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Notion of Equity
367
CHAPTER 30 N O TIO N
3 0.1 .
OF
E Q U IT Y
IN TR OD UCT ION
Th e
n o tio n
of
-
e q u ity
portan ce .
The
sh ip m e n ts
fro m
verned
lo n g -te rm
lo ts .
by
D EFINITIO N
tra d e
of
a same
A c c o rd in g
to
producer
agreem ents th e s e
is
d e te rm in e d
in
In
com m ercial
s a m p lin g ,
te rm s
com ponent w hich of
th e
th e
co m m e rcia l
to
a same
p ro v id in g
agreem ents,
of
dry
we
ta ken
assume
w e ig h in g ,
o n ly
th is
in
ta re
re s id u a l
u n d o u b te d ly pe o f
is
in
co m m o d itie s
th e
w e ig h ts
can
in to
a
account
in
w here
a lw a ys
is
of
e x tre m e
c o n s is ts
of
s e rie s
consum er.
Such
fo r
hundreds
te n s ,
purchase
and
d e fin e
s a m p lin g
n e a rly
c ritic a l
th e
s h ip m e n ts
p ric e
"c ritic a l
i t
of
or each
co nte nts c o nte nt"
purchase
are
even
u s u a lly
lo t
p ric e
th e
of
or
of
s h ip m e n t on
sam ples.
p ro p o rtio n
th e
go
th o u s a n d s
e v a lu a te d as
im of
m a te ria l
of
a
o b je c t
co ntract.
We s h a l l gross
a ris e s m in e ra l
very
th e
fo llo w in g
w e ig h in g ,
errors
b e in g
o p tim is tic
book.
It
w ou ld
s e c tio n s
m o is tu re th e
s a m p lin g
sa m p lin g
h ypotheses
anyway
th a t
but
re q u ire
re le v a n t and
e rrors th e ir
m easurem ents
a n a ly s is ,
in
th e
a ss a yin g
are
sense.
These
w id e r
d is c u s s io n
a m u lti-v o lu m e
such
fa lls
te x tb o o k
o u ts id e s till
to
as e xact, are
th e
sco
be w r i t
te n . We s h a l l in s ta n c e , chase ly
riu m -
-
iro n
in
such m a in
T h is
as,
e tc
p e n a liz e d
fo r
in
p e n a liz e d e tc
c ritic a l
com ponent,
a s in g le
te rm
on
ju s tifie d In
in s ta n c e component
a b a rite
v a lu a b le
by
th e
a g iv e n
fa c t
of
th a t
co m m od ity,
th e
m ain
th e
fo rm u la
a ll
te rm s
several
of
v a lu a b le
component
d e te rm in in g th e
th e
fo rm u la
com ponents
are
fo r
pur
are
u s u a l
u s u a lly
c ri
: :
iro n
in
co nce n tra te ,
co m ponent(s)
:
an
iro n
e tc
ore,
z in c
in
a z in c
c o n c e n tra te ,
ba
...
g o ld ,
s ilv e r,
in
a
lead
or
copper
con
... im p u ritie s
c o n c e n tra te ,
ore,
is
a s in g le w ords
same w a y .
v a lu a b le
secondary
on
o th e r
th e
s u lp h a te
ce ntrate ,
rite
in
p ric e .
b u ilt
tic a l
re aso n
or
e tc
o ff-s iz e ...
:
a rs e n ic
or
z in c
in
a
lea d
co nc e n tra te ,
o v e rs iz e
or
u n d e rs iz e
s ilic a
in
a ba
... m a te ria l
:
to
a g iv e n
mesh
in
an
368
We s h a l l
:
c a ll
p^n
:
th e
tru e
p^n
:
th e
s e ttle m e n t
te n t
m arket
a ^ n as
p ric e p ric e
of
a g iv e n
of
as
lo t
w ith
c a lc u la te d
a random e s t i m a t o r
of
a ^ .
c ritic a l
when
Th e
u s in g
p ric e
content
th e
p^n
a^n
sam ple
is
c ritic a l
a random
con
e s tim a to r
of PLn:
VEn
th e
"re la tiv e
%
“ d
VEn
is
the
cu m u la tiv e
^
s e ttle m e n t
,E
:
e rror"
“ Sn -
.
