1.

151

1.126

1.

107

I .092

I .0

80

1.07

1

1.05

6 1.

046

I .039

1.0

33

1.028

1.02

1 1.

015

1.01

0 I .0

05

1.00

1

1 .Ooo

-

1.36

3 1.

255

1.19

5 1.

157

1.130

1.1

10

I .095

I .0

83

1.07

3

,058

.0

47

.QQo

.0

34

.029

.022

1.

016

1 .Ol

O I .0

05

I .00

1

1 .Ooo

-

1.379

1.2

63

1.20

1 1.

161

1.13

3 1.

113

1.09

7 I .0

84

1.074

1.05

9 1.

048

I .040

1.03

4 I .0

29

I .022

1.

016

1.01

0 I .

005

1.00

1

I .Ooo

I .520

1.

318

1.23

2 1.

182

1.14

8

1.12

4 1.

106

1.09

2 1.

080

1.07

1

1.05

7 1.

047

1.03

9 1.

033

1.02

9

1.02

2 1.

016

1.01

0 1.

005

1.00

1

I .Ooo

1.56

3 I .3

35

I .242

1.

189

1.15

1

1.12

8 1.

109

1.09

4 1.

083

1.07

3

1.05

9 1.

048

I .w

1.03

4 I .0

30

1.02

3 1.

016

1.01

0 1.

005

1.00

1

I .Ooo

xs,,

136.

982

143.

246

148.

829

155.

4%

160.

146

I .607

I .3

5 1

1.25

I 1.

195

1.15

8

1.13

2 1.

112

1.097

1.08

5 1.0

75

1.06

0 1.

049

1.04

1 1.

035

1.03

0

I .02

3 1.

016

1.01

0 I .005

1.00

2

1 .Ooo

I .664

1.37

1 1.

263

1.20

3 1.

164

1.13

6 1.

116

1.10

0 1.

087

1.07

7

1.06

2 1.

05 1

I .04

2 1.

036

1.03

1

1.02

4 1.

017

1.01

I 1.

005

1.00

2

1 .Ooo

I .70

8 1.

387

1.27

1 1.

208

1.16

8

1.13

9 1.

118

1.10

2 I .

089

1.07

9

1.06

3 1.

051

I .04

3 1.

037

I .03

2

1.02

4 1.

017

1.01

1 1.

005

1.00

2

I .OOo

146.

7241

15

3.19

79 1

58.%

24

165.

8410

170

.634

1

1-

-

2 1.

326

1.34

3 3

1.24

0 1.

250

4 1.

189

1.1%

5

1.15

5 1.

160

6 1.

130

1.13

4 7

1.11

1 1.

115

8 1.09

7 1.

099

9 1.

085

1.08

7 10

1.

075

1.07

7

12

1.06

1 1.

062

14

1.05

0 1.

051

16

1.04

2 1.

043

18

1.03

6 1.

037

20

1.03

1 1.

032

24

1.02

4 1.

024

30

1.01

7 1.

017

40

1.01

1 1.

011

60

1.00

5 1.

006

120

1.00

2 1.

002

00

1.OOo

1.O

Oo

-

-

-

1.35

9 -

-

1.25

9 1.

271

1.27

9 1.

202

1.21

0 1.

216

1.16

5 1.

170

1.17

4

1.138

1.

142

1.14

5 1.

118

1.12

1 1.

124

1.10

2 1.

105

1.10

7 1.

089

1.09

2 1.

094

1.07

9 1.

081

1.08

3

1.06

4 1.

065

1.06

6 1.

052

1.05

4 1.

055

1.04

4 1.

045

1.04

6 1.

037

1.03

8 1.

039

1.03

2 1.

033

1.03

4

1.02

5 1.

025

1.02

6 1.

018

1.01

8 1.

018

1.01

1 1.

012

1.01

2 1.

006

1.00

6 1.

006

1.00

2 1.

002

1.00

2

1.OOo

1.O

Oo

1.OOo

Tab

le 9

(C

mrh

ued

)

m=

9

-

r=15

r=

16

a

0.10

0 0.

050

0.02

5 0.

010

0.00

5 0.

100

0.05

0 0.

025

0.01

0 0.

005

1.53

6 1.

579

1.62

2 1.

679

1.72

2 1.

335

1.35

2 1.

248

1.25

8 1.

1%

1.20

3 1.

161

1.16

6

1.13

6 1.

140

1.11

7 1.

120

1.10

2 1.

104

1.08

9 1.

092

1.07

9 1.

082

1.06

4 1.

066

1.053

1.

054

1.04

5 1.

046

1.03

8 1.

039

1.03

3 1.

034

1.02

6 1.

026

1.01

8 1.

019

1.01

2 1.

012

1.00

6 1.

006

1.00

2 1.

002

1.oo

o 1.O

Oo

1.36

8 1.

