IS THE SINGLE-SAMPLE TEST FOR THE NEGATIVE
EXPONENTIAL DISTRIBUTION ROBUST?
By
D. Holt
University of Southampton
and
J. G. Fryer
University of Exeter and
University of North Carolina at Chapel Hill
Department of Biostatistics
Institute of Statistics Mimeo Series No. 955
OCTOBER 1974
Summary
The robustness of the power function of the standard one-sample
parametric test for the mean of the negative exponential distribution is
examined.
The main form of departure from the exponential assumption is
a mixture of negative exponential components although an alternative Gamma
distribution is also examined.
It is found that the test is sensitive to
these departures although the effect of mixtures with short-tails is less
dramatic than those with long-tails.
An alternative test for large sample
sizes is given and shown not only to be robust but also asymptotically
efficient.
- 2 -
1.
Introduction
The usual parametric test that the mean p of a negative exponential
(K.E.) distribution is equal to some prescribed value Po is based upon
the fact that
x the
sample mean then has a X2 distri.bution.
In fact
with probability density function given by
f(x;
1
lJ) =
lJ
e
-x/lJ
x,
(1)
> 0
lJ
the lest againRt the one-sided alternative lJ > lJ
region for sample
where X2 2
a, n
S1.ze
O
relies upon the critical
n given by
(2)
x
>
1S
the upper IOOa 7. point of the X2 distribution with 2n degrees
of freedom.
The critical region for the two-sided alternative lJ
+ lJ O may
be similarly specified either using a central acceptance region with equal
rejection probabilities in each tailor adjusting the tail probabilities
to achieve an unbiased test.
However one can seldom be sure of the precise form of the underlying
probability distribution and there are many instances with small or moderate
sample
Sl.zes
where we would be unlikely to detect the difference bet\o..cen
the sample generated from the assumed form or some similar alternative.
For the one-sample problem described above lclen and Dannemiller
[6J
have investigated tile robustness of the test when the probability distribution
actually g('ner:lting the data is of the Weibull form.
In this note we examine
in det.1i 1 the powe1 function of the test when the deviation from the N.E.
'I:WlIIlljltion
ifl
in
the form of a mixture of two N.l':. components.
(),,)
(
0
I',X;P'I'
..
,,-~
0 c
1
-x/O
1 +
.9
(j 2
-x/e
e
2
x. 0 l' O > (); 0 < Jl < 1;
2
p + q '" 1
nnd the
par;lmeter of interest. the wean, tak(!s the form
~
co
pOl + q02.
(3)
e·
- 3 -
In addition we examine the robustness of the test when the departure
frolll the N.E. assumption is in the form of a Gannna distribution with
shape parameter y close to uuity,
h(x; Y, 0) =
x
y-l
e
-xiS
x, y, S
(4)
> 0
sy f(y)
where the parameter of interest
1.S
\l
= yeo
In the case of the mixture of N.E. components, since we are concerned
with the situation
when the investigator is unknmlingly dealing with a
mixture, we have concentrated our attention on situations where a large
proportion of one N.E. component is mixed with a small proportion of the
other (p ~ 1 or q ~ 1).
\-lithout loss of generality we assume 02 > 01
leading to "long-tailed contamination" when p
. 1 since the small
~
proportion of tIle second component has a larger scale parameter and
conversely "short-u.liled contamination" when q
~
like this with p close to zero or unity that it
to delect the
pre~ence
of the mixture.
1.
1.S
It is in situations
particularly difficult
In fact, with regard to the
numerical results presented, it is generally true that much larger sample
sizes t.han those considered, would be needed in order to detect the
presence of the second N.E. component.
It is perhaps appropriate at this point to make a gelleral comment
abol1t
t.he
accuracy of numerical
rc~ults.
We have relied upon numerical
integration, sOHJPtimcs of cOlnplicated functions, using an iterative form
of Sill.pson's Rule with succel-isively smaller step lengths until the
pro('(~d\ln'
illJd
cIJC!ckr;
converged.
This was found to be quite
illdic;(!e tllal
n'f;t11t:s h:lve ,1n
c£fici.(~nt
iICClll-::Cy
<!('clllial places }lu'r;ented in the tables of results.
l'.n~alfT
for our purpose
th~lIl L1H~
thret'
- 4 -
Results for a mixture of two N.E. components
2.
For a mixture of ttllO N.E. components the probability density E;:(x)
of the sample mean x is given in the appendix.
15
gIven in Holt
[3J.
A complete derivation
E;: (x) is a function of n, p, 8
1
and 02 and our
approach has been to use numerical int.egration techniques on it for given
values of the parameters to obtain points on the power function.
thr~
calculations it
\oTaS
assumed without loss of generality that the mean
of the distribution under the null hypothesis was
possible
case of
~ance
p= p
*
For
Po = 1.
This is
if we consider the power function in the more general
when the underlying distribution is a mixture as in (3) and
the critical rCglon
IS
set as if for a N.E. distribution as
1U (2)
then
the pm,'cr of the test is precisely as if the mixture was
g(x)
==
p f(x;
°1
P
test
lJ == lJ
O
*
8
) + q f(x;
lJ
2
*
)
and the critical region was set to
e.
= 1.
We now turn our attention to the numerical results of which only a
snmple are presented for reasons of space.
We will present results only
for tests which have a nominal size of a :: 0.05.
Other values of a have
been examined with a similar pattern of numerical values resulting.
As
we stnted before we have concentrated our effort on mixtures which are
p;'rticul[.rly difficult to det('ct frOla the assumed single N.E. form.
In
Llct we have t;lk('11 tIll' set. of values for p : {O.99, 0.95, 0.90, 0.1, O.OS, O.O})
\ he f jr~t thn'(~ vn}II<'H lt~nding to "long-tailed contamina'tion" anci the
n~m:Jit\(l('r \0 "~;hort-tHiled
tll(~ 1;Pt
of. the
of
tlrlO
V;:tl\lPfi
02/O}
contaminat.ion" under 02 >
: {3, 5,
lO}
°1 •
We have examined
for the ratio of sc."l1e paramcten;
components.
Tables 1
orlf>·~;jcl(!d t(~sl
;Hld
')
present points ou the power functiol1r-; of the usual.
\vith allernative 11 > IJ
O
haGed
upon
the N.E. distributi.on
and WlOn tIl(' Vllri0l1l.l mixture 0] I"(>rnatives we hav(' cXilminecl.
Tctble 1 deals
•
- 5 -
with samples of size n
=5
and Table 2 with those of size n
~
20.
