Characterization and tabulation of a density giving significance levels in variance testing for the
bivariate normal case
by Raymond P Hitchcock
A THESIS Submitted to the Graduate Faculty in partial fulfillment of the requirements for the degree
of Master of Science in Mathematics
Montana State University
© Copyright by Raymond P Hitchcock (1959)
Abstract:
This thesis is concerned with an examination of significance levels in variance testing in the case of
sampling from a bivariate normal density with a nonzero correlation coefficient. The density related to
this situation is derived from the Wishart distribution. Using this basic density, a characterization is
made with the emphasis being placed on the behavior of the density for special cases. A tabulation of
the density follows, and conclusions are drawn as to the effect of the correlation coefficient on
significance levels in testing for homogeneity of variance. The numerical analysis techniques, tables,
and IBM 650 programs necessary for the numerical examinations are included. CHARACTERIZATION A N D TABULATION OF A DENSITY
GIVING SIGNIFICANCE: LEVELS IN VARIANCE
TESTING F O R THE BIVARIATE NORMAL CASE
by
RAYMOND P. HITCHCOCK
A THESIS
Submitted to the Graduate Faculty
>
.
:
;
partial fulfillment of the requirements
for the degree o f
!foster o f Science in Mathematics
at
Montana State College
Approved:
/Head, Major Department
Chairmi
Examining Comffitti
Bozeman, Montana
■
fV318
^^35c
G*p. 2.
Page
ACKNOWLEDGEMENT
3
ABSTRACT
4
I.
INTRODUCTION A N D STATEMENT OF THE PROBLEM
5
II.
ANALYTIC EXAMINATIONS OF THE DENSITY FUNCTION
7
A. Derivation of the Density
7
IH.
B.
Verification of the Reciprocal Relationship
for Percentage Points
16
C.
A n Equation for the Mode
18
D.
Behavior of the Density for Small n
20
E.
Moments of the Density
22
NUMERICAL ANALYSIS
A.
Formula for Programning the IBM
B.
Numerical Methods
25
650
25
27
IV.
NUMERICAL EXAMINATIONS OF DATA AN D CONCLUSIONS
31
V.
TABLES
38
Table I - Upper Percentage Points
38
Table II -F(u) for Upper Percentage Points of
the F Distribution
4o
APPENDIX
42
A.
Program for Numerical Integration
43
B.
Program for Interpolation of Percentages
47
C.
Program for Numerical Differentiation
50
D.
Tabulation of the Density and Cumulative Distribution
54
VI.
VII. LITERATURE CITED
117
136908
3
ACKNOWLEDGEMENT
The writer wishes to express his thanks tp Dr. Franklin. S. McFeely
for suggesting this problem a n d for his aid and encouragement in the
completion of this thesis.
Thanks are also due to the Computer Laborar
tory of the Mathematics Department at Montana State College for the use
of the IBM
650 without, wh i c h the thesis would have been impossible, and
to Mrs. Janet Bierrum for h e r careful typing and for making a number of
valuable suggestions.
4
ABSTRACT
This thesis is concerned with an examination of significance levels
in variance testing in the case of sampling from a bivariate normal den­
sity w i t h a nonzero correlation coefficient.
The density related to this
situation is derived from the Wishart distribution. Using this basic
density, a characterization is.made with the emphasis be i n g placed on
the behavior of the density for special c a s e s . A tabulation, o f the den­
sity follows, and conclusions are drawn as to the effect of the correla­
tion coefficient oh significance levels in testing for homogeneity of
variance. The numerical analysis techniques, tables, and. IBM 650 p r o ­
grams necessary for the numerical examinations are included.
5
I.
IKTRODUCTIOK A H D STATEMENT OF ,THE PROBLEM
A n important,phase of statistical inference is testing for homogene­
ity of v a r i a n c e .
The assumption of homogeneity of variance is necessary
in analysis of variance techniques for tests of hypotheses, e.g«, u
H g = , » <.=
=
. and, if there is question as to the validity of this
assumption, the test for homogeneity should be made before proceeding to
the tests of hypotheses concerning location parameters.
There are also
-many times when the test for homogeneity of variance is itself the prin­
cipal object of the experiment.
In the general cage of k variances, a n approximate test due to
Bartlett is available.
In the case of two variances, an exact test
made using the F distribution.
In the usual variance testing situation,
because of the assumption of stochastic independence, correlation co­
efficients are irrelevant in null a nd alternative hypothesis distribu­
tions.
If, however, sampling is from a multivariate normal distribu­
tion, the. various correlation coefficients appear in the joint distri­
bution of the sample variances and covariances.
In the case of sampling
from a bivariate population, there is a single correlation coefficient,,
p, to b e considered;, a n d we no longer have two independent samples, with
the resulting independent estimates of two variances, b ut rather a single
bivariate sample from which we obtain two estimates of possibly different
1
f
variances whose joint distribution involves p.
■
•
:
. . . . . .
•
,
6.
<_
A test for equality of variances is sometimes required when w e have
a single sample from a bivariate normal distribution.
The question
then
arises, as to the effect of p upon the true significance levels if the usual
,
■
'
' ' .r\
test is made for equality of variances taking the critical points from
the F distribution, which, of course, is not the appropriate one in this
case.
The particular distribution required is that of the quotient of
the ratio of each unbiased sample variance to the corresponding popula­
tion variance/ a n d this distribution involves p, the ratio o f the true
variances, and n (the bivariate sample size minus. One),
Investigation
of significance l e v e l s .implies examination o f the null hypothesis dis­
tribution/ i.e,, the case of equal variances, so that the ra t i o ;o f the
true variances becomes unity, and only p and the sample size appear as
parameters in the distribution.
'
The p roblem is then (i) that of tabulating the applicable distri­
but i o n for various values o f p and n in order to obtain percentage points
and probabilities associated w i t h percentage points of the F distribu­
tion, and (ii) examining analytically and characterizing the distribu■
•
:•
tion.
The distribution is not a simple function, and its tabulation was
a task of a magnitude sufficient to require extensive u se o f a n electronic
digital computer.
7
II.
AMLYTIC examinations of t h e density function
A.
-Derivation of the Density
We w i s h to find the joint distribution of r, the sample correlation
V i
coefficient, a n d v = s / a
g/ 2
where s
and s
are the unbiased sample
fcfc
v
jlX-,
variances, from the bivariate normal population and c
11
corresponding population variances.
a nd
o
22
are the
This distribution will b e one in
wh i c h the population correlation coefficient is not necessarily zero.
Upon, simplification, the distribution will b e reduced to a function of
v along*,
We shall be g i n w i t h the Wishart distribution in the form taken from
Anderson [l] .
This form is
n
q
(I)
I V xSS " xV
„ 1)
. . P rCT1 1 , T 1 2 , T 2 2 )
• =xp
2
a:,
(T11T 22 - T 1 2 ) .
+ 2V
i
2 +
V 22B
. ST1 1 d T ^ ST22
where the v
denote the sample variances a n d covariances, and X. . deij
-I
note the elements of
I
and
)
^ I
is the dispersion matrix, i.e., the
m a t r i x of the population variances and covariances.
8,
From the definition of sample and population correlation coeffi­
cients, we have
(2 )
r . — ,
k
(Tl l V )5
—X
^xn X22 ^
a n d the density m a y "be simplified "by the change of variables
U1
?
U2
T:u ° " h i ' v22 " "55 5 Ta-2 =
^X11X2 2 ^
where the ranges of the n e w variables are
(4)
G < U <
I
0
-CO
CO
< U 2 < OO
<C W
"C CO
e
Upon substituting these values into the Wishart distribution, we,obtain
n
(5)
e«p«(u^,, u ^ w ) = (l - p2 ) 2 CuiU g - w 2 )
n -5
2
exp Q - V 2 C111 - 2pw + U g Q
/ 4ic P ( n - I )
du ^du dw
2
V
9.
F r o m (2) a n d (3 ) it is apparent that
_1_
2
(6)
r = WZ(U1U 2 )
_L
2
d w = (U1Ug)
dr.