267

1.21

0 1.

171

1.14

4 1.1

23

1.107

1.

094

1.08

3

1.06

7 1.

056

1.04

7 1.

040

1.03

5

1.02

7 1.

019

1.01

2 1.

006

1.00

2

1 .Ooo

1.38

9 1.

279

1.21

8 1.

177

1.14

8 1.

127

1.1

10

1.09

7 1.

086

1.06

9 1.

057

1.04

8 1.

041

1.03

5

1.02

7 1.

020

1.01

2 1.

006

1.00

2

1 .OOo

1.40

4 1.

288

I .223

1.

181

1.15

2 1.

130

1.1

12

I .099

1.08

7

1.07

0 1.

058

1.04

9 1.

042

1.03

6

1.02

8 1.

020

1.013

1.

006

1.00

2

1 .OOo

xs

, 15

6.44

0 16

3.1

16

169.

056

176.

138

181.

070

166.

1318

173

.004

1 17

9.11

37

186.

3930

191

.458

5

m=

lO

r=lO

r=

ll

0.05

0 0.

025

0.01

0 0.

005

0.10

0 0.

050

0.02

5 0.

010

0.00

5

I 1.

4%

2 1.2

91

3 1.

208

4 1.

160

5 1.

128

6 1.

106

7 1.

090

8 1.

077

9 1.

067

10

1.05

9

12

1.04

7 14

1.

038

16

1.03

1 18

1.

027

20

1.02

3

24

1.01

7 30

1.

012

40

1.00

8 60

1.00

4 12

0 1.

001

M

1.Ooo

I .54

0 I .3

08

1.21

7 1.

166

1.13

3

1.11

0 I .0

93

1.07

9 1.

069

1.06

1

1.04

8 I .0

39

1.03

2 1.

027

1.02

3

1.01

8 1.

01 3

1.00

8 1.

004

1.00

1

I .Ooo

1.58

4 1.

324

1.2.26

1.

172

1.13

7

1.11

3 1.

095

I .082

1.

07 1

1.06

2

1.04

9 1 .oQo

1.03

3 1.

028

1.02

4

1.01

8 1.

013

I .008

1 .m

1.00

1

I .Ooo

1.64

1 1.

345

1.23

8 1.

180

1.14

3

1.11

7 1 .o

w

I .OM

1.07

3 1.

064

1.05

1 1.

041

1.03

4 1.

029

1.02

5

1.01

9 1.

01 3

1.00

8 1.

004

1.00

1

1 .Ooo

1.58

6 I .

360

1.24

6 1.

185

1.14

7

1.12

0 1.

101

I .08

6 1.

075

1.06

6

1.05

2 1.

042

1.03

5 1.

029

1.02

5

1.01

9 1.

013

1.00

8 1.

004

1.00

1

1 .Ooo

x:,,,

18.4

980

124.

3421

129

.561

2 13

5.80

67 1

40.1

695

-

1.2%

1.

213

1.16

5 1.

133

1.11

1 1.

094

1.08

I I .

070

1.06

2

I .04

9 1.

040

1.03

4 1.

028

I .02

4

1.01

9 1.

013

I .008

1.00

4 1.

001

1 .Ooo

12

9.38

5

-

1.31

3 1.

222

1.17

1 1.

138

1.1

14

1.09

7 1.

083

1.07

2 1.

064

I .05

1 1-

041

1.03

4 1.

029

1.02

5

1.01

9 I .

013

I .00

8 1.

004

1 .00

1

1 .Ooo

135.

480

-

I .32

9 1.

231

1.17

7 1.

I42

1.1

18

1.09

9 1.

085

1.07

4 1.

065

1.05

2 I .0

42

1.03

5 1.

030

1.02

6

I .02

0 1.

014

1.00

9 1.

004

1.00

1

1 .Ooo

14

0.91

7

-

-

1.34

9 -

1.24

3 1.

251

1.18

5 1.

190

1.14

8 1.

152

1.12

2 1.

125

1.10

3 1.

105

1.08

8 1.

090

1.07

7 1.

078

1.06

7 1.

069

1.05

4 1.

055

1.04

4 1.

044

1.03

6 1.

037

1.03

1 1.

031

1.02

6 1.

027

1.02

0 1.

020

1.01

4 1.

014

1.00

9 1.

009

1.00

4 1.

005

1.00

1 1.

001

1.OOo

1.

Ooo

147.

414

151.

948

Tab

le 9

(C

onri

nu

ed)

~ ~

~~

~

m=

lO

r=1

2

r=14

a

0.10

0 0.

050

0.02

5 0.

010

@.0

05

0.10

0 0.050

0.02

5 0.

010

3.005

I 1.

500

1.54

3 1.

585

1.64

1 1.

684

1.50

9 1.

551

1.59

3 1.