Points of interest are as follows:
(i)
There 1S distortion of the power function both at the nominal
s1ze level and at other points on the power function.
For the
cases vc consider actual S1zes occur as large as 0.137 with
n
~
5 and 0.185 with n
nominal size of a
(ii)
= 20
{p
= 0.9, e2 /8 1 = 10}
rather than the
= 0.05.
The overall distortion in the power function is to lessen
power causing estimated sample sizes for a particular test performance
to be too small.
Of course locally near the null hypothesis power
tends to be increased since this is the effect on the actual SLze
of the test but for larger values of the mean
p
the
actual power
function is uniformly lower than that based upon the N.E. distribution.
(i if)
.
As one might expect "short-tailed contamination" (p ~ 0)
is comparatively less of a problem, the larger distortions to the
actual size of the test and the remainder of the power function
occurring in the cases of "long-tailed contamination".
Nevertheless
distortion of the size and power function does occur and it is
perhaps surprising to note that whether contamination is short or
long-tailed (p close to zero or unity) the effect on the actual size
is always to inflate it.
(iv)
1S
Corresponding results for the one-siued case when the alternati.ve
~ < ~O
are similar in
D
qualitative sense to those for
p >
PO' but
a larger pcrccntilf,e change in the mean of the distribution is nceucd
lw!()re tIle loss in power showfo up.
- 6 -
Tables 3 and 4 present results for the two-sided test corresponding
to sample sizes of 5 and 20 respectively where the acceptance region 1S
specified with equal probabilities in each tail (central interval).
Here
W,' must cxannne the bias of the test as well as its actual size and the
(~ffect
on the rem:linder of the pO\vcr function.
Points of interest are as follows:
The stand;nd N.E. t.est
(i)
IS
sliehtly biased, although a comparison of
'1';11>1(-s 3 and 4 Sllgf,CSts that the bias rapidly disappears.
of a
~ixture
In the presence
of N.E. compo00nts there is evidence that the range of
~
10
\-ihich the power function is bel 0\01 0.05 1S enlarged and also shifted in an
upwards direction.
For example for n ; 5 the mixture p ; 0.9 02/01
=
10
lws PQver function below the actual size in the range of ].1(1.0, 1.59) with
a rr1111lmllm value of 0.2.27 (93.0% of 0.244).
IS
Thus while the drop in power
relatively slight it docs persist over a wide range of].1.
e-
Again
as n Increases this range shortens and the loss in power belm-] the actual
size of the test becomes less.
Table 5 presents
for each mixture the
range of \l in "lhich the power function is below the actual size and the
lowest level reached as a percentage of the actual size.
(ii)
I t would appear that in other respects the two-sided test is
considerably less robust than the one-sided version.
The actual size of
the tes t reaches such extremes as 0.2/,4 when n ..., 5 and 0.383 when n
(I'
.c
O. ~ t O~/Ol ,'"" 10).
=.
20
For a nominal test si?e of 0.05 such results arc
dramatic anll pve" less extreme mixtures exhibit large distortions to the
nominal
(iii)
U~~;t
!\!;
size.
with t.he on('-sidf'd test the over.all effect for more extreme
va1.ll('S of II is to reduce
t111~
power of the test.
Of course with such
till' power fUllctions in the n£.'ll'.llhollrhood of the null
];tl"/'.c ('j fpels
(,1\
hYI'0tl)('I;is it
takes l'<~lativ(;ly large values of. II in some cases hefore
•
- 7 -
the overall reduction in power is apparent.
However the usc of the
pm.,er function in estimating required sample sizes is frequently from points
on the pClwr function in the range of power::; 0.9 or 0.95 so that type I and
type II errors are sinlilar in magnitude.
IS
By this stage the power function
almot;t always less than that for the N.E. distribution for the mixtures
we have considered.
TableR 6 and 7 correspond to Tables 3 and 4 except that following
[4]
Ramachandran
L10
unhia~~d
the tail areas of the test have been adjusted to yield
tef;L of size a ::;: 0.05 for the caGe of the N.E. distribution
(,...ll1.ch pH~\;iously Shovlf~d hias below
}i
=,,~.
For the less extreme mixtures
the effect appears to be to reduce the bias.
lIm"ever for the more
raixtures the range of values in which the test is biased is
('xtrC:TIW
incrcilscd 2ud the percentage of the actual size to which the power function
falls at its ndnimum is lower too.
of bias is
Thus for p
= 0.9
6 /6 == 10 the range
2 1
(1.00 - 1.8) and the lowest value of the power function
IIOW
1S 87.9% of the actual slze.In other respects the results are similar
to those
3.
1n
I~esul ts
Tahles 3 and 4 and the main points arc still appropriate.
for the Gamma dist.ribution
When the actual distribution i.s of the GaImna form (I~) then x the
~:llllp]('
m.-:an al!;() follows thi!. same form x .. hex; ny, (J/n) and it
:;lr:tlghtforw:ll,l tIl:ing
of tllf' tcst.
Tahle
e
is
nUlIlCTjcal techniques to c>:mlllne the po\-rer function
present::: pojnts
Oil
the pm,,('r functions derived from
a v:,riety of Gn,lUlla distributions y "" {O.5, O.B, 0.9, 0.95, 0.99, 1.01,
1.0'i, 1.),
t~st
J .2, ]. ')}
for sr\l'lplcs of size 5 :1nd 20 for the one-sided
(\I > lJ ) ann Table 9 presents the corresponding results for the
O
two--si (kd tps I
•
- 8 -
The main points are as follows:
(i)
Unlike tile mixture situation the actual si2e of the test can be
inflated or reduced depending on whether y'is less t.han or greater than
one.
(ii)
The value of y also detenlines the overall effect on the extremes
of the power function.
Thus for
y > I the tendency is for the actual
~
power to be greal.pr than that indicated under the N.E. assumption
\-iherens for
y < 1 the reven;c is true t
(but this does not actually
up in the t\,Jo·-sidcd resul ts for n : .;: 5 for the range we have
ShOH
presented).
(iii)
Distortion to the nominal Size of the test can be great.
the one sided case
(y
= 1.5) to 0.11?
(iv)
lJ > lJ
(y
e-
' for example, actual sizes range from 0.024
O
= 0.5)
for n
~
20.
As in the mixture case the effect on the two-sided test seems to be
worse than that for the corresponding one-sided procedure.
tt~st
the
In
For example
sizes corresponding to those i.n point (iii) above are 0.016 and
0.167 respectively for the sa.me sample size.