"
:
p. y l tj
a random
(V
fo r
v a lu e ),
re la tiv e
d iffe re n c e
between
p Ln
'Ln
v a ria b le . tru e
m arket
p ric e
of
a s e rie s
of
N lo ts
L
= 1,
(n
n
2,
...
N)
ρίΣ =Σ ρίη n
ΡςΣ :
the
cu m u la tiv e
s e ttle m e n t
:
th e
cu m u la tiv e
of
th e
s e ttle m e n t vc
error
d e fin e d
PS Z - P LJ:
?
is
th e
v a lu e s , is
w e ig h te d
we
equal
s h a ll
to
th e
me an
a d m it
:
E q u ity
:
3 0.2.
:
as
1 =—
E
:
by
J
th e
s h ip m e n ts
p ra c tic a l
u s u a lly
purposes
have
th is
a<*n ,
v a lu a b le
at
Fo rm u la s
pressed
p ric e
le a s t
by
lin k in g
or
fro m
th e
ores p^n
p^n up
one
most
and
is
w ith in
com ponents,
c om m odity
me an
:
n is
s a id
to
d e fin itio n w hich
be of
"e q u ita b le " VE
n
:
in v o lv e s
when
th e
me an
of
th e
: m(Psn)
= p Ln
FUNCTION OF THE
CONTENT
m e ta l-b e a rin g
s e ttle m e n t
c o m p a ra b le
w e ig h te d
ve
proce d u re zero
m e an
n
N
As a ll
PR OPER TI ES OF THE S ETTLEMENT P R IC E ASSUMED TO BE A L I N E A R
W ith th e
is
n
fo r
m(VEn ) = 0
CRITICAL
te n t
VE
VEn ·
a rith m e tic a l
A s a m p lin g
v a ria b le
of
th a t
Σ
random
:
N lo ts
ξ ρ ίη
ve
D e fin itio n
of
PL n VEn
Σ ν Ε Σ
s e rie s
" Σ pSn
p Sz VE
p ric e
n e a rly
th e is
th e
a lw a y s
lim its
p o s itiv e . p ric e
co ntract
general
c o n ce n tra te s
to
of
fo rm u la ,
lin e a r
a g iv e n
W ith
p^n to th e
a
as w e l l
th e
as w i t h fu n c tio n
range
p e n a liz e d c ritic a l
fix e d
in d u s tria l
m in e ra ls ,
of
th e
c ritic a l
by
th e
co n tra ct.
com ponents,
p^n
co n te n t
may v a r y
next
but
as
a general
v a lid
fo r
a s in g le
a ^ ru le ,
component
is
th e y
:
con W ith
n e g a tiv e .
can
fro m be
one ex
:
369
Psn
= ML n [ ( a Sn
ML n
:
dry w weight of tthe lot ei* 9 ^ lie
pQ
:
p ric e
of
'
one
p e n a liz e d by
W
to n
0
Pln
and
s m e ltin g
of
m e ta l,
a c ritic a l
a n > t>n ,
* 1 "
and
pure
m e ta l
m in e ra l
:
'
w ith
P2n
:
*-n '
co nte nt
p 2n
^
b n ) ( p On ’
or
a^
m in e ra l
(p o s itiv e )
(n e g a tiv e )
in
or
p e n a lty
a m a te ria l
per
to n
of
c h a ra c te riz e d
,
p ara m e te rs
p ro c e s s in g
or
m a te ria l
a c c o u n tin g
charges,
e tc
fo r
...
s m e ltin g
Some
of
or
p ro c e s s in g
lo s s e s ,
these
p a ra m e te rs
can
be
zero. W h ether th e
m e ta l
producer th is
is
by
th e
beyond
We c a n
PL n
th e se
th e
P in )
as
th is
procedure
is
th e
th e y
same
Pln >
.