U p o n making these substitutions in (3 ), the Wishart becomes
(T) e.p.^, u2, r) =(l -
jn
2v 2
p*)'*
• exp ) -
* d ^
n -3
n -2
(I - r2) 2 ( u ^ ) ^ / Iktf1(U-I)
2 u I " 2 Pr (u Xu 2 )
+ u2
du2 dr .
Xt is convenient at this time.to recall the relationships
1
2
(8)
sX a^Cv 1 1 / 11)
B2 .=.
(V2 2 A
)1
which.gives the value of v as
2
" I
(9) v = (Vu ) (V22)
O22Z b 11 .
10
Also, from the matrix relationship
1
(IP)
X12
22
0I l 0IS
X11 X12
a
a
12 22
X
D
- X.
11
X
12
22
D
where D = X 1 1Xpp - X 1p and from the definition of equality o f matrices we
have
_ I
(11.)
O2 Za1
I
5
2
22^
(0Il^
" (Xll
) /X2 2 ^ ^
" (xH ^
1
I
2
n
(X22^
so that
.2
2
(12)
(13)
.v,= (X11V 1 1 )
(X22V 22;)
1
2
I
-I
=
(U1 )
(u2 )
U i = u gv
I
(U1U 2 ) = U 2 V
du
I
= 2u v dv .
2
Making these substitutions in
(14)
e*p*(v, u j, r)
c
(rJ)
yields
u
n-3
P 2
2 12
n-1 n -1
= (I - p ) (I - r )
v
u
/ 2jt P ( n - 1)
2
11
* exp
2
2
2
(v - 2p r v + I)
dv dUg dr
I1Imt
over its range the joint distribution
of v and r
a n d u p o n integrating
is obtained.
If a change o f variable is made b y letting
(15 )
t =s U g (v
0 < t < CO
- 2p r v + I ) / 2
dUg = 2 dt / (v
- 2 p r v + I)
and substituting into (l4) gives
(16 )
n
n -3
g o
2
2
n-1 n -1 n-1
e.p.(v, t, r) == (I - p ). (I - r )
2
v
t
-
—
2
at P ( n - l)(y
-1
I
n
- 2prv 4- l)_]
_t
e
dv dr dt
The integral over t is P (n), hence the joint density of v and r becomes
n
(IT)
e»p.(y,r) =s . ( n - l ) 2
n -3
(l-p2 ) 2 (l - r2 )”^ ~ v
‘[\t(v2 - 2prv + 1)^]
dv dr .
To’ f i n d the distribution of v> r must b e integrated out of (IT) •
Asking the substitution^
g
g
) - (I - u) (I - 2pv 4- v )
r =
g:
,
.
•gX
u ( l 4- 2p v t v )■+ (l - u) (l - 2pv 4- v )
u ( l 4- 2p V ■» v
(18 )
the transformation "becomes
(19 )
r = (2au - a + "b)(2fbu + a - "b)
dr = 2(aP - b 2 )(2bu + a - b )
du
where the range on r is -I < r < I.
When r = — I, w e have 2u(a + "b) = .0 w h i c h yields the possibilities
that u = O if a + b.
y2
0 since a + "b = 0 is a trivial case.
g
4- 2pv 4 -1 = 0 and u p o n examining the discriminant (4p
That is, .
I
- 4 )^
we
see that it is real only if
|p| = I. When r = I, we have after simplifi
cation 2 u ( a - h ) = 2 ( a - b).
Since u = 1>. if a - b ^ 0, w e have a case
amenable to the same argument as,above* i.e., the discriminant is the
same in this case.
2
Appearing in the density are the quantities (l - r
a n d u p o n evaluating these it is found that
2
(20)
I - r
2
= 4u(l - u ) ( a
- b
2 '
-2
)(2bu,4- a - b )
a - b r = (a2 - b 2 )(2bu + a - b ) -"*".
Substituting these values in
(17) yields
) an d (a - br),
1H + 1
-t
(21)
e«pe(vy u) = k 2° 2 Vn 1 (a^ - T^)
n -1
n - g
n- 5
rt - 5
2
* / 2bu
*
(I - u)
+ (a - "b)
U
2
(1-u). 2
du dv .
U p o n Integrating u over its range,, the Beta function is obtained,,
and the density becomes
V a j j A
(22)
e„p,(v)
= k 2 l"2 V^-1 (a2 - t,2 )
n+1
n-I
>■
2
2 b B(
*'■'
2
n-1
) + (a - b ) B(-
2
n -1
'r
2
)
Simplifying the quantity in brackets in (22) gives
(23)
ap*
(2 ^ i ) n ‘V
l
|
a n d u p o n substituting for k, e»p»(v) becomes
(2k)
2*-1(n - 1)(1
-P2 )2 2n "2 Vn -1 a ^
e.p.(v)
n+1
«(a2 - b2 ), 2 P 7(U -I)
Using E u l e r tS relation , we find that
2 n-i;
(25) P
(—
)
k-2n. p
it 2
P2(n-1)
r 2 (i)
d v
.
14.
Replaoing the values of a and h h y those of (IS)j, and making use of the
relationship "between the b e t a and gamma functions, we have
2(1 - Pg )2
(26 )
Va '1
T m
I - 4p 2 V 2 .
e.p.(v)
B(#,§)
a
..
(l+v2)2
+ v^j
The formula (26 ) is a density having the correlation coefficient, p,
as a factor*
2 ,2
' I
The quantity v 2 = sI
is o f little u se since the popu-
'■£/*
lation variances are generally not known*
2
2
2
0^ = cr , we let 0^ =
null hypothesis distribution in testing
v
2
Since we are concerned with the
becomes the ratio of two sample variances.
In order to simplify the form of (26), let
(27 )
u = V6
du
2v dv = 2u
dv
a n d upon substituting these values into (26 ) we have
n
(28 )
where
/ X
e.p.(u) = L u
-2
2
-n
2
I - 4p u(l + u)
(I + u)
n
n
a
(I - P2 )2
- o2 )5
H w
(29)
=B(M)
"
V
i
T
"
-2
i¥)
2
and then
15*
W i t h some mi n o r simplifications, we have the density that is to be
considered in this thesis, i.e..
-/2+1'
‘)
n -2
(30)
f(u) d u = e.p.(u) = L u 2 '
(l 4- u)[ja^ + (2 - Ij-p2 ) u + ij
du .
16.
B.
Verification o f the Reciprocal Relationship for Percentage Points,
As stated in the numerical methods section, only the 90, 95, 97.5,
99, a n d 99«5 percentage points are isolated since, the values o f the lower
percentage points are simply the reciprocals of the corresponding upper
percentage points. F o r example, suppose that, for some particular values
o f the parameters n and p, the value of u which has 97»5 percent of the
area under the curve to the left o f it is u = 5«
Then the value of u
that has 2.5 percent of the area to the left of it is equal to l/5 or
<20
.
The nature of the statistic considered indicates that its reciprocal
should have the same density as the statistic itself.
That this is actu­
ally the case is easily verified h y making the. change of variable W = U 1
in (50) of Section II, Part A.
The differential becomes du .= w
■*2
substituting these values into (50) of Section II, Part A, we have
L w
(I)
'(2 F y
'■ '
(l + w
-1 X -2
)w
e.p.(w) =.
■
n*l
- gr-2 + ( 2 + V
W
"
1 + 5
2
Simplifying the denominator yields the form
"f+l
e.p.M=
.
_g n-fl
£ £ --- , , (I + "" ? "
^ _
dv
n +1
4- (2 - 4
)w + l|
^
a -1
(2 )
Lw
(1 + w )
+ (2 - 4p ) w
dw
dw, and
17»
The form of (2) is the same as that of (5Q) of Section II, Part
and this verifies the fact that the lower percentage points, are the
reciprocals of the upper percentage points for a fixed n and p.
J
A,
j
C.
18
A n Equation for. the Mode
Since the density is unimodal, the nonzero value of u for which the
first derivative of the density equals zero is the modal v a l u e .
In this
part, w e w i l l examine the first derivative and find an equation for the
mode.