(v)
\JhiJ st the lest procedure in the two-sided case is biased the evidence
is that this rem,:nTLS virtually unchanged over the range of y '"hich ,,/e hav('
considf~l'cd.
For f:amples of size 5, the range of
function is bl'1Il\-.. the actual
di:;tribill'j(lllS ('o\lsidcn~d.
t(':;1.
Till'
for whicll the power
test size is frol1l 0.87 - l.00 for all Gmmna
Hltilst the percentage which fall belo", the actual
:.izl' v;lri,':. fro\ll casl.' to case it is never less than 89.6% (y'~ 1.5).
c.1~H~ 15
fl.J<.
(J',,:)!. ",- tIl('
to
~
().(.J(.
-
hi:wed ov('r the
;1('(11;11
I.OO
II/Ill'
/;)Z\'.
J'O\'
!::lmc
:;111111'1,.'/;
Ill' ]o..'c::l point of
r'Hlgc with
or
lilly
size /10
of th('
n
minimum
tl,,~
11111;',('
)()Wl'r
p(~n'cIltag{'
1::IS
of
s!lortl'I1('r!
fUllctiOlW is
e
- 9 -
the procedure of Rml1.1chandran will remove the bias of the test not only
in the N.E. case but also to a very large extent in the test based upon
the
G<i.n:rna
distribution. leaving the rest of the conclusions for the
biascd test practically as they stand.
4.
Tl~.t,w-samplc
test
The test corresponding to (2) for the equality of means based
upon samples from two N.E. distributions is based upon
the s2.li!ple means.
i/i
the ratio of
This ratio has an F distribution with 2m and 2n degrees
of frC'cdom on the null hypo the:., is ,,,,here m and n are the sizes of the t,,,o
independent samples.
Gehan and Thomas (1969) have examined the robustnesf>
of the procedure against the Weibull alternative.
\~len
the two samples are actually drawn from mixtures of N.E.
C0l!lpOnents the ratio of sample means has a very complex distribution.
In the case when both samples came from the same mixture (a special case
of the null hypother.is) the distribution of the ratio has been derived
by Hal t [3J and is given in the Appendix.
Again using numerical techniques
we arc able to examine the actual size of the test in this special
restricted case of tile null hypothesis.'
Table 10 presents the actual sizes of the nominal a
varju\l~;
lllix1.ures (.f N.E. compolll'nLs and sample sizcs.
C'x<JIllinl'd the l/hn] (' of
hav i Ill', LIle same
'-OblH;llll'SS
II!I'
1",v.'('T
111(';10
illil
test for
Hhilst we have not
the )lO\,'cr function nol' samples from different mixtures
the re:>u It s
arc not cncouraglng in terms of tile
of" the lIumin,,] test: size.
[tinct
= 0.05
it
seclns
reasonable to conclude that:
wil I ;t1:;o he distorted at lea:;t in the immC'tliatc
nClgJI1)()urhood of the null hypothesis.
In view of the complexity of the
CitlCII];ILiow; involved and the discouraging ref,ulU, obtained we have not
- 10 -
pursued lIds study.
Our experience would suggest that a better met.hod
of attack for a fuller investigation would be using computer simulation
techl1iquc~~
but our feeling is that the few results
\ole
have obtained are
enough to indi(:.c:U· that the t\w-sample test is no more robust than its
one-sample counterpart.
5.
Alternative test statistics
th~
In view of
from the
1Ja~ic
sensitivity of the N.E. tests to even slight departures
assumptions one is led to Inok for c.lternative approaches.
Gdwn and Tholilas [2] investigated the performance of certain non-parametric
allernatives for Weibull departures and found them to be clear improvements.
It seems likely that suitable non-parametric competitors would also be
s,!tisfactory for the
t\oJO
alternative distributional forms which we consider,
and for small Eumples there seems 1 ittle else to reconnnelld.
larg(~ s2.mpl('~
c~(n
",e
e.
However for
take advantage of an optimal parametric test statistic
'/hi eh rem;:: 1.llS the saIne function of the data for samples from the single
c:xponcnti"d., a mixture of K
cir.triLutions.
N.E. components or even a mixture of K
Gamma
It may be> argued that roLust large samplc tcsts are of littlc
v;!lue her. since as the sample increases the form of the underlying distribution
",ill ber.f'l"f": clear.
However as we have pointed out, mixtures are usually
difficult to clet(~ct even for lnrgc samples unless the components are quite
clj~.:;jlJli
1.11<11
a
I;,r
dl1(l
dj~;cll~:.;j(\n
('ollr;j(",,.
in substantial proportions.
;11:;0
of ]arr.c s<Imple
Diz('s
For this reason \.,c feel
is far from being valueless.
We
IIf'I"(' tile theory [or til<' one-sample problem Lut that for the two-
'1'11(· l>;l!;io of
1I)('lltjOl1f~d ,d)OVl~
1.:'>
all
optimal parametric teat for all of distributions
Ihe fact tha! -; the flUlllple mC'nn
1.5
tlte
maXllllUii\
liLclihood
- 11 -
estimate of the mean of the distribution in all cases.
A proof of this
is given In Fryer and Holt (1970). Thus the Wald [5] large-sample
statistic for the hypothesis
II
II
o
:: 0
takes the form
11(;;
1l)2
___0
t
ano t has an asymptotic X2 distribution with one degree of freedom on H •
O
Here ~2 is the maximum likelihood estimator of the variance of the
distribution and in general depends upon the form of the distribution and
the value of any parameters involved.
by
52
t.he sample varIance gIves a new statistic t
equivalent to t.
t
1/2
1
llowever the replacement of
if it
IS
l
;2
which is asymptotically
Clearly appropriate one-tailed tests can be based upon
properly signed.
Acknowledgement
Dr. Fryer gratefully acknowledges support from Institute of General
Medical Sciences Grant No. GM-12868 during the preparation of this paper.
APPENDIX
1.
Distr·ibutie..n of y
expanl':'nti~l
~
x the sample I;\ean of a mixture of two negative
components.
Fran; Holt (J %9) the probability density function
E;(y)
Il
_
~1.
.._y
n-1
n
_
n-r
r.
l' (n)
Pi
P
r==O
r
z
~s g~ven
by
K(r,n,ny~)
co;d:] ucnt hypogcometric function
15 l:l;!.T\er' s
K{r, n, nyy) =
r.
i=O
r(r + 1) ••• (r + R. -
(r) t -
2.
DisU:ir\\ltion of u ~
s<:,mp} (·s of size
In
"X/y
1)
for example.
the ratio of sample means for two independent
and n frOl'l the same mixture of two negative
expo:1I..:nti<l1 CO;;tpouents, also from Holt (1969).
m
l;
r.