,
( a Sn -
aLn}
s e c tio n
th e
e x a c tly
‘
p ro p o s itio n are
w ork
u s u a lly in
a lto g e th e r im po se d
fa vo u r
of
:
upon
th e
in th e
buyer
but
fo rm u la
:
p 2n] I
,
(lin e a r
m(VEn )
= 0
th e
e q u ita b le very
re s u lt
N becomes
of
h y p o th e s is ),
reduces
e q u ita b le
fo r
he
has
what
bn ,
its e lf
p Qn a n d
p ln
to
:
is
u n b ia se d .
re m ain
s a m p lin g
is
of
as
o n ly it
i f
it
u s u a lly
does· w i t h
s ta tis tic s
th a t
a
VE
When
th e
lo n g -ra n g e te n d s
tow ards
: :
VΕ Σ
re c e iv e d ,
in te re s ts
and
la rg e ,
in fin ite
is
The
i f
m a th e m a tic a l
sa m p lin g
s h ip p e d . i f
is
becomes
a known
as
both
in e q u ita b le
te n d s
th e
s e lle r
p a rtie s :
tow ards has
have
VE^, t e n d s
m(VEn ) been
been
= 0.
p a id
e q u a lly
to w a rd s
m(VEn )
The
buyer
e x a c tly
fo r
has what
re sp e cte d .
φ
0.
T wo
cases
m ay
: m(VEn )
is
v a lu e .
Th e
in te re s ts
tru e th e
to
'
c o n d itio n
s h ip m e n ts
it
Then,
i f
has
-
in
e q u ity
ru le ,
th e y
book.
b n ) ( p 0n
ML n
a n o th e r
= a Ln
m(VEn ) . 1)
assumed
th e
s a m p lin g
co n tra ct,
th e
th is
is
in d u s trie s ,
a general
n p Ln
number N o f
tru e
* 1 ’
bn)
The
-
W
-
m( a Sn>
a ris e
"e q u ita b le "
a c c o rd in g
:
When,
2)
'
p^n
As of
we d e d u c e
co nsta nt,
p a id
scope
c a lc u la te
n =
are
m e ta llu rg ic a l
w hich
vr
he
and
consum er.
= ML n [ ( a L n
fro m
fo rm u la s
m in in g
p o s itiv e
is
v a lu e .
The
in te re s ts
sa m p lin g
of
m(VEn )
:
th e
s a m p lin g th e
s e ttle m e n t
in e q u ity
buyer
n e g a tiv e
of
th e
:
are th e
in
s e ttle m e n t
are
w orks
is
fa v o u r
s ig n ific a n tly
s y s te m a tic a lly
h ig h e r
th a n
th e
of
In
long
run,
th e
s e lle r.
th e
je o p a rd iz e d .
p ric e in
is
fa vo u r
s ig n ific a n tly
in e q u ity
s e lle r
w orks
p ric e
s y s te m a tic a lly of
th e
buyer.
je o p a rd iz e d .
lo w e r In
th e
th a n
th e
lon g
run,
370
As
fa r
as
sa m p lin g
a~
is
are
u n b ia s e d .
3 0.3.
its e lf
is
concerned
e q u ita b le ,
the
and
ina sm u ch
s e ttle m e n t
is
as
th e
e q u ita b le
fo rm u la
i f
and
lin k in g
o n ly
i f
PROPER TI ES OF THE SETTLEMENT P R I C E ASSUMED TO BE A N O N - L I N E A R THE C R I T I C A L
Fo rm u la s tra c t.
up
th e se
m e te rs.