648

1.69

0 2

1.30

2 1.

318

1.33

4 1.

354

3 1.

219

1.22

8 1.

237

1.24

8 4

1.17

0 1.

177

1.18

3 1.

190

5 1.

138

1.14

3 1.

148

1.15

3

6 1.

115

1.11

9 1.

123

1.12

7 7

1.09

8 1.

101

1.10

4 1.

107

8 1.

085

1.08

7 1.

090

1.09

2 9

1.07

4 1.

076

1.07

8 1.

081

10

1.06

5 1.

067

1.06

9 1.

071

12

1.05

2 1.

054

1.05

5 1.

057

14

1.04

3 1.

044

1.04

5 1.

046

16

1.03

6 1.

037

1.03

8 1.

039

I8

1.03

0 1.

031

1.03

2 1.

033

20

1.02

6 1.

027

1.02

7 1.

028

24

1.02

0 1.

020

1.02

1 1.

021

30

1.01

4 1.

015

1.01

5 1.

015

40

1.00

9 1.

009

1.00

9 1.

010

60

1.00

4 1.

005

1.00

5 1.

005

120

1.00

1 1.

001

1.00

1 1.

001

30

1.OOo

1.O

Oo

1.OOo

1.O

Oo

xf,

140.

2326

14

6.56

74

152.

21 1

4 15

8.95

02

1.36

9 1.

257

1.1%

1.

157

1.13

0 1.

110

1.09

4 1.

082

1.07

2

1.05

8 1.

047

I .03

9 1.

033

1.02

9

1.02

2 1.

015

1.01

0 1.

005

1.00

1

1 .OoO

163.

6482

1.31

5 I .

232

1.18

2 1.

149

1.12

6 1.

107

1.09

3 1.

082

1.07

3

I .058

1.

048

i .040

1.03

5 1.

030

1.02

3 1.

016

1.01

0 1.

005

I .00

2

1 .O

oo

1.33

1 1.

241

1.18

9 1.

154

1.12

9 1.

111

1.09

6 1 .OM

1.07

5

1.06

0 1.

049

1.04

2 1.

035

I .03

1

1.02

4 1.

017

1.01

1

1.00

5 1.

002

1 .ooo

1.34

7 1.

250

1.19

5 1.

159

1.13

3 1.

113

1.09

8 1.

086

1.07

6

1.06

1 1.

05 1

I .

042

1.03

6 1.0

3 1

1.02

4 1.

017

1.01

1

1.00

6 1.

002

I .OOo

1.36

7 1.

382

1.26

1 1.

269

1.20

3 1.

208

1.16

4 1.

168

1.13

7 1.

141

1.11

7 1.

119

1.10

1 1.1

03

1.08

9 1.

090

1.07

8 1.

080

1.06

3 1.

064

1.05

2 1.

053

1.04

3 1.

044

1.03

7 1.

038

1.03

2 1.

033

1.02

5 1.0

25

1.01

8 1.

018

1.01

1 1.

011

1.00

6 1.

006

1.00

2 1.

002

1.OOo

1.O

Oo

161.

8270

16

8.61

30 1

74.6

478

181.

8403

186.

8468

m=12

r=12

a

0.10

0 0.

050

0.025

0.010

0.005

1 2 3 4 5 6 7 8 9

10

12

14

16

18

20

24

30

40

60

120 00

2 X

rm

1.495

1.30

6 1.225

1.145

1.122

1.104

1.091

1.080

1.07 1

1.057

I .047

1.039

1.034

1.029

1.022

1.016

1.010

1.005

1.001

1 .ooo

1.m

1.322

1.234

1.184

1.15

0

1.126

1.107

1.093

1.082

1.072

1.058

1.04

8 1.

040

1.034

1.030

1.023

1.016

1.010

1.005

1.002

1 .Ooo

1.576

1.337

1.243

1.190

1.154

1.129

1.110

1.095

1.084

1.074

1.060

1.049

1.041

1.035

1.030

1.023

1.017

1.01 1

1.005

1.002

1 .Ooo

1.630

1.356

1.25

4 1.197

1.16

0

1.133

1.1 14

I .09

8 1.086

1.076

1.061

1.05

0 1.042

1.036

1.031

1.024

1.017

1.01

1 1.005

1.00

2

I .OOo

1.67 1

1.371

1262

1.20

2 1.163

1.136

1.116

1.100

1.088

1.078

1.062

1.05 1

1.043

1.037

1.032

1.024

1.017

1.01 1

1.

006

1.002

1 .OOo

166.1318 173.0041 179.1137 186.3930 191.4585

aspects of multivariate statistical theory - robb muirhead (appendix)

Documents

p x p matrix

p x q matrix

q x p matrix

p x r matrix

p x q matrices

square matrix of order

identity matrix of order

matrix theory