(u)
r=O
where
K
;tnd
I:
{~.!-}m+n+t
1
/.
n
L
5""0
mu+n
t.
I:
,1'.' ()
r(m+n+t) K
2
(t, u, r, s)
e.
REFERENCES
[1]
Fryer, J.G. and Holt, D. (1970)
On the rohuBtness of the standard
estimates of the exponential mean to contamination.
[21
Gehan, LA. and Thomas, D.G. (1969)
Biometl'ika~
57, 64l-6!,8,
The performance of some two-
sample tests in small samples, with and without censoring.
[3J
Holt, D.
Shlg~c
(1969), Some c:ontl'-ibut1.:Ons to the stai;isticaZ anal-ys'is of
and mi:ced ('.x[xmential- dZ:stl'·1:butions.
UllpublisLed Ph.D. thesis,
University of Exeter.
[4 ]
Ramachandran, K. V. (1958)
[5J
Hald, A. (1943)
A test of variances'.
J. Amer. Statist.
Tests of hypotheses concerning several par2.meters
when the mllnber of observations is large.
TransactionG of the
Amer1:can J.1athe?ra.t1:caZ Society, 54, 426-482.
[6]
Zelcn,
B.
and Dannemiller,
M.e.
(1961)
The robustness of life
testing procedures derived from the exponential distribution.
TccllJLol!?(;tric3~
3, 1.9-'~9.
~
As!]. ,
1\1.ble 1.
Points on the
Po~~e~ Fun~tiG!". ~;~
t~;.~
G~:c-Sid,:;d
'fest £or
~
=
1.0
for Mixtures of N.E. Components
= 0.05
---------,---------_._--------------------------------------------------;
Sample size of S. Nominal size of test
Mean of Distribution
Parameter Values
p
a
I
,- II
I
I 2.8
1
I
I 807
I
I
!
Ii:
1.0
1.2
1.4
1.6
1.8
2.0
050
123
219
324
426
518
053
124
218
320
420
511
591
659
5
056
122
212
311
409
499
579
647! 705
10
056
III
190
281
375
464
544
614
I 674
3
064
130
215
310
403
491
569
636
694
742
782
816
1844
5
079
132
200
281
364
445
520
588
647
698
742
779
10
101
132
171
222
277
339
403
466
525
580
630
i
I
;3
074
138
217
304
392
474
549
616
673
722
I . 0.90
5
100
152
210
275
344
413
480
542
599
10
137
175
211
245
281
320
361
404
3
051
124
220
324
425
517
596
5
051
124
220
325
426
517
596
6/6 1
Negative Exponential
3
0.99
0.95
I
I
G.Ol
10
I
!
2.2
2.4
597
2.6
u51
221
124
325
426
517
596
I
I 3.6
I
; 3.8
4.0;
'903
913!
833
860
! 885
I 882
326
853
i 876
S9~
S10
803
833! 858
! 379
396
86S
8S7
903
: 811
538
861
820
674
1
, 714
.
748
779
805
763
798
528
553
674
sn
650
695
734
768
793
U_"T
Q" '.
846
447
491
533
574
_612
647
680
710
664
720
767
805
837
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720
766
804
836
852
333
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663
I
758
I 762 801
753
793
I 725 I 768
I
715
I
719
766
804
835
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j
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900: 915
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,
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i
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3
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127
223
325
425
514
593
659
715
I 761
: 799
831
657
i 87S
597
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5
054
129
224
327
425
514
591
657
712
I
758
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796
827
853
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I
I
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513
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655
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on thl' Power FlIuctil1!l ()f the l)l1/,-sidl'd 'l'p!;t
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Mean of Distribution
1.2
Negative EXJ10nential
223
050
-
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0.99
2.0
846
926
693
839
921
223
457
675
825
913
205
406
618
781
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233
458
667
1-816
422
607
365
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085
5
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10
1.64
I ;~;
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260
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905
758
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861
605
i
720
I
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0.90
2/~3
452
649
5
128
264
423
10
185
295
403
888
580
715
818
504
592
673
22ft
478
700
I 845
925
5
051
225
478
699
844
924
051
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699
844
924
3
053
227
477
696
840
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229
477
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837
918
10
056
231
477
692
835
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3
057
231
476
691
834
916
060
235
476
686
828
911
476
683
239
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823
906
0.10
Decimal
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OS)
10
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:
3
_-~----_.-
().os
I
795
,
---+_..
--
1.8
472
066
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701
1.78 __.
.-
225
5
i
II :. 1.0
10
point~
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omitted:
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n t)~ .;.... l.-, .__. . o>
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for
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....
of NeE.
t _n\::o
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r
.,
•• u
-
Co~poncn~s
S<!r.;ple size of 5. No:,,1.nal Te::;t Size a ;: 0.05 (C",,,t:-al Inten'Jl)
II
Pararr,eter Values
i
P
,0.3
0.4
! 629
383
Negative Exponential
-I
0.99
.
I
I
3
II
5
10
633
II·
644
I
i
0.510.6 jO.7/C.S 10.911.0
I 228
I
I
r-
I
.6 2 /6 1
}~ean
389
234
245
I
678 1440 1279
I
1
402
I
i
I
143
1 092
0.90
1033-
Il-s-TIs4
194
i 237
i 060 ! 060
071
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118 1151
189
i' 139
172
'I
181 1124 091
I
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,
I
I
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1075 i 070
111
46213031203
1 100
1098
10
759
590
' 174 1 159
1156
I 255 1161
I
5
69LI
Ii 741
1629
'38~
324
27511791123 1094 1083
I. 623
I
I
1
I
I
I
,
527 ! 430 1354 i 301
,I
I
I
1135
i 266
!084
!~ 151
1132 1137
I
I 244
,I
I
! 232
1'.384
j
_
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I
274
315
36:-
209
248
289
33J
3"7 ~ I,
I195
234; 275
317
35'?
400
160! 188
219 [253
288.325: 361 I
164 1176
191,210
231: 25"
i 200
236: 273
312
245! 255
1541195
240
154
196 ; 240
23.0 1141
I: G?Q
i 062 I1051 li 052 ~ 065 :. 088
, .
118
155
196
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l
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350· 339 !
231 ;237
118
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279 ,
230: 257 '285 ! 31.:. i 344 '
ii 087
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I
416 ,
405 i
183! 205
266. 279
!
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1-0~91~-5-11~~-4-mI0861117
1
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i 227 ! 227
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231
167
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372
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1095 11141138
!