An e x a m p l e
c e rta in
iro n
ore
p^n
to
sam ples
FUNCTION OF
CONTENT
lin k in g
O u tsid e
up
th e
p~
w ill
is
to
lim its ,
a^
are
th e
illu s tra te
p a id
lin e a r
fo rm u la th is
as
fo llo w s
o n ly
rem ains p o in t.
fo r
one
w ith in lin e a r
A c c o rd in g
to n
of
d ry
lim its
d e fin e d
b u t w ith to
a c e rta in
ore
:
per
1
in
d iffe re n t
th e
con-
para
c o n tra ct,
a
-
N ormal
:
65
% Fe
> a^n >
62
% Fe
:
21
-
Low g r a d e
1
:
62
% Fe
> a^n >
61
%
Fe
:
p e n a lty
of
42
cents
per
1
% belo w
62
%
-
Low g r a d e
2
:
61
% Fe
> a^n >
60
% Fe
:
p e n a lty
of
70
cents
per
1
% below
61
%
T h is
13.2-
range
fo rm u la ,
p Sn
us
w hich
$ per
may seem c o m p l e x ,
is
cents
illu s tra te d
in
%
F ig .
Fe
30.1.
to n
-----------
13.1-
1 13.0-
A
I I1
12.9-
I I1
/ 1 12.8-
/ /
12.7-
/
I
1 1
1
\I
1
1
1 19 1C. £0 -
12.5-
.. -·-
■'''
J
/
G
______________
/
12.3-
/
12.2/ / 12.1-
11 9 -
1 1 Β
1 I
|
1
1
/
12.4- /
12.0-
/ S
/ 60
/
//
/
/
/ 1 1 1 I 1
1 // / 1 J 1/ 1 1
!
1
1
1
1
1
1
!
1 I 1 1 | 1 1 I 1
fX
λ
!
\\
V \i l
i1
1 1
'
/i
\
1 1 \
/
1
\
J w
\\
1 \
1 1 1
1 e>1
t32
a. Sn
---------- 1---------- 1---------- 1------% Fe 63
Flg. 30.1. Example of non-tineaA neZoutionshtp between the settlement pnlce of an Inon one and I t s c/UXlcal content Fe 1 .
371
L e t's the
suppose
broken
te n t
a^n
sae. mal
lin e ,
is ,
fo r
A s su m in g and
th a t
th e
apex
A fo r
th e
th e
sake
th e
both
c o n fid e n c e
re s p o n d in g
s e ttle m e n t
Due
change
to
ves, th e is
th e
th e
lo w e r
o rd in a te th e
m (pS n)
of
co u ld
and
a p p lie d
be
s h o u ld
o u r e x a m p le , be
th e
a lit tle
be
to th e
best
we
can
even
th e n
th e
w ill
ever
be
3 0.4 .
TE
η
:
:
in to
c a ll
as
is
by
fa c to r is
sa m p lin g
is
of
of
sam ple th e
p ra c tic a lly broken
th e
of
con
a b s c is nor
lin e th e
co r
o rd in a te s .
d is s y m m e tric a l The
th e
curve is
tru e
p ric e
h a l p^n
is
s e ttle m e n t
p ric e
w h ic h ,
to
due
its
:
< 0
of
is
In
b ia s
1.0 0 2
(p o in t
is
th a t
The
a ll
of
th e
a^n
is
d e p a rtu re
e xam ples
fig .
fa c to r
of
when
in
3 0 .1 .,
th is
c o n tra ct,
la rg e r
order near
of 62
th is
th a n
%.
1
%
0 .2 It
in
s h o u ld
B ).
concerned, and
in
an e q u i t a b l e
a c o rre c tiv e
th e
As
S in c e
lik e
buyer.
be
in e q u ita b le .
p ^ .
downwards
a correct
in e q u ita b le .
pS n ,
consequence
u s in g
%
61
of
one.
of
fre q u e n cy
s h o u ld
apexes
th e
th e
tw o
th e
As
near
of
The
th e
co nte nt
is
on
a x is
upper
m e an
th e
of
d is trib u tio n
th e made
th e
s e ttle m e n t
of
of
a x is
curve
arrow s
p ^ .
concave
a g a in s t
is
th e
G of
d e fin itio n
fa v o u r
a c h ie v e is
th e
The
fre q u e n c y
th e
one
can
we
th e re fo re
th in g s
seem v e r y
are,
do n o t h in g u nb ia sed
th e
about
s a m p lin g
chances
th a t
it
:
and
th is
p o in t
rem ote.