Taking the derivative of (30) Section II, Part A, w e have
(l)
f*(u) = R (An^ + Bu2 + Cu + D)
where
n
n -4
(I- C2 )5
(2)
R =
_
.
h+1
|_ua + (2- 4p2.)u + lj 2
(n + 2)
A =
2
4p2 - 6 - n
B =
C =
;
2
- 3 + 6p2 + 1
CU
I C
a
D =
Upon setting f *(u ) e q u a l .to zero, it is seen that R = O only if u = 0 or
p '= + 1«
The values o f p = + I represent the degenerate cases.
This leaves only the possibility of the cubic expression
(3)
Au
3
+Bu
2
+ Cu + D -
0
\
19.
This equation is one which, when solved for u, yields the mod a l value, for
a n y fixed p and n > 5«
cussion
The cases. n < 3 w i l l h e considered in a later dis­
20
Do
Behavior of the Density for Small n
In this part of the thesis we will examine the. behavior of the density
for small values of n as we let the variable u become small«
We shall con-
aider three cases for small u»
The first case is, w h e n n = I, what the limit of f (u) wi l l be as
u -> Oo
That is, since
§
(I)
B n- 2
(I - p2 ) ■ r ( n )
J
f(u)
r
(|)
(u
(u- + u ^
n +1
* Pu +
1) 2
where B = 2 - 4p , we w i s h to examine
2
(2) Iim ;f(u) =
u ■> 0
p .
I
-I
U ln J L J L 2 ---u-J-0 (u -+
+ l)
)
Since this limit becomes infinite as u -> 0, we see that for the case when
n = I, f(u) approaches infinity as u
0, i.e,, the density roughly.re­
sembles the upper half of the hyperbola x y - A>
The second case is w h e n n = .2.»
limit o f f ( u ) as u
0.
Again, we are interested in the
Setting n = 2 in (l), we haye
(I -O2 ) P(2)
(3) Iim f(u)
.
u -> 0
i
"■
.............
(I + u)
Iim
u •> 0
O
P O )
(u
+ Pu + I)
Since the limit approaches I as u -> 0, w e see that for the case n = 2,
'
■/
.
f(u) approaches I -
p*
as u becomes small.
21.
The third case to he considered is what the limit of f (u) will he
when Ti-J-O for n > 5-
Putting n = 3 in (l )y we have
(I - p2 )2
(4)
Iim f ( u )
u-J-0
P(3)
3
Z
Iim
u -> 0
I
2
u + u
^ ' '
"
2"
(u + Pu + I)
Since u appears as a factor in each term in the numerator of the fraction,
w e see that for n > 3, f (u) approaches zero as u becomes small,
n- 4 ■
In (2) Section II, Part C, the quantity ^ 2.
appears as a factor
in the expression for the first derivative of the function.
This raises
the question as to the method of approach of the function to the origin.
As noted above, for the case n = I, the function becomes infinite as
u-J-0 so. that in this case the first derivative is meaningless^
For
n = 2, the function approaches I - p^ a nd f ’(u) approaches infinity^
hence, w e have tangency to the f(u) axis.
F o r n = 3» the function ap-,
proaches zero as u -» 0, but again f 1 (u) approaches infinity which im­
plies we h ave tangency to the f(u) axis.
F o r n = 4, the function a p ­
proaches zero, a n d f ’(u) is finite a nd positive implying intersection with
b o t h axes at the origin.
Finally, for n > 5# the function approaches zero
as u -> 0; and f * ( u ) is zero, implying that the function is tangent to the
u axis w i t h the order of contact increasing as n increases.
This completes the discussion for the various cases as u
0, and we
arrive at the conclusion that the density Exhibits the same characteris­
tics for small n as the F distribution.
Eo
Moments of the Density
-Since all moments exist for the F distribution, the questions arises
as to'whether the moments exist w h e n p
0.
We will show that after a
certain point, f(u) of (30), Section II, Part A, is less than g(u) where
g(u) is the F distribution, and hence that moments of all orders exist.
'Recalling
w e see that the ratio f (u) / g(u) is
f(u) _
g(u')
H
n + 1
2x2
(I - PC ) (I 4 U )
t
(2)
,n+1
Q u 2 + (2 - 4p2 ) u 4- I^j 2
Taking the limit as u becomes large gives
(3)
Iim
U
-> 0»
= (I -
P2 f
Iim
U
^
■> oo
Ql2 •
______
2 - •Tl
!.-2V
n+1
since the highest power of the polynomial in the numerator is the same as
the highest power of the polynomial in the denominator.
2%
The quantity (l - p )
is always less than one, and this implies
that f(u) is hounded h y g(u) for all u greater than some value say a.
I
-
r
It is apparent that the introduction of the factor u
in f(u) and g(u)
of (l) w i l l produce the same result in the limit.
Referring to Widder [%] w e have as a theorem on the .convergence of
improper integrals
(4)
Theorem
I.
f(u), g ( u ) £ C
a < u < Co
2»
G < f(u) < g(u)
a < u < co
3.
dU < CO
T o show that condition ohe holds, let us examine f(u) a n d g (u j for continu­
ity.
Since the ratio of two polynomials is continuous a t a ll points ex­
cept where the denominator is equal to zero, let us solve the quadratic
in the denominator of f (u).
Solving h y the quadratic formula, w e see that
are the only possible roots.
in the.interval
u =
|p| < 1 ,
P
^
However, the. quantity (p - l) is imaginary
so a l l of the roots are imaginary.
Similarly,
'.-1 is 'the only root of the denominator of g(u); and, since u is defined
to have the range
defined.
0 < u < .<», there are no points where g(u) becomes u n ­
Thus, f(u) and g(u) are continuous.
Since f ( u ) is a density function, it is greater than zero/ and we have
shown in (3) above that, for u sufficiently large, f(u) < g(u).
condition 2 is satisfied.
Thus,
A s the moments of the F distribution are known
to exist, we have satisfied all of the hypotheses of the theorem/ there­
fore, we m a y conclude that moments of all orders, of the density exist.
.
25.
III.
A*
HDMERICAL A M L Y S I S
Formula for Programming the XBH
650
Due to the complexity of the form o f e*p*(u) in (50) o f Section II,
Part A, a closed form could not he found for the integral, of the density.
a n d numerical methods are u s e d to find the critical percentage points of
the cumulative distribution.
To facilitate the programming o f the IBM
10 of the density are taken.
650, logarithms to the base
;This form of the density is
(I)
The integration program of the Appe n d i x uses this formula as the inte­
grand,
A s a check on the validity of the derivations preceding (l), we m ay
substitute the value of p = O into (50) o f Section II, Part. A.
The
e.p.(u) should reduce to e , p , ( F ) where F is the ratio of two independent
chi-square variables..
Proceeding with p = 0 , w e find that
where
26
/,X
’
j
_
P(n)
= P2(I)
for this special case.
The F distribution is
&
P / nI + n 2\
(4)
I \
e .p. ( F ) =
2
J
U1
-ZnI ^ n 2
^&
;
^
Qg
F
-(n2 + n £ F )
n 2\
2
'
'
where
(5)
0 < F < O0
+ I s'-number o f -observations in sample I
n_ + I = number of observations in sample
2
'
2
2
F = Si /S2
,
2
Recalling the fact that we have let
2
- aQ
our BuU.
hypothesis distribution, it is seen that F = u and n = n. = n ® Making
J.
C
these substitutions in (4) gives the relationship '
n .-2
(6 )
e.p.(F) = e.p. (u)
.PM.
u
i£'
t + u )\‘n du
(I
f 2 -§
Therefore, for the,special.case, p = O 3 w e have that e«.p.(u) = e.p.(F).
27»
B=
ETumerical Methods
A n important portion of the problem is to evaluate the integral of
the density in (30) of Section II, Part A.
Due to the complexity of
the density, numerical methods, are employed in the isolation of critical
points of the cumulative distribution.
Since numerical analysis techniques are employed, errors are intro­
duced into the tabulated values; however, b y taking increments of u
to b e .01, it is believed that reasonably accurate results are obtained.
In order to assess the accuracy of the tabulation, a table of values
of the cumulative distribution calculated when p =
0 has b e e n included.
It is seen that these values agree w i t h the values, of the cumulative
F distribution.