282 1327
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1
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628 1384 I 230 \140 1089 1061 1050 1052 : 064
5
10
080 1068 1070
i
\ 504 1352 1.1249 1186 ,151
II 637
I
076 i 090
682
3
I
I
377
I
I
067
5
1655 . 432
I
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~ 239 i 285 i332
!194
1106410531054
!' 081 I1100 Ii 127
104 i 117 i 136
3
1.6 !L7 :1.2
I '
412
437
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085 '1'116 1153
647
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3
10
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1.211.3
,
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0.95
I;
I
153 \101 1072
I
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1381087 1060 ,'048
11.1
of Distributic,n
292
285 :331 :377 :421
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332
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1 ...... -
237
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;
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201
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I
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203
I
:l69
I
i 17~/J
e
~2ci[;:~1 points
C::il.
t t cd :
ent~ies
stould be divided by 18CO.
e
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l"U::.':.... . t].on
..
~-~-_._--_._-_._--------------------_._-
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~'-'.L
ures
l/·..
..... ~x
Parameter Values
P
Ne6ati~.:e
a/8
0.99
Ii
~:r F
""4~CI'
)-.
!
j......
I
0.3 10.0-+
I
I~
--'_nn
i 1000
I
I;
999
997
981
10
I
Oe95
:'l"?
r.,..I ..
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t=
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I
0 . / :I 0.8
' I '
-
I
983
841
561
30l
978
838
566
310
967
835
580
330
943
843
636
401
~ 7'.~
5.1.-..'1,..;
0
-F~'
....
,...
.1
C __
r
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,. =
.:..x,
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n r'· \.,;~_.J)
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("":1-"1 .... ,..."..
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I 141
I
: 0 ....Q
I
l.0
I
i
066
050
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167
i
I
231
'1
1.1
I1.2!i
•
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-
,"
: !
.....
:".1
...
Q
':'.v;
: 602
: 699
I
366
484
'59:
; 690
; 769
248
355
469
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673
753
228
317
417
. 5i~
615
7CO
i
109
118 I 159
i
'
490
097
140
,:1.)
369
072
.
1 I
_ ...
II
-:
OSO! 150 I 251
036 I i54--I 252
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1
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3
998
965! 824
577
336
176
097
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lOS
171
259
362
469
57~
665
743
5
939
933. 804
610
403
247
153
199
265
345
'432
522
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682
669
556
443
162 I 136
!
351 i 293
270
275
293
334
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423
. ':'83
54iJ
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i
: ;,.~;::
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. b utlon
.
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3
C(tu~.L"~~
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1
Exponential
r-
0
f
10
956
i
876
i
775
I
!
0.90
I
I
0.01
8.05
0.10
L~ciT:~1
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57;
c~c
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4 7 ~~
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370
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698
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37C
45;0
601
697
775
......... C0..:..
t93
i70
771
812
581
355
198
118
100
126
136
268
363
463
5
987
918
780
605
. , . _, 4000
'':l,)
296
214
184
195
235
293
3&1
0.;..,)4
10
956
861
742
628
:> ... :>
-
464
414
383
368
369
333
':;02
068, 051
081
IS1
252
081
e:. ,
JJ.
?-?
-::>-
3
j
723
956
.
I
6~6
997
1----·--1--·-- -
I
559
3
5
1000
11000
I
--.---~;
j
:
I
982 : 840 . 561
t
.'
i 142
:
303' 143
i
i
:
068
052
r?
l
152
':' c; ')
--- ....
370
• 490
,.
:\
3'J3
144
069
O::> ..
307
148
072
055
085
155
255
371
459
J
.... '-i"
...;c;
561
309
150
075
058
088
158
257
.-.~7?
'::;89
-~
:;;;:
6C'
833
561
310
:'53
077
eGO '
090
160
259
~.~
"'-.\:":,
~S9
:l:;
.-
979
832
562
312
154
078
061.
091
160
259
:'ii.
4SS
:')'
1000
'"'7-I
~.
328
561
316
160
083
r
~oo
. 096
166
263
375
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...)
lGG'J
S75
V~"i'
SSC)
319
iSS
089
071
102
177
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-,
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)-:;-
lOOO
982
8£;0
561
3
1000
981
837
'1
5 OJ.
5
, 1("/JO
9bO
835
10
, 1000
97S
3
11000
5
10
o~itted:
entries
~.,
I
£~o~~ld
I
, , ,
.,
- -
;
982 ; 840 : 561 ' 302
10
?oints
~':l
:"
~
b2 divided by lOCO.
I
,...,~
.
__
,~
.....
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ICC
,
V,V
:.:
- ,-
..- -
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. G
I
5S~
::.:.~
~
v_
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J
~,
V~lllCS
Size of the Test and Range of
Under a Hixturl.> of N. E.
,---;
l'<1r:::l
p
I
;. -'-'-"-'---Ii
i
letel'~;
8 /8
2 1
...•. -. --.-.,..--,--
_
NLg<ltiV(~IL'XpC)ll(~nLia 1
I
I
i
.---_. -----------
Range
I
(l.9S
Ii
I
i
-i)':'8 j-1. 00
I
I
I
95.8
0.96-1.00
99.0
97.7
o. 98~1.00
99.6
10
0.99-1.00
100.0
1.00--1.05
..
99.1
3
0.88-1.00
97.0
0.96-1.00
99.3
5
0.95··1.00
99.7
1.00-1.01
100.0
LOO-1.35
95.0
1.00-1.28
91.9
...
_._--
i
I
_....._. -"---j'--
----_.~_.
3
0.87-1.00
97.4
0.97-1.00
99.6
5
0.97-1.00
99.9
1.00-1.04
99.7
10
1.00-1.59
93.0
1.00-1. 30
95.7
3
0.87-1.00
95.3
0.96-1.00
5
0.87-1.00
95.2
a •96~·1.00
98.8
0.87-1.00
95.4
0.97-1.00
98.8
3
o .87-1.00
95.5
0.96-1.00
99.6
5
o .87-l.00
96.0
0.96-1.00
99.0
0.87-1.00
96.1
a .96-1.00
99.0
----
10
I
_:.l:'_J
98.8
0.89-1.00
__ 'A. _ _ '_ _
I
() .0:>
0.96-1.00
5
-··----i -_. '\
\
.__.... - . - - i -
95.2
0.88-1.00
_ . ~ _
(J,.en
s~ze
3
10
I
0.90
of
- - - - - - - - -_ _ _ _ _ _1 _ -
I
!