AND RANDOM ERROR I N
COMMERCIAL
S AM PL IN G
:
Tc
_ a Sn
'
error
c o m m itte d
on
lo t
L
n
.
By
d e fin itio n
:
aLn
a,L n
be
lo ts
of
th e
regarded
as
v a ria n c e c u m u la tiv e
T|-
asI
Σ As
than
th e
IMPORTANCE OF B I A S
s a m p lin g
th e
g ra v ity
a lo n g
a lo n g
than
but
m(VEn )
of
in
The
curve
la rg e r
c o n s id e ra tio n
to ta l
tio n a ry ς
to
th e
When
Τ Ε
fa r
is
ta ke n
" can
a^n
s e ttle m e n t
RELATIVE
We s h a l l
when
do
of
lin e
sam ple
th e
d o lla rs ),
s m a lle r
p ro v id e d
as
A,
w ith
d is trib u tio n
fre q u e n c y Fe.
re p re se n te d
u n b ia s e d ,
c o rre c tiv e
la rg e r
is
o b v io u s ly
w orks
th e
U n fo rtu n a te ly , th e
broken
a lw a y s
%
± 2 σ ( a^n ).
in v o lv e s is
th e
a ^ n = 62
p o in t
(1 3.02
a consequence
e q u ity
to
at
ce ntre
s a m p lin g
is
p o s s e s s io n
la c k
th e
w h ic h
th e
e q u ity
A
corresponds fre q u e n c y
u n b ia s e d , of
p Sn
n e c e s s a rily
< PL n
Though
our
of
is
be
in te rv a l
b e in g
p o in t
o rd in a te
d is s ym m e try ,
fro m
h a lf
of
to
s lo p e
a Ln The
c o n v e n ie n c e ,re p re s e n te d
sid e s
p ric e
of
co n te n t
in s ta n c e .
of
s a m p lin g
s y m m e tric a l,
d e lim it
tru e
" a
th e
s a me m a t e r i a l a random
are
v a ria b le
sa m ple d w ith
under
ro u tin e
a s ta tio n a ry
m e an
c o n d itio n s , m(TE)
and
TEn
a s ta
σ 2 (ΤΕ). s a m p lin g
e rro r
co m m itte d
on
th e
set
of
N lo ts
Ln
(n
= 1,
2,
aLZ
LZ
w e ig h ts
ML n
of
th e
lo ts
are
very
o fte n
c o m p a ra b le ,
we m ay w r i t e
:
..N )
372 1 m (TE_)
= m( T E )
and
σ 2(ΤΕ_ )
Σ
= - σ 2(ΤΕ) Ν
Σ
The
% p ro b a b ility
95
For
a s in g le
lo t
Ln
For
a set
N lo ts
c o n fid e n c e
in te rv a ls
: Τ Ε ρ
= m(TE)
± 2 σ( ΤΕ)
: TE
= m (TE)
± —
are
:
2
In
th e
m itte d nary
of
lin e a r
on
means
and
m(VEn )
= m(TE)
m( VE
= m( VE
τ
)
(see
lo t
L
on
3 0 .2 .)
a set
expressed
= m (TE)
% p ro b a b ility
95
s e c tio n
and
n
v a ria n c e s
)
η
The
range
a s in g le
σ( ΤΕ)
as
of
th e
s e ttle m e n ts
N lo ts
fo llo w s
are
a 2 (VEn )
= σ 2(ΤΕ)
and
a2 (V E J σ
1 = — a2 (V E J N η
in te rv a ls
For
a s in g le
lo t
:
VEn
= m (TE)
± 2 σ( ΤΕ)
For
a set
N lo ts
:
VE„
= m(TE)
± —
VEn a n d
v a ria b le s
V EZ c o m
w ith
s ta tio -
:
and
c o n fid e n c e
errors
random
are
1 =— σ 2(ΤΕ) N
th e re fo re
:
2 of
Expressed hand,
th o se
These very
re la tiv e TEn a n d
fo rm u la s
la rg e ,
standard the
in of
d e v ia tio n
b ia s ,
im p o rta n t
p o rta n t
can
become
3 0 .4 .1 .