F o r the numerical integration, S i m p s o n ’s rule has b e e n selected
because of the simple coefficients involved.
A n h value o f .01 was
chosen,and cumulative probabilities are tabulated.
.Values of the cumu­
lative distribution are obtained for n =
6, 8, 10, 12, 15, 2Q» 25, 30
w i t h p = 0.0(0.l)0.5(0.3)0=8 for each n.
Since p appears only, as p^
in (30) of Section II, Part A, only positive, values o f p are considered,
and the tabulated values are applicable for negative p.
The p rogram to accomplish the integration is Program A of the
Appendix.
The formula programmed is S i m p s o n ’s r u l e ,with (l) of Section
III, Part A, bfing u s e d to evaluate e.p.(u).
28.
After the tabulation of the cumulative areas for a given n and
p,
the next step is to isolate the critical percentage points of the cumula­
tive distribution.
The 90* 95, 97 -5, 99, and 99.5 points are relevant
in the testing of hypotheses..
As shorn in Section II* Part B* the density
-I
of w = u
is of the same form as that of u* hence* only the upper per­
centage points are tabulated.
The lower.points are found by taking the
reciprocal of the tabulated upper percentage points.
To isolate the percentage points, it is necessary to use a numerical
technique for inverse interpolation.
Lagrange’s interpolation formula
has been selected with the primary,reason, again, being the ease in the
programming of the IBE 650»
A third degree polynomial is used with the''
formula being taken from Kunz [2J .
The formula is
(y - E0 X e -
~
Gr1 - E0 Ky1 - ^ X y 1- y5) 1
(y - F0 H r - E1 H y - yg ) x
(E2 - E 0 K y 2 - E1 X y 2 - E 5 )
A
C
E
Ce 5 - E0 )(e 5 - E1Ky5- E2)
G
= B X0 + D X 1 + F Xg + H X5
where the correspondence is:
(2)
y = percentage point desired
x '= u value of percentage point desired
29
du
y±
= u values corresponding to the
B j, . . o 'H, = notation-, used, in I B M 650. program.
Two values of
point.
are chosen on each side of the desired percentage
The program to accomplish the interpolation is.Program B of
the Appendix.
■The values of x% and y ^ necessary for the program are
obtained from Program A of the Appendix.
The maximum values of the density are also the values of the mode
for each h and p.
To facilitate graphing .and to aid in the study of
the characteristics of the d e n s i t y t h e first derivative of the density
must be found.
Upon examination of the values obtained in the tabula­
tion of the density, it is seen that there is only one m a x i m u m for a
given n and p.
Therefore', it is sufficient to find the value o f u for
which the first derivative changes, sign.
The formula selected is taken from Uielsen jjjJ and was derived b y
taking the first derivative of Uewton *s Forward Interpolation formula.
The simplified form is
(3)
y'(u) = l / h 21 D1 A^ y0
where
(W.D
'i+i
(w - I)
+ (w - i 4- 1) bji
" 5 1
u.
I
30«
bI-S-I = ' A I bI (v - i + ij]
bI ~
~
and the eorrespondehce is:
(5)
U - U q
w = — 57--- I
h = .1^
u = initial guessj
u
0
= value corresponding to v ;
0
y i ■= S-Cuj.) -
Three values o f .y n- are chosen on each side of the estimated value of
f(u) where the first derivative is equal to zero.
The program to accom­
plish the differentiation is given as Program C of the Appendix.
values of y 1 are the evaluation of f (u) as given in
Part A.
(30)
The
of Section II,
31
BT-
MJMERIGAL EXAMINATIONS.OF DATA A ND CONCHJSIONS
In Table I, the percentage points for n = 6, 8, 10; 12; 1 % 20s 25;
30 with p =s 0.0(0.1)0.5(0.3)0.8 are tabulated. As n increases for a
fixed p and a fixed percentage point; the value of u required in a test
of a hypothesis becomes smaller. For a fixed n and percentage point;
the value of u decreases as p increases.
This implies that; if one is testing at the CX level, of significance;
the upper critical point for a two-tailed test decreases. For example,
suppose OnerWas making a test at the 5 percent level of significance and
had taken samples of size 7® Realizing that the tables are tabulated for
the sample size minus I, it is seen in Table I that for p = 0 and n = 6,
the upper critical point is 5®82b However, if there is reason to believe
that [pI actually is equal to 0.5, then the critical point for the test
should be 4.78.
For small values of p, the differences in the values of u for a fixed
n and percentage point are very small. However, as p increases the differ­
ences become quite appreciable.' When a test is made at a higher level of
significance, the relative differences in u become greater. If there is
reason to believe that |p| is as great.as
0.4,
then the critical points
for rejection are smaller than those for the case when p = 0.
As n takes on integral values beginning with one, the shape of the
graphs of the density show considerable variation. As pointed out in
32
Section II, Part D on the analytic examination of the density, for
n = I the density become infinite as u ->
0 and for n = 2 the density
2
assumes the value of I - p
approaches zero as u ->
as u
Cu
F p r values of n > 3, the density,
0.
These facts have b e e n represented graphically in Figure I at the
end. of this section.
F o r n = I and n = 2, the graph of the density
w o u l d show that there is no maximum value.
For n > 3? the maximum value,
o r mode o f the density, is given b y the equation (3) of Section II, Part C.
It is further observed that for large values of n the dispersion appears
to b e decreased.
F o r the values of n and p tabulated, the m o d e and its
abscissa increase as n and p increase.
Table II shows the effect of p upon the cumulative distribution for
a given u a n d n.
The first two columns o f the body of the table give u
a n d the percentage points' it represents for the F distribution.
The r e ­
mainder of the table gives the associated cumulative probabilities for
the indicated value o f p.
Upon examination of these values, it is seen that if one uses the
value of u for the ease when p =
0, the test is actually at a lower level
of significanceJ i.e., the test is more stringent, when there is reason
to believe that p ^ 0.
For example, if a sample of size 7 is taken, so
that n =s.6, the value of u' to b e used as the upper critical point at the
5 percent level of significance when p = 0 is $.82.
actually
0.
the test would b e made at the
However, if ]p| is
5 percent level of significance
33
These facts have "been illustrated b y Figure 2 at the end of this
section.
The values of
for the case n = 10.
a
have been plotted against varying p values
F r o m the graph, it is seen that a decreases as
n increases.
F o r example, a. 10 percent level test is actually con­
ducted at the
7.6 percent level for |p| equal to 0 .%.
In Figure 5 at the end of this section, the function f(u’) has been
plotted for varying values of u when n = 1 0 .
F o r a fixed n, the dis­
persion of the density decreases as p increases.
the mode and its abscissa, increase.
bution, and for n =
Also, as p increases,
The curve p = 0 is the F distri­
10 the curves p = 0,3 and p = 0.8 are bounded b y the
F distribution as u becomes large.
This is as expected from our analytic
considerations of the density.
In conclusion, this thesis has been concerned with the analytic and
tabular examinations of a density where the correlation coefficient between
two random variables is not necessarily zero.
The following results have
b e e n obtained:
1. The density satisfies a reciprocal relationship for percentage
points. This requires tabulation of only upper-tail critical
points.
2.
The density exhibits fundamental changes in character for small
values of n.
F o r the ease n = I, the function becomes infinite
p
as u -> Gj for n = 2, the function approaches (I = p ) as u -> Oj
and for n > 3j the function approaches zero as u -> 0.
54
3»
The median of the cumulative distribution is at u == I 5 r e ­
gardless of the value of n and
p.
A s n and
p
increase, the
value of the mode and its. abscissa b o t h increase.
The'
equation for the mode is a cubic, and the density has n o
mode for n = I and n = 2.
density decreases as
4.
p
In general, the variance o f the
increases for a fixed n.
When making tests of hypotheses o n the equality of variances,
if the assumption is made that
p=
Q when actually
p / 0,
then the test is being made at a lower level of significance.
Also, the critical point for the rejection of the hypothesis
is greater w h e n
p
= G than when
p^ 0
for a fixed n.
35.