-----~
Rauge
MInI.mum %
of s~ze
l'iinimum .%
i
i
I
20
S
I
r·----~--
[-
Com_~~.~t.~.
Sample Size
r----------- -_.
I
!
for which Test is Biased
10
-_.
....
__..
---
I
98.8
3
() .87-1.00
95.9
0.96-1.00
98.8
5
o •88-l.00
96.4
o .9i'-1.00
99.1
10
0.88-1.00
--"-
96.7
o .97-1.00
99.2
---_.- "---'-
...
~-
I
--~
I
I
I!
I
I
-1
I
•
Table 6.
Points
the Pow~-::" Function of \."c 'I'~'o-Si~eG Test for ~ ,.., l.O
0:1
for Mixtures of N.E.
Cor:r~)onents
- Tail Areas Adjusted to
;<d~:e ;~.E.
of
.
1 Test Size a =
5. ".. ~o~.l.na
O~OS.
Sample Size
Parameter
P
-,------------
--
("\v.3:I 0.'+, Il'v.::>
n 8 /8
2 - - -1- - - ,
!
I
:
!
CI.99
i
II
:
:
j
I
!
i
:
j
0.95
I0.:>:
(.! 0.7
1697
i 449
/278
1.
,()
i 0.0
i
-~:--_._--~~
--
-----'-
I
I
j
i
i
!
I
i!
!:
i
'
1709! 477 \307
5
: 736
! 525
199
1357
133
244. i 174
i
713
5
738
561
218
406
292
- -
i,
-
!-eJ
218
.'. ~
,)
,
,
:. 0
r
tJ
~.7
.J... ..
1055
070,093
12(,! 159
i
2(.J i 234 . -;;27 ; 37C
2.98
' - - : _ _'
i 060
074! 096
062! 065
075
:.9 ; 2eO
2t
I
125; 160
198
;'25
].37
-: c"'-
031: 090 : 117
l:t~
,":..75
078' 098
!
'
i 074
2S1
229
j
323
i 3S6
:
'
~.:
L .J L
~~-
L--~.....
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73
I
J.~ -'
·...,I
'""
247
~
.... -
-
_' ' -
..
'J::::
.....J - i
I
j
;
071! 075
088
108
134
164
199: 236
274
3:3
i
101! 102
109
123
141
164
19:
221
252
?3S 132C
172!I 160
155
155
159
167
179: 193
210
090
102
:'.21
145
1.'- .;,
136
, '"
I 4L
154
169
245
236
;'""'""'.,
232 IL.JL.
133
111
231
i
l
111
195
I
085
091
172
i
,j:)J
i
I
150
, ,
'_.4
1
i
373!f 288
328
494
2
.L
055
I
I
3
....
096 1076
!
794 : 644 'I 493
10
'1
1
I !
'
3
,j_.~,
1 I
- ---,----
! 050
j 700 1455 1285 1179 1116 080 060
I
i
I
!
Iii
5 ! 709 : 468 I 297 '190 i 126 : 088 : 069
!
i:
!
I
10
1739: 507 i 333 221 i 151 110: 086
Distri9~ticn
of
.... 0
174 i H1 1075 1056
3
~
0.90
'"
o.~
'-- I
- . ~ ~ ,- - - ~ - - - - . - - - . - ~ - - . - - - . - , -
I Negative :r.:xponentia1
I
II
~ean
Values
125': l:nb-i2.sed.
137
148
1
I
Z2?! 251
.
I
]!. 5
:204 . 237
273 ; ~·C9
187 '20S
':1,
" - - ....
256 : 223
1
..... ~.::..
2·_ 2
257
i 278
I
~
1
,~.
I.,)"\'"
1
763
10
671
570
475
395
2 0')
.,'-
335
264
!
I
""~
..:.. -,'-,
L~\"";·
i
;
3
i
I
Ii,
0.01
005
•
I 696
175
450! 280
Ii3
076
057
051
176
113
077
058
052
450: 281 : 177
114
077
~S8
052
I
I
I
056
072, O?(J ; ::..::0 '~_)l
2CJ
Ie.\.
I'
5
I1694
45 4288'1°,
.
I CJ."I
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L.
084
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'
I
,I
I.
!I
I
,
I'
J
i 692
'I
454 i 290
3
1694
457 ''''92
5
(:.91
459 '- J
,J
I !
I (- (") q
. (.. n
f)"'"
1 J::''~ .' \.\..:.' '-' '"
10
1
---L-
,
.
... ()
•L ......
I!
Dec~·,::l :,oints OJ~1tte~~:
I
I 'I
I:Q8
I
I
Iu;.:
....
4
I
125
i
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C6S
!
I
I( "'f,4
U... .
;
1
'I'
062
1",
"'lu9 :Q13'--2'
.
2(,,...,
.:. -./ L
.. "'\ t'\
J..
..,j ;'
I
l
po ',C',:'~7!.:.I_
\C)7':
,.- ,0 U"oe;
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J
......
01
...
'-'
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079:'()'1.13~'·e:.-1
: ..I...
~
L
1.L v
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022: leS
I
'
I
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r-.
.L..-
c--t;~·:C'~~ ~.1.:()_:..~Jd l~c ~~5...,~·:':.·-=·(_~
~. l·_/~'
("! 11
Ii
,
l, /
_I
-:-:-1
' -
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~,~
;
'-
__
-.
",'::;'")"
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i.. ...... ~
, .
~~~ :~"':'
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:
:
Q
A
()
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:
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_).-...
.)
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7:;:"~':C:"'7'"
_'.J.l..
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.... u:)
I
:':'9
~: ...
::'9l: 332
'
'I
I .
(\7~'"
)-::-
:
0'2'1 i 067 ! 0"-1:
0(.,6: e81 11C~{. ,,"'/, :'(,0
v
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"06 7- ,: v . )08
I
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I
J. 4 ~
• , _
:
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;
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'n -. ,
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l./':j 1.
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1" :..;"
...... c' ~
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! ::'35 11. iC : ::':3
'
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!
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i "
! 067
10-4
2:'.:2
...., ..... , ....
---,------..,.-----:- - - 1 - - - - ,
I !I
i '
'100
I
CSS
095 '1:'.5,
I
~)58
V
oJ
"
2::-<:'
1'~
1U
i
--;--1
100..
E?
. ::'l: 2-3.:. '32-3 : 370
l~-j
453 i')85
1'~1
4..
.1. v
'
~~
095 ,125
'
872.
3 ! 69;
I
:
071.
057
:i
:
f
0.10
!