R e la tiv e
s h a ll
random e r r o r s ,
present
buyer.
know n
to
We s h a l l rie n c e ) is the
assume th a t
σ 2(ΤΕ)
re a d e r must
e xam p le
s ix th a t
th e
and
s a me o r d e r
of
w ith
v a lu e s
s m a lle r w h ils t
(s e c tio n
VΕ Σ
on
th e
one
3 0 .4 .1 .)
and
o f N lik e ly
to
be
c h a ra c te riz e d
by
th e
th e
unchanged.
of
und e rsta n d
σ 2(ΤΕ).
u ran ium
th e lb s
te rm
in v o lv in g
We s h a l l on
show
a no the r
how
on one unim
th o se
order
to
of
100
lo ts
c a lc in e s ,
was
s ign e d
c o n ta in s
pound
In
co m m ercia l
an
average
(Decem ber
errors
1978).
of
of
c a lc in e d between
lo t
(h y p o th e s is
lo ts
have
is
v a ria n c e equal
L e t's
n il
th e
is
b ia s
m a g n itu d e ).
: me an
th a t
th is
zero
p o in t,
we
uran ium
o x id e ,
each
a s s o c ia te d
s ta tis tic a l
assume
now t h a t
a g iv e n
% U^Og.
93
Each
th e
s a m p lin g
w ith
illu s tra te
to
s a m p lin g
sa m p lin g
c o n c e n tra te s .
s h ip m e n t of
in
d is trib u te d
th e
VEn a n d
random e r r o r ,
tw in -s a m p le s
to ta l
th a t
of
id e n tic a l.
rem ains
random e r r o r s
v a ria n c e
US $ p e r of
of
th e
The m a t e r i a l
s e rie s
th e
and
m(TE)
b ia s
in te rv a ls
are
c o n tra cts,
s m a lle r
th e
c o n c e rn in g
be 4 3
te rm
in v o lv in g
me an
th e
hand
random e r r o r .
a b o u t 20,000
g iv e n
s in g le
by
th e
A co n tra ct c o n ta in in g
th e
u n im p o rta n c e
ch a ra c te riz e d
lo n g
te rm
be
c o n fid e n c e
o th e r
becomes
by
can
th e
th e
th e
on
th e
σ( Τ Ε ) ,
c h a ra c te riz e d
how
By
th a t
m ore,
e xam ple
are
v a lu e ,
TE^, o n
show
100 o r
σ( ΤΕ)
S
Σ
The to
p a rtie s
w ith
w e ig h ts th e
p ric e
be or
w e ll each
of
kept
in
U^Og
th e
a is
by
a
re serve .
su pp o rte d of
and
re p re s e n te d
(p h y s ic a l
tru e
s e lle r
by expe
tw in -s a m p le s w e ig h ts
(unknown)
of
c ritic a l
373
co nte nt p^n
a^n o f
= 2 0,000
th e
to
02 ( a σ2 (ΤΕ)
u n b ia s e d , lo t
Ln
to
be
= --------------=
tru e
$ o r,
in
we
kn ow
(unknow n) round
th e
about ± 0 .4
95
%
% U^Og,
m arket
fig u re s
0 .8
p ro b a b ility fro m
w hich
p ric e
of
x
$
10^
is
c o n fid e n c e we
can
in te rv a l
e s tim a te
4.62 X
10
σ(ΤΕ)
and
= 2 .1 5
x
σ(ΤΕ) have
th e
we
can
= 0 .8
x
assumed
chances
com pute
:
106 x 2 .1 5
th e
are,
x
sa m p lin g
as w e l l
1 0 ' 3 = 1720
to
fo r
be
th e
$
c o rre c tly s e lle r
or
c a rrie d th e
out
buyer,
and
and
th e re fo re
fo r
a s in g le
:
%
to
g a in
or
lo o s e
more
5160
$
2 .1 5
%
to
g a in
or
lo o s e
between
2 and
3 σ(ρ^ η )
or
:
between
3440
and
5160
$
13.59
%
to
g a in
or
lo o s e
between
1 and
2 σ(ρ^ η )
or
: between
1720
and
3440
$
34.13
% to
g a in
or
lo o s e
between
0
1 σ(ρ^ η )
or
:
0 and
1720
$
T h is
c a lls
As of
tiv e
v a lu e ,
2)
th e
s o me
S ince
th e
fo r
lon g
5 ,000
dard
th e
s a m p lin g
1
The
pLz
at
now c o n s i d e r
1
= —
a(VE
a(VE
& to ta l
or
v a lu e
seem t o is
be
very
u n b ia s e d ,
: m ore
th a n
between
of
800,000
e x c e s s iv e . s m a ll
th e
maxim um p o s s i b l e
re pre se n ts
%
0.6 5
in
re la
in d e e d .