Figure I
Graph of f(u) for p = 0.5
- - n = 2 j
------ n = 10
y
-
-
- n = 50
36
P
Figure 2
Graph of a versus p for n = 10
37
Figure 3
Graph u versus f(u) for n = 10
P = o ^
_____ p =
0.3 ;
- • — • — p = o.8
V.
TABLES
Table I - Upper Percentage Points
u
F(u)
F
P
900
0.0
.1
.2
*5
A
►5
0.0
.1
.2
950
.3
A
*5
.8
975
'
0»0
.1
*2
.3
.4
»5
.8
n
6
3.05
3.04
2.99
2.91
2.8o
2.66
2.00
8
2.59
2.58
2.54
2.49
2.4o
2.30
1.79
4.28
4.26
4.18
4.o4
3.86
3.44
3.61
2.51
2.96
2.16
5.82
4>43
4.41
4.32
4.18
3.98
3*73
2.57
5.78
5.66
5.45
5.16
4.78
3.12
3.42
3.36
3.27
3.13
.
10
12
15
20
25
30
2.32
2.31
2.29
2.15
2.14
2.12
1*60
2.08
1*79
1.79
1.77
1.75
1.68
1.68
2.24
2.17
2.09'
1.67
1.97
1.97
1*95
1.91
1.67
2.02
1.87
1.71
1.64
1.6l
1.95
1.59
1.8l
1.66
1.42
1.57
1.37
2.98
2.96
2.92
2*84
2.74
2*6o
1*96
2.692.67
2.64
2.40
2.39
1.96
2.58
2.31
2.24
2.15
1.71
2.12
2.12
2.09
2.03
2.00
1.93
I .58
3.72
3.70
3*28
3.26
2.86
2.46
2.85
2*81
2.45
3.63
3*52
3*37
3*17
2.28
1*51
2.36
2.49
2.37
1.84
.
3.21
3.12
3.00
2.83
2.09
.
2.74
2.64
2.51
1.91
2.42
2.37
2.30
2.20
1.74
1.95
1.93
1.90
1.39
1*57
1.54
1.51
1.33
1.84
1.84
1.82
1.79
1.30
1.79
1*75
1.70
1,44
2.23
2.22
2.20
2,15
2.09
2.01
1.63
2.07
2.07
2.04
2.01
1.-96
1.89
I .56
1.85
.
1.6l
Table I (Continued)
F
P
990
0.0
.1
*2
n
•?
ih
.5
.8
995
0.0
.1
.2
»3
. A
•5.
.8
'
6
8.4?
8.4o
8.20
7.86
7-40
6.79
4.13
11.0
10.9
10.7
10 i2
8
6^03
5-99
5.86
5.64
5.34
4.94
3.20
7.50
7.44
7.27
10
12
15
20
25
.30
4.85
4.82
4.72
4.56
4.16
3.52
3.50
3.44
3.34
3*21
3.02
2.19
2.94
2.-92
2.88
2.60
2.39
2.81
2.70
2.57
1.95
2.50
2.42
2.31
4.07
4*05
3*97
3.85
3.67
3.45
2.42
3.32
3.30
3.25
3.16
3.03
2.87
2.11
4.34
4.o4
2.73
4.13
4.05
3.93
3 <*75
3.51
2.45
5.81
5.68
4.91
4.87
4.78
5.85
5.47
4.6l
8.76
6.98
6.58
6.06
5.18
4.81
4.09
5.10
3.76
3.13
2.76
9-59
4.38
2.59
2.38
2.56
1.80
2.35
2.30
2.23
2.13
1.70
2.90
2.88
2.84
2.77
2.63
2,62
2.58
2*52
2.67
2.54
2.44
2.33
1.93
1.81
4o
Table II - F(u) for Upper Percentage Points o f the F Distribution
n
6
P
u
3.05
4.28
5.82
8
10
0.3
900
901
904
930
951
975
990
995.
11.0
995
2.59
3.44
4.45
6.03
7.50
950
975
990
2.32
5.85
2.15
2.69
3.28
4.16
900
995
900
950
9.75
990
995
900
950
. 975
990
4,SL
995
1.97
2.4o
900
950
2.86
5.52
4,07
20
0.2
975
990
3.72
4.85
15
0.1
8.47
2.98
12.
0.0
1.79
2.12
975
990
995
980
.2.4^
950
975-
2.94
3.32
990
995
0.4
0.5
0.8
909
916
926
953
977
991
956
979
992
961
981
967
973
991
995
996
993
997
901
951
975
990
995
904
953
977
991
995
909
956
979
992
901
951
975
990
904
953
977
991
992 .
982
994
995
996
996
904
953
977
991
910
957
979
992
901
951
975
990
995
901
951
976
990
995
901
951
976
990
995
996
910
. 957
979
985
995
997
999
997
100
917
927
962
982
968
976
993
986
998
997
995
998
100
917 ' ■
928
977
962
969
986
994
998
997
995
998
100
100
918
928
978
962:
983
969
986
994
998
100
993
999
994
996
995
996
997
998
ioo
904
953
910
957
918
962
929
977
991
979
992
996
996
983 .
994 -997
979
995
999
100,
904
953
977
991
910
957
996
996
■
980
992
918
963
983
994
997
969
987
996
998
100
929
980
.970
987
996
998
995
999
100
100
41.
Table
n
25
U
1*68
1.96
2*23
2*60
2.90
30
l*6l
1.84
2.07
2*39
2.63
II (Continued)
0*0
0.1
0*2
0.3
0.4
0*5
0.8
900
901
951
976
990
995
904
953
977
991
996
910
957
918
929
970
987
906
998
981
950
975
990
995
900
950
975
990
995
901
951
976
990
995
904
953
977
991
99&
,
980
99(2
996
910
957
980
992
996
963
983
994
997
918
963
983
994
997
996
999
loo
100
930
970
987
982
996
996
100
100
998
999
■ 42.
VI..
The IBM
APEBEDK
650 programs of Parts A> B> and C are written using the
B e l l Statistical Interpretive System.
The advantages o f using this system
are the floating decimal point type of arithmetic which eliminates the
computer scaling problems a n d the built-in subroutines such as the gamma
function, logarithms, and looping.
Part G gives a tabulation of the density and cumulative distribu­
tion.
The listing was done b y the IBM 402 accounting machine whi c h is
not equipped w i t h a decimal symbol.
Therefore* each element in the p
column is actually preceded b y a decimal*.
actually 10 u ,
i.e., 15 is
function o f (30) Section
Similarly, the u column is
1 .5 . The f(u) column gives the value of the
Part A, and F(u) is the cumulative area from
0 to u inclusive.
The decimal point, is located b y the exponent following
the column, e.g.,
6525 4-7 is actually .006525 and 12345 50 should be read
1.2345.
43
I
<
Loc.
O0j
H°
A.
Program for Numerical Integration
B
C
Remarks
20 0
O
40 0
OOI
0 05
Read a card
20 1
2
OOI
05 2
10 3
n
20 2
3
OOI
050
IOO
n /2
20 3
2
OOO
05 2
10 2
n /2 - I
IOO
050
OOO
2
0
4
I
-
I
n /2 + 1/2
OOO
IOI
20 6
O
30 9
10 3
111
PM
207
O
30 9
10 2
112
P (n /2 )
20 8
O
35 2
111
114
lo g P (n )
20 9
O
35 2
112
115
log P(n/2 )
2 10
3
OOO
054
12 1
2 11
3
00 2
0 02
10 4
-2 l o g P (n /2 )
2
P
2 12
3
OOO
053
10 6
4p2
2 13
2
05 1
OOO
10 7
2 14
2
05 2
10 4
10 5
H
2 15
O
35 2
10 5
113
lo g ( I
2 16
3
OOO
IOO
119
n /2 lo g ( l
2 17
I
00 3
052
12 7
I +
2 18
O
35 2
OOO
1 17
lo g ( I +
2 19
3
10 7
0 03
I OS
22 0
3
00 3
0 03
10 9
d
-
"a
30 3
I
O
CU
-(n /2 + 1 /2 )
20 5
I
cV
-
p2 )
U
u
(2 - 4p2 )u
2
U
2
- p )
)
H°
ro°
I
>
a.