450;I 280
:
I
I
, I
('\81: at:,' :I 055:
"'::;0-..
."
I \;-
i
l
II
I,
696
5 ! 696
10
!,
I'
r-
i
;,...
':'7: ~ . ~c:
ie',,', '0;:-'
i l~l"lt !I:~~~ :.'~~::: ::~ j :~~ !:~; ;=~:.
n
..
:
'._. __
~~
t
-
......
_ - J
~
__
: L.:J L.
"-:
~
, _ __
0_ :: :..:
'J _ -
_' _ : : :
, .J
I
"
_~
I
.-
:
_
T.:-i_hle 7 ..
PO,_7_:-:.:: S en
tl:e
l'o,;~"~r Funct:o::1 0 ..··
,----------,
t:_e 'i"...·o-S~(~Cd '':1::5: fer
for Mixtures of N.E. Components - Tail Areas Adjusted
L3
Sample Size of 20. Nominal Test Size a
=
P
6 2 /° 1
0.31°.410.5 10.610'.710.810.911.0 ILl
I Negative Exponential !
I
0.99
I.
0.05
I
1000
! 986
861
,
!592
1328
1 157
j 074 I 050 1 071
!
I
I
I
! 574 i 674
i8711
756 ,':21
,
I
I
!
336
081
057
078
138
229
340
456
567
666
748
_1 ....
'II
5
997
971 1852
609
356 i 184 1098
072
090
144
228
331
442
550
648
731
799 : 852 i
I
10
982
9471856; 660
i249
111
114
149
211
295
392
-'.93
59~
677
:50
S~l
080
100
156
238
338
L.43
5:"7 ,, 641.
722
~
~
660
729
787
523
500
635
: " '-,?
0.90
'
i362
,
i 192
631
959 1882
784
680! 570 1458
829
606: 380
i 9251794
623 j 451
3
998
5
988
959
I 961
1
1150
1105
I
138
148
i 363
301
215
126
312
223
326
411
272: 273
292 i 325
367
099
119
172
i
249 1341 :439
186
191
226
280 1345
0.01
I
t
.
,
J
5
1000
10
1000
I--~----~T
!
I
! 1000
3
---r
i 986 i
1
i
!
J
-~
: 108':)
984' 856
592
473
421
387
159
075
051' 072
592
329
159
075
051
073
592
329
160
076
051
073
591
!
I
i 582
.-
c,' ?
89
70'2-
- .. ~
. Co..;
416 :I 489
560
627
i~89 ; 7:'4
503
541
579
673 ,755
::.20
870 :
:, 461 ~ 574 ; 672 • 754
135: 229 iI~143 ; '61 ' 5"'4 i 6" 'J 75 ...
319
S69
I
3lS
'-;:""
c~,
!
379 ,402
368
I
i
432 • 466
!
I
i
228 ~ 342 : 461 . 574
134
135
229
!i 342
I
!.--
;
.
077
139: 232
i
I.,...~'"
,-
i
I
i 616
i
I
'..
I
1079 I 055
333
I
!
344 : 461 ! 572
669 ' 75C
":::5 . S65
I
i
I
370
498
•
,
1
623
I
329
, bo-.
535
:
860
! 9861859
i 985 ! 859
J
250
i
i
3
188
I
I 869 I 752 ! 639 ! 545
,- r '
i
I
1264 .171
818
i 426
I'
.
I 939
991
10
1 166
-I- - - - ;
998 1970 ; 841 : 605
5
427
.
.
r-~~'-~-
i!
I
!
'
596
0.95
0.05
!
982 I 856
10
I
133 1227 1342 1461
'
1.8 '1. ~ i 2.0 !
1000
3
I
!1. 5 1 L611.7
3
-~-T-
!
L211.3 11.4
!
l
I
leG
Mean of Distribution
Fara=eter Values
I
=
,---
Mak2 S.E. Test Unbizsed
r,i
t-
5
lOCO
984
854
591
334
167 1082
057
079
141
234
345
461 : 571
667 : 748
,-,4._
,.:"'! .,
862
10
1000
933
852
590
336
169
i 085
060
082
144
236 : 347
461 : 570
666 , 7",6
':;10
S59
I
I
3
G.10
,
5
10
,
e
1337 :170
S:3::', i SSl 59l
I ,
lOCO 981 0:'7 590 1 341
176
091
066
CSS
i 344
181
096 i 071
093
l':;C;J
i
1000 ; 9f,;C
Decin~l poi~ts o~itted:
844
538
entries should
085
be~vided
060! 032
by 1000.
144: 236 ! 346
I
I
150 124: 1349
L.60
.560' 66.:.
'7:':':"'~09 : ~59
.
461
1~5J~4_'5_!_3_5_2_462
567
661! 739
:003
566
658
-9S: S':'S
735
e
I
E)::'
I
i.,-~D;.e ().
~}.=.i;:it3 ..) r l
t...>(; l.Jo·.",,'c:r Fl:i:ction ::.; ......._~~e
One--S.~.(j('d
for Garr.:ra Distributions wi ttl S<'--':f:fl1e Sizes of Sand 2.0.
I!
I
Sc:mple size n
y
~':::J;'.1ir,c.l
-
j 1.0
1.4
1.6
1.8
050
219
324
426
324
3""
.::..)
321
Size of T~:t a ::: 0.05.
Di~tribution
Mean of
Parameters
Test i~--::-,r;..: .::; 1.0
I"..:..0'
2.8
oJ.V
! 722
768
c~;
2.2
2.4
518
597
00.)
426
427
428
429
518
521
524
5%
666
603
672
723
, 729
608
617
678
736
"
; f'
, ... ':J
433
546
6':'2
,U
6 Qrl
7::1
517
514
510
502
470
506
cb4
592
658
586
651
636
334
425
4"-)
423
420
406 '
701
846
926
~oo
702
707
712
723
751
848
853
859
871
927
931
936
945
964
966
969
972
978
':l
'
r-------~-~-~-~
I Negative Exponential
~-l~l1.05
5
r
i
I
1.10
I
!
1.20
i
1.50;
5
!
--
0.50
122
: 119
219
216
043
038
025
; 114
'
i 107
: oS7
212
206
187
I
CSl
054
I
I
5
1
058
066
103
I
II
'I Negative Exponential
,
20
: 050
~
I
!
LOI
1.05
1.10
1. 20
318
309
530
,~
.-
20
u49
046
043
037
1.50
024
0.99
0.95
0.90
0.80
051
054
058
068
124
128
132
142
220
178
?c;""1
_.J/
223
324
326
227
327
?")r
330
"-~'+
I 223 ~ 478 :
I !