i t
is
th e re fo re
or
lo o s e
co n tra ct
of
100
to
$,
It
w in
lik e ly
an
e q u ita b le
w hich
a p p re c ia b le
means
th a t
p ro p o rtio n
of
sta k e .
of
w h o le the
cu m ula te d
1
= 2 .1 5
x
error
10
lo ts
(N
co m m itte d
= 100). on
th e
The
set
of
is
:
re la tiv e 100
lo ts
s ta n is
:
-4
*
10
p ric e
of
= 80
a b s o lu te
is is
= ------ σ(ΤΕ)
)
and
co m m e rcia l not
th e
)
n
= 100 p Ln The
nobody
3 σ(ρ^ η )
:
p ro b a b ility
money
d e v ia tio n
σ ( VE )
th e
$ does
its
run,
th a n
re m arks
and
amount o f L e t's
tw o
com pared w it h
loss
in
:
10 J
0 .1 3
1)
:
(93)2
= σ (TE)
= PL n we
= 799,800
The
(0.2)2
)
a Ln As cr ( VEn )
S in ce
% ^O g.
e x p e rie n c e ,
co nte nt
= ----------—
σ ( ρ 5 η)
93
% U308
0.2
=
is
x 43
past
c ritic a l
a (aSn)
Ln
x 0.9 3
A c c o rd in g of
lo t
x
th e
100
lo ts
is
of
d e v ia tio n
of
th e
th e
order
of
:
1C)6 *
sta n d a rd
c u m u la tiv e
p ric e
p Lj;
1 σ ( pS z } Th e s in g le cept or
=
pL e
g(VE
re la tiv e lo t,
%of
as
= 100
s ta n d ard
w h ils t
3 σ(ρ^ Σ )
0.0 6 5
}
th e
th e
th e
p Ln
d e v ia tio n
a b s o lu te
maxim um
to ta l
Χ ^
σ( ΤΕ) is
te n
sta n d a rd
p o s s ib le
co m m e rcia l
lo s s
v a lu e
= 10 ö ( p S n ^ tim e s
s m a lle r
d e v ia tio n of of
e ith e r th e
= 1 7 ’ 200
is
fo r
te n
p a rty,
*
100
tim e s it
co n c e n tra te s
lo ts
th a n
la rg e r.
am ounts o b je c t
of
fo r
I f
to
a
we a c
5 1,600
th e
$
c o n tra ct.
374
Such
a maxim um
u n re a lis tic neous
m a te ria ls
a q u ite
b a b ility
fo r
th e w ho le
such
pass
d u stry
is
What
is
w ith
th e
is out
to
mes
b ia s
s a m p lin g
o fte n
succeed.
3 0.5 .
CONCLUSIONS
Who c a n th e of
a ffo rd
in tro d u c tio n a much m o r e
o n ly
co rrect p lin g
and a p t
th a n
u s u a lly but
in
to o
s a m p lin g
to
d e liv e r
te c h n ic a l
s m a ll
data
is
of
th e
la rg e
to
enough.
in
h is
n a tu re
ch ap ters
17,
to
s a me
th e are
g a in
th e
th e
very
le t's
is
o n ly
any
n ow
loss
w o u ld
% pro
a 0 .1 3
F u rth e rm o re ,
amount o f
on
homoge
assume
th e re
is
money s i n c e
ra n g in g fa c t
th a t
u n lik e ly or
lo o s e
s a m p lin g
fro m
is
1 to
th e
to
in
observed).