B
C
I
OOO
10 8
I IO
I
OOO
052
I28
, 2,
)u
2
,2.
u + (2 - 4p )u + I
22 3
O
35 2
OOO
I I8
log [ji + (2 - 4p2 )u + l]
22 4
3
OOO
IOl
I
-(n/2 + 1/2) Iogjjt2 + (2-4p2 )u+lj
22 5
O
35 2
0 03
I I6
log U
22 6
3
OOO
10 2
I2
(n/2 - I) log u
22 7
I
119
114
O OO
228
I
OOO
12 1
O
OO
N
229
I
OOO
12 2
O OO
(
23 0
I
OOO
117
O
OO
23 1
I
OOO
12 4
I
25
23 2
O
35 1
12 5
3 2 O
XO1oiCftuL
23 3
I
00 3
3 12
O O 3
u + ih
23 4
O
00 6
232
O O I
Alter Store
23 5
O
IOO
O il
2 I7
loop
23 6
9
OOO
3 13
O O4
Initialize / f(u) du
23 7
3
30 0
320
O OO
GiYi
238
I
OOO
0 04
O O4
239
O
IIO
Oil
2 3 7
Xoop
24 0
4
00 4
3 11
O O 4
h/3 ^ f 1Z i W
24 1
I
OOO
005
O O5
24 2
2
00 3
3 16
O O3
Loc .
22 1
22 2
Remarks
2
u
2 4
2
+ (2 -
n/2 log(l - p2 ) + log p(n)
log
I
f(u) - 7l
U2
U1
^ f ( u ) du
f(u ) du
1
Correct u + ih
45
O O O
3 3 0
O O S
f(u)
2-4 4
O
4 10
O O I
0 0 6
Punch
2-4 5
O
0 0 7
2 3 2
O i l
Initialize Store Address
2 4 6
O
6 0 0
3 14
O O I
Loop box for p
2 4 T
6
3 14
0 2 0
2 17
Variable loop for p
2 4 6
O
7 0 0
3 14
0 2 0
Initialize loop box for
2 4 9
O
2 0 3
O O O
2 0 0
Repeat
0 5 0
5
O O O
O O O
O 4 9
0 5 1
2
O O O
O O O
0 5 0
0 5 2
I
O O O
O O O
0 5 0
0 5 3
4
O O O
O O O
0 5 0
0 5 4
2
O O O
O O O
0 5 0
3 0 0
I
O O O
O O O
0 5 0
3 0 1
4
O O O
O O O
0 5 0
3 0 2
2
O O O
O O O
0 5 0
3 0 3
4
O O O
O O O
0 5 0
3 0 4
2
O O O
O O O
0 5 0
3 0 5
4
O O O
O O O
0 5 0
I
9
C
0H
2 4 3
ow
Loc.
B
C
Remarks
X
Constants
y
H°
o°
Ocu
c3
c4
C5
2
O O O
O O O
0 5 0
3 0 7
4
O O O
O O O
0 5 0
3 0 6“
2
O O O
O O O
0 5 0
c6
cT
O
OO
3 0 6
46
0I
O2 - A
B
Remarks
C ,
O
VO
Loc.
3 0 9
4
OOO
OOO
0 5 0
3 10
I
OOO
OOO
0 5 0
3 11
3
OOO
OOO
0 5 1
h/ 3
3 19
I
OOO
OOO
0 4 9
Constant
3 13
O
OOO
OOO
O O O^j
3 14
O
OOO
OOO
O OO
3 15
O
OOO
OOO
O OO I
3 16
1
OOO
OOO
0 4 8
cio
L
Initializing Zeroes
h
47.
B. Program for Interpolation of Percentages
0H
Loc .
O2 - A
B
C
24 6
O
40 0
OOI
0 06
Read Cards
2 47
9
OOO
003
OIO
Move Xi
2 4 a
9
OOO
0 05
0 15
Move yi
249
O
OOI
0 05
2 4 6
Loop 5 times
25 0
2
0 15
0 16
0 20
y - y0
25 1
2
0 15
0 17
0 2 1
y - y-L
25 2
2
0 15
0 18
0 2 2
Y-Y2
253
2
0 15
0 19
0 23
y - y,
25 4
2
0 16
0 17
0 24
y0 - yi
25 5
O
308
024
0 25
25 6
2
0 16
0 18
0 26
25 7
O
308
026
0 27
258
2
0 16
0 19
0 28
259
O
308
028
0 29
? ! " yO
yO - y2
y2
-
yO
?0 y -y
3
o
26 0
2
0 17
0 18
0 30
26 1
O
30 8
030
0 3 1
26 2
2
0 17
26 3
O
308
0 32
0 3 3
26 4
2
0 18
0 19
0 3 4
yg - y ,
26 5
O
308
0 34
0 35
y,
26 6
3
02 2
0 23
0 36
A
yi-y^
?2 - ? i
0 19
0 3 2
yI
'
-
y3
y2
3
26S
C
2 6-7
Ocu
Loc.
H°
48
C
02 0
02 1
0 3 7
3
036
02 1
0 38
269
3
0 24
026
OOO
B
27 0
3
OOO
028
OOO
B
27 1
4
038
OOO
OOO
A/B
272
3
OOO
Oil
0 39
A/B
I
B
Remarks
A
x
O
C
27 3
3
036
020
OOO
4
3
OOO
0 12
0 4 0
27 5
3
025
030
O o o'
276
3
OOO
032
OOO
277
4
040
OOO
0 4 1
Cx 1Zd
039
04 1
0 4 2
A/B
E
27
D
—V
278
I
xq
279
3
03 7
023
OOO
28 0
3
OOO
0 13
0 43
28 1
3
02 7
0 3 1
O OO j
28 2
3
OOO
03 4
OOO
28 3
4
04 3
OOO
0 4 4
E/F Xg
3
03 7
022
OOO
G
28 5
3
OOO
0 14
0 4 5
G x^
28 6
3
029
0 33
O o o"
28 7
3
OOO
035
OOO
288
4
045
OOO
0 46
F
2S
-4
H
G/H
x
3
+
C/D
X1
(
49.
0H
Loc.
Og-A
B
Remarks
C
2 8 9
I
O
2
0 4 4
0 4 7
A/B
xq
+ C/D X1 + E/F Xg
2 9 0
I
0 4 7
0 4 6
0 0 3
A/B
x
q+
2 9 1
9
O O O
0 15
0 0 4
Move Percentage
2 9 2
O
4 10
O O I
0 0 4
Punch n, p, u,Percentage
2 9 3
O
2 0 3
O O O
2 4 6
Repeat
C/D X
1+ E/F X
g+ G/H x^
50
C. Program for'Numerical Differentiation
C
Remarks
Ocy
C
0
H
Loc.
008
O
40 0
OOl
0 0 6
Read Master
090
9
00 6
OOl
OTI
Move Master
IOO
O
40 0
OOI
0 06
Read Details
IOI
9
00 6
OOI
Oil
Move Details
10 2
O
00 6
IOI
OIO
Change Move
10 3
O
IOO
OOT
IOO
Read 7 cards
10 4
9
OOO
4 08
IOI
Initialize move
10 5
2
0 26
0 16
2 00
10 6
3
0 26
4 0 1
OOO
IOT
2
03 6
OOO
OOO
IOS
I
OOO
0 16
2 0 1
10 9
3
036
4 02
2 02
IIO
2
046
2 02
2 02
111
3
02 6
4 02
OOO
B
Ayo
2yI
y2 - 2yI
A2yo
>2
yJ - > 2
3yI
.y5 - 5y2 + Sy1
112
I
20 2
OOO
OOO
113
2
OOO
0 16
2 02
11-4
I
056
0 16
2 03
115
3
405
036
OOO
116
I
OOO
2 03
2 0 3
IIT
I
0 26
046
OOO
P1 + y,
118
3
OOO
4 03
OOO
M y 1 + y5)
a3
y0
^
+ ?0 + 4/2
51
I
C
Oj
H
O
Loc.
B
C
Tic?