,
,
! 222
! 219
I
~
214
: L,78
479
430
! 206 ,481
! 134
485
900
r ... ,
574
527
577
,"
984
~.'"
985
!
9S6
933
C01
.."J~
988
996
965
%2
957
946
801
934
082
I
I
II
I
!
I
20
"
: 224
2:::8
I
1233
,243
n:: i 277
0.50
Decim~l
770
be::
776
784
Bli+
u.;::...
,c..,)
797
832
720
714
767
760
8e.)
'>, .
7
706
751
733
659
f~~
322
Q'
'
,Sj~
---~
I
I
i
I
~----~--'
0.99
0.95
0.90
0.80
0/+9
047
t"
points ocitted:
e~tTies
478
4,7
476
474
463
should be
700
G95
689
845! 925
1339 \ 920
~31
S14
675
625
1515
899
748! 834
~ivicied
i
I
by 1000.
~79
')72
929
688
-~t'
t.:v
-'~
I
,
78:
/ "';
-- ~
6 e.:
'" -
Table 9.
Points on the Power Function of ti:e T"l.Jo-Siue:d Test for
for G3.r.'".rna Distributions
wi~h
Sa,;Jple Sizes of 5 and 20.
Parameters
y
~;omin;).l
Negative Exponential
0.4 I 0.5
0.7
228
138 ! 087
383
5
0.9
0.8
060 : 048
! 1.0 I l.l
I 050
5
I
381
377
371
361
334
226 1137
220
131
213 I, 124
199! III
164
081
I
0.99
0.95
0.90
0.80
0.50
Negative
1.01
1.05
1.10
1.20
1.50
0.99
0.95
0.90
0.80
0.50
I: 384
1
j
5
.
Exponential
20
'I
i 982
I
I
I
i 9dO
20
!
978
1971
i 938
Decimal points omitted:
e
2~5
264
339
841
842
846
I'
I
155
174
259
Size of Test a. = C.05.
058
054
049
040
023
047;
043'
033
030
015
061
066
073
091
180
050
054
061
078
169
049
045
040
032
016
?
L.~
1.3
1.4
1.6
i1.7
239
i 285
238
284
282
279
274
259
085
116
153
062
057
052
043
026
084
079
074
065
045
115
110
105
095
073
152 I
147
142,,
133 I
110
.
051! 064
056
063
080
170
,
069
076
093
181'
\
i
087
092
099
116 i
200'
193
I 189
I
2~5
185
176
155
':l· ,
2 _1.
223
205
i
I
117
122
129
145
221
154
159
165
180!
250
I
! 561
I
301
141
.
195
200
2C5
218
278
240
2~3
2~8
256
3G3
285 I
28S I
291!
299 I
335 I
1
066
050
080
I i ;
561 I 300
140
065
049; 078
I 562 I 296
I
I 290
860! 562 : 280
884 1 563
252
iI
150
134
128
061
045
055
040
068
148
144
138
116
08S
046 I 032
028 i 016
059
039
128
104
074
251
369
250
369
246: 367
242
365
23~
362
212
352'
:
I
840 ( 561 ! 303
143
C68
051
081
151
836 I
820!
820 II
776
149
156
174
251
073 i 056
080
063
097
080
183
167
C86
093
110
193
156
561
561
I
S62
565
307
I 314
327
1
379
163
178
247
370
252
255
260
271
371
373
378
320
400
490
490
491
492
6C2
I oC3
i
6C~
610
494. t:7
500 :
,
"e"
-
'
49\): . tJl
489 i 5LJ:3
I .ss-+
485; ):0
481 i 553
488
entries should be divided by 1000.
e
I'
I
I
851! 562
I
,
I
I
I
I
i_ _
089
094
102
121
209
236 ~ 146
I
983
985
986
939
995
086
081
075
064
041
i
I 140
--r---r
I
I
20
229
I 389
\ 395
I 408
459
T- I 983 :
I
~
1.0
! 063
I
I
=
Mean of Distribution
n
1.01
1.05
1.10
1.20
1.50
j..l
e
699
i
i
700
7J4
710
7::'0 ,
"7
I.~
J
..........
i
,
,
I
(...;:'7
,,;
. I
I
693
636 :I
073 Ii
627 !
1e
l 'a b _
0
1. L .
/. r
t" Ucl.-.
, ~ 1
.l._ ...
c:", . .J..4-Ie
~ ~
of Sizes
ill
c: .', 1 e "" .c.~ " ~
f "~ ... ,,-,.~.l.n...........
~.,,'
1 5 <"
. -, ,~l
-.'
/_ I. . to
~~
,r - ...... ~l11lJ
n
1 -....)
'i
0
a~d
n Fr02 tte
Sa~e ~ixture
...":J ...
.!.
.:. -
J... ..)~ •
-In.L.c,pL:~
J " . . •, n'"
". t
.. L.el.
'['.
-.'
.....~I_.J
C ~?
oJ2.......
~ ~_
a ,
..
of Ncg~civc Exponential
ComponeLts and Vurious Values of 8 /8 "
2
.
1
097
062
152
"''''
v
i
~
¥'
....
ii
0
J
z-
'-
C81
069
lJ~ ...
l ??
--
oss
062
:37
IS:
079
-==r===
•....... 1
--~-==
i
054
C66
C78
052
062
:.
C81
059
055
C83
(';-
v/.;..
0.56
053
,
.l.:>v
::"0:
,
.:.""\;
065
.-. '"
0.:)':
'i
057
082
123
061
095
'i 118
101
j'
i
I !
10
;:
Ii
192
~-=~-l=--==~f
I 20 I
!
I I
j
I
.
I
""'0
VI
1
.,:
070
061
052
106
:I 198
081
057
3
::
5
10
!I
.;
')1 "
139
Decin:.al points o",i ttcd:
I
070
162
.....
-----;·---~1·-
il
1
I13'7
. --
._
._._._.
.
_
,,_
079
...1. ..
_
'i
079
C67
i
I 054
':
125
096
!
2:"7
1....
IiI'
I:
;l
I
"
I
-/0
,: 210
, ::...
~I
.. v",
-,.-'
",-:C..,.
,~:;-:.~~-=---: -,~...;:~--~-=.-
0&2
O<c.
0,)~
v",
-, -
I
061
I
12-:"
102
063
,v...
OQ'J
i? ') ')
1_JJ
:"
I
:.
087
I
I
er:t::ies sL:::ald be divided by 1000.
~f'\
"....