2
20 3
OOO
2 0 3
12 0
2
06 6
0 16
2 0 4
12 1
2
0 26
056
OOO
Remarks
yI " y4
12 2
3
OOO
4 04
OOO
12 3
I
20 4
OOO
2 04
5Cy1
-y
i
j
.
)
y5'yc + 5(yi - yV
12-4
2
04 6
036
OOO
12 5
3
OOO
4 06
OOO
204
OOO
2 04
12 6
I
I2 T
2
07 1
0 13
OOO
I2 ©
4
OOO
4 07
4 2 0
ioCy^ - yg)
U -u
h
12 9
2
420
4 00
4 2 1
W-I
13 0
2
4 20
4 0 1
4 2 2
w -2
13 1
2
42 0
4 02
4 2 3
w - 3
13 2
2
42 0
4 03
42 4
w-lf-
13 3
4
42 0
4 0 1
4 3 0
b2
13-4
3
430
4 2 1
OOO
13 5
4
OOO
4 02
43 I
13 6
4
43 1
4 03
OOO
13 7
3
OOO
422
4 3 2
13 8
9
OOO
4 00
2 5 0
D1
13 9
I
42 0
42 1
OOO
W
14 0
4
OOO
4 0 1
25 1
b2 • (w - I)
S
Ta3A
\
+ (
w - I)
|
j
L
/
2
w + (w - I)] = D2
Remarks
Loc.
0H
M°
I
>
52
14 1
3
4 2 2
2
1
2 5 2
(v - 2) D2
2
3
4 2 1
4 3 0
OOO
(v -
14 3
I
OOO
2 5 2
OOO
(v
14 4
4
OOO
4 0 2
2 5 2
1/5
(V - 2 ) D 2
14 5
3
4 2 3
2 5 2
2 5 3
(w -
5) D 3
6
3
4 2 2
4 3 1
OOO
(w -
2) b 3
14 7
I
OOO
2 5 3
OOO
(v -
3)D
14 6
4
OOO
4 0 3
2 5 3
1/4 [(V
14 9
3
4 2 4
2 5 3
2 5 4
(w - 4)D4
15 0
3
4 2 3
4 3 2
OOO
(w -
15
I
OOO
2 5 4
OOO
(w
15 2
4
OOO
4 0 4
2 5 4
l/5
15 3
9
O O O
4 10
3 0 0
Initialize 300
15 4
3
20 0
2 5 0
2 6 0
D1 AL yQ (Negative Instruction)
15 5
I
2 6 0
3 00
3 0 0
Z=:
I5 6
O
I I O
0 0 5
15 4
Loop 5 times
15 7
9
O O O
0 7 1
0 0 3
Move
15 6
I
0 7 1
4 0 9
0 7 1
u + Au
15 9
4
3 0 0
4 0 7
0 0 4
V h ^ D 1 A1 y0 = y'(u)
16 0
O
2 0 1
12 7
16
Transfer until negative
16 1
O
4 10
O O l
0 0 4
Punch n, p,
16 2
O
2 0 3
O O O
0 9 8
Read new cards
14
14
1
C
B
5
1
l)b2
- 2) Dg +
-
+
-
(w - I) bg
(v
+
-
(w -I
Jhgj = D 5
2) b
3)D5 +(w - 2)13^]
=
D4
5)bu
4)D4
+
(v - 3)b4
Rv - 4) d ^ +
(w - 3 )l RJ
a1 *0
U
u,
y' (u )
=
D
53
C
I
ow
40 0
0H
Loc.
OOO
B
OOO
C
Remarks
O 5 0\
\
40 1
2
OOO
OOO
0 5 0
•
40 2
3
OOO
OOO
O 5 O
40 3
4
OOO
OOO
0 50
40 4
5
OOO
OOO
0 5 0
40 5
6
OOO
OOO
0 50
I
^
Constants
40 6
I
OOO
OOO
O 5^y
4OT
I
OOO
OOO
0 49
h
00 6
OOI
Oil
Initialization of 101
OOO
OOO
0 48
Au
OOO
OOO
OOO
Initialize zero
408
40 9
4 10
9
I
O
54
D.
n
p
Tabulation o f the Density and Cumulative Distribution
u
F(u)
f(u)
I
16 9 3 4 2 I 5
49
6525928V
4V
2
4 0 18 V V 5 4
49
3549388
48
8428
3
559 3 V 5VO
49
4
6 3V 4 a 9 4 5
4 9
5
65 8 4 3 6 2 I
6
643 V3 OO S
T
S
I
12 0 1
48
144V 01 6 6
4 9
49
2098V659
4 9
49
2V520V5S
4 9
609 O O89 8
49
3 3 V 9 6 153
49
56 4 5 O 29 3
49
39668T52
49
5 16 5 I 6 9 S
4 9
450V4896
49
IO
4 68 V4 99 I
49
5 0 0 0 00 0 I
49
II
423 24 39 9
49
5445V542
4 9
I2
3 8 1O I9 8 5
49
5S4V588V
49
I3
342 4a 44 9
49
620902V4
4 9
I4
3O T 68 V46
49
65338059
49
I5
2V6 4 V 99 2
I6
248 6
IT
223V8 V8O
49
V32383VS
49
IS
2 0 1V O 5 6 2
4 9
V5363686
49
I9
18 2 O V O 8 4
49
VV28064V
49
2O
16 4 6 O 9 O 5
49
V 9 O 12 3 4 4
49
I
I4 3
49
68255999
49
4 9
V 08TST92
4 9
2 I
14 9 O 6 9 8 I
49
805V9234
49
22
13 5 2 2 V 9 9
49
81999396
4 9
23
12 2 8 a 3 3 V
49
832S8V83
4 9
24
111 8 5
4
49
8446
1462
49
25
IOI 9 9 8 2 V
4 9
8552983V
4 9
26
9 3 16 5 O 2 4
48
86504850
49
2V
8 5 2 39 O 3 I
4S
8V396160
4 9
aa
2S
V 8 I I5 2 2 a
4 8
86212303
49
29
T I V O IV 2 5
48
88960831
49
3O
6 5 9 IV 9 5 I
4 8
89648435
49
3 I
606 93 40 5
48
9 0 2 8 I053
49
32
5 5 9 6 6 14 6
48
90863960
49
33
5 16 8 I 8 9 2
48
91401852
49
34
4VV 9 28 36
4 8
9 1 8 9 8 9 15
49
35
44 25V O 3 4
48
92358886
49
36
4 10 3 V 3 2 I
48
9 2 V 8 5 I 09
49
3T
3 8 1 O I0 9 9
4a
93
4 9
I8 O 5 V 8
3a
3 5 4 I9 3 6 8
48
9354V9V9
4 9
39
329 6 6 5V3
4 S
93889V
49
28
55
n
p
u
f(u)
F(u)
4 O
3 0 7
A
A
A
I
2 8 6 5 9
2
2 6 7 6 6 9 9 4
4 8
9 4 7 8
I 7 4 7
4 9
3
2 5 0 2 6 6 4 9
4 8
9 5 O 4 O5 9 5
4 9
4 4
2 3 4 2 4
I 6 4
4 8
9 5 2 8 2 7 4 O
4 9
4 5
2 19
4 6 7 3 9
4 8
9 5 S O9 4 9 5
4 9
4 S
2 0 5 8 2 9 9 2
4 a
9 5 7 2 2 O5 3
4 9
4 T
19 3 2 2 6 9 9
4 8
9 S 9
4 9
4 S
18
4 8
9 6
4 9
1 7 0 7 6 6 OO
4 8
9 6 2 8 4 9
I 9
4 9
I 5
4 9
I 9 9 9 9
4 4 O
1 5 6 6 4 9
4 8
9 4 2 O7 9 9 8
4 9
4 8
9 4 5 O4 7 4 8
4 9
2 I 4 9 9
I O8 S 2 I
4 9
I OI
4 8
9 6 4 5 O6
1 4 5 4 5 8
4 8
9 6 6 O6 6 6
I 6 2 a
4 8
9 6 7 5 3 7 4 4
4 9
4 8
9 6 8 9 2 4 9 5
4 9
1 2 7 3 OO I 8
4
9 7 O2 3 4 9
I
4 9
12 0 3 2 7 9 5
4 8
9 7
I 4 7 2 6 4
4 9
4 6
9 7 2 6 4 3 O3
4 9
4 a
9 7 3 7 S O5 6
4
9 7
5 O
16 0 7 5
5 I
15
5 2
14 2 8
5 3
13 4 7 8
5 4
5 5
5 &
113
5 T
10 7 7 5
5 S
10 2 O7 6 O4
a
2
I 2 9
3 9 I
I 2 O
a
I
4 9
9
6 2
8 2 7 7 6 9 8 5
6 3
7 8 6 8 O I 5 4
4
4
4
4
4
4
6 4
7 4 8 3 2 6
4 7
9 8 OO5
4
I 2
4 9
6 5
7 12
I 6 4 6
7
9 8 O7 8 4
i a
S 6
6 7 8
I S 2 7 I
7
9 8
I 7
4
4
6 T
6 4 6
I 3 9 3 7
7
9 8 2
I 4 I
I 6
4 9
6 S
6 15 9 8 6 7 9
7
9 8 2 7 7 2
O 7
6 9
5 8 7 5 6 6 I 9
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
7
9 8 3 3 7 3 7
5 9
9 6 7 6 7 7 4 I
6 O
9 17 9 8
6 I
8 7 1 4 2 6 5 8
4 4
O
I 2
3
TO
5 6 0 7 6 O3 5
T I
5 3 5 4 6
T2
5 11
I O7
5 6 7 6 8
T3
4 8 8 9 8
T4
4 6 7 6 3 6 9 3
8
I I
T5
4 4 7 4 3 4 9 5
T6
4 2 8 3 O9 4 9
T7
4 10
7 S
3 9 3 O I 9 8 7
I 9
I 9 2
7 9
3 7 6 7 3 4 5 I
S O
3 6 1 2 8
S I
3 4 6 6
8 2
3 3 2 6 7 6 I 8
8 3
3 19
I 9 O
I I 3 7
4 3 2 8 7
8
4
7 9 9 3 8
4 9
7
9 7 5 7 9 3 3 I
4 9
7
9 7 6 7 3 5 8 7
4
7
9 7 7 6 3 O3 2
4 9
7
9 7 8 4 7 9 6 9
7
9 7 9 2 8 6 7 6
4
4
I 4 7 9
I
7
9 8 3 9 4 7 7 4
7
9 8
7
9 8 5 O I 9
7
9 a 5 5
7
9 8 5 9 9 7 5 I
4
4 9 5 7 3
I 3
I 9 3 O
4
4
9
9
9
9
9
9
4 9
4
4
4
4
4
4
4
4
4
9
9
9
9
9
7
9 a 6 4 5
7
9 8 6 8 9 2 7 4
7
9 a 7 3
I I 9 I
7
9 8 7 7
I
7
9 a 8 0 9 8 2 5
4 9
7
9 a 8 4 6 7
I 9
4 9
7
9
a 8 8 2 I O7
4 9
7
9 a 9
7
9 8 9
9 5
3 4 4
16 0 6 5
4 8
6 6 5
9
9
9
9
4 9
4
9
56.
f(u)
S
O
F(u)
6-4
30SS-4 0 5 1
4 Y
989Y99Y3
85
2 9 4 a e 10 5
4 Y
990
10053
49
86
283
4Y
99038964
49
4
5 9 16
8Y
2 Y 2 6 O'
88
262 25
4 9
304
4 Y
99066Y63
49
a s s
4 Y
99093502
4 9
a9
252 4 O0 38
4 Y
99
9O
242 99 99 6
4 Y
99 I 4 3 9 9 Y
49
9 I
234 O 3 235
4 Y
99
I6 Y 8 4 5
49
92
225 4Y 3 TO
4Y
99
I 908
9 3
2 IY 3 O 2 0 9
4 Y
99212953
9 4
209 49 6 4 6
4 Y
99234290
49
95
202 O 3YT9
4 Y
99254864
4 9
I 19231
IY
4 9
49
4 9
96
19 4 9 O Y 4 S
4Y
992Y4Y09
49
9 Y
18 8 O 8 8 3 1
4Y
99293856
49
9a
18 15 6 4 4 4
4 Y
99312336
49
99
IY 5 3 2 0 4 6
4 Y
9 9 3 3 0 1 YS
4 9
49
IO O
16 9 3
4 2 12
4 Y
9934Y409
IO I
16 3 6 I 6 0 9
4Y
99364055
49
IO 2
15 8 I 2 9 Y 6
4 Y
99380140
4 9
49
IO 3
I5 2 8 Y 0 8 9
4 Y
99395688
IO 4
I4 Y a 2
868
4Y
9 9 4 IO Y 2 I
49
IO 5
14 2 9 9
2 28
18 6
Y S S
4 Y
99425260
4 9
4 Y
99439326
4 9
4 Y
99 4 5 2 9 3Y
49
IO 6
13 8 3 5
IO Y
13 3 8 9
IO S
12 9 6 2 14 3
4Y
9 9 4 6 6 I 12
49
IO 9
12 5 5 I 4 10
4 Y
994Y886Y
4 9
I IO
12 1 5 6 Y 9 2
4 Y
99491220
49
I II
IIY Y Y 5 5 4
4 Y
99503186
49
I
16 6 2 2 Y 8 6
49
63SY9580
4Y
2
3 9 Y 6 I9 2 3
49
34958
4a
3
556 46 863
4 9
83 3Y 5 38 I
48
4
636 5 5 Y Y4
49
14360408
49
S
6 5 9 I6 0 2 4
49
208YY300
49
6
645 5 4 5 42
49
2Y 4 2 3 5 6 6
49
Y
6 11 4 O 3 6 1
49
33Y 2 O38 Y
49
8
56Y O9 9 0 2
49
39618196
4 9
9
5 19 O 6
49
S I Y
19 1
49
45050235
IO
4 Y I I I 12 8
49
49999990
49
II
425 34 20 3
4 9
544Y9S6Y
4 9
49
49
5 8 5 IY Y 4 8
4 9
6 2 1 4 8 Y 55
49
308 93 400
49
65410346
4 9
2 Y Y 4 a 0 14
49
68 3 3 9 4 9 Y
4 9
I2
382 8 2Y4S
I3
34 4 O O
I4
I5
282
57
n
P
u
f(u)
F(u)
I 6
2 4 9 3 9 4 0 6
4 9
7 0 9 7
I 7
2 2 4 3 8
15 3
4 9
7 3 3 3 7 6
I 6
2 0 2
1 3 7 7 4
4 9
7 5 4 6 8 0 3 8
4 9
I 9
18 2 3 6 6 3 1
4 9
7 7 3 8 8 6
17
4 9
I 2 2 6 7 9
4 9
I 18 0
23
4 9
4 9
2 O
16 4 7 9 0 0 6
4 9
79
2 I
14 9
1 5 5 7 5
4 9
8 0 6 9 0 8 8 9
4 9
2 2
13 5 2 3 5 7 5
4 9
8 2 1 1 1 5 0 6
4 9
2 3
12 2 8 2 7 2 4
4 9
8 3 4 0 0 6 3 9
4 9
2 4
111
7 5
4 9
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VIIo
LITERATURE CITED
1.
Anderson, T. ¥« 1958. A H H T R O D U C T I O H . T O MULTIVARIATE STATISTICAL
ANALYSIS, p. 67. John Wiley and Sons, Inc.,. Hew, Y o r k * .
2.
Kunz,-Eaiser S. 1 9 5 7 < NUMERICAL ANALYSIS, p* 9k.
B o o k Company,' Inc., ,New York.
5.
Nielsen, Eaj L* lgp6. METHODS I N NUMERICAL ANALYSIS, p. 102*
MacMillan Company, N e w York.
4.
.Widder,' David" V.' 1947.
Inc., N e w York.
.ADVANCED CALCULUS, p.
McGraw-Hill
The
269. Prentice-Hall,
MONTANA STATE UNIVFRsttv i
IlMlIlllllIIII Illl
*
3 1762 1
4329 4
136908