% as
0.0 6 5
2 % c o u ld
u ran ium
be
b ia s e d
com pared
and
i f
p o s s ib ility
your
and
fa v o u r.
The
tru th
th e
is
of
th e
tin
m a tte r
m ine
co m m ercia l A c c o rd in g
fa c ilitie s
be
can
perhaps
number o f fro m
in
can
in s p e c tio n
th e
th e lon g
d e te ct of
s till
th e s e
m e n tio n e d
to
re ga rd e d le s s
our as
in
in e v ita b le run.
th e se
a
i f
th e
sa m p lin g
and
c h e ckin g
w h e th e r
re s p e cte d ,
e x p e rie n c e
1 to
sam
2 % b ia s
s ta tis tic a l
th e
number o f
p re p a ra tio n
rem ains
is
random e r r o r s
a s p e c ia lis t a c tu a lly
in
b ia s
a b s o lu te ly
A c a re fu l
b ia s e s
th e y
co m m e rcia l
in tro d u ce
a
s o m e ti th a t
s a m p lin g ,
random e r r o r s .
31)
are
lik e
YOU ? I n
d e trim e n ta l
27
th ro u g h
c o n d itio n s
sa m ple s,
by
money
sa m p lin g
d is tin g u is h e d
out
lo o s in g
o wn
A fa ir
chapter
of
th e ir
p ro d u c tio n
or
u n b ia s e d
s a m p lin g .
18 a nd
th e re
b ia s e s
b ia s e s
? Can
th a n
d e v ic e s
c a rrie d
but
exceeded.
e m p h a s iz e
2 % i f
to
p art
A c ritic a l
when
and
th e
th a t
ris k
im pose
9 % of
(see
g a in
fo u n d e d
are
Th e m aximum p o s s i b l e
s lig h tly
to
such
1 or
o wn
c le a rly
re p ro d u c ib le
v a lu e
not
e q u ita b le .
th e ir
s ig n ific a n tly re s u lts
m ent and m e th o d s , s ta te d
be
very
is
c o n ce n tra te s
s id e .
to
e ig h th
of
n e v e rth e le s s
a n a ly s is
in
tre a ch e ro u s
a m in o rity
lo o se
try
lo o s e th is
or
we s h o u l d
w rong
c o n c lu s io n
in te rv a l.
o cca sio n s
s e n s itiz e d
th e y
to to
be
one w he re
to
so
th a t
th e
to
Our
u ran ium
to ta l
a sym m e trica l
th e
s a m p lin g
to
is
chance)
v a rio u s
of
b ia s e d
th e
reached
(or
(th o u g h
on
are
be
o n ly
ris k
be
Some C o m p a n i e s
to
on
th e
s y s te m a tic
th a t
sa m p lin g
assumed
im p o rta n c e
happens
triv ia l.
happens
%of
0 .1 6
ris k
u nd e te cte d
th e
so
th e ir
a lo s s
p ro b a b ly
a b s o lu te ly i t
1 % U^Og c o n f i d e n c e
an e q u a l
p o in te d
e a s ily
:
$ or
procedure
N o w , we
is
th a t ±
129,000
h y p o th e s is
p a rty
and
im p ro b a b le
n o t exceed
by
ris k
h y p o th e s is
th e
th e
e q u ip
c o n d itio n s
best
b ia s
de
te c to r. L arger
b ia s e s
e x p e rim e n ta l are
im p le m e n te d
p o i n t w hich
in
w ill
o p p o rtu n itie s te d w hich
are
lik e ly
m e thods
does
to
be
of
arouse fo r
accordance w ith d e a lt w ith
c ome
not
to
c h e ck in g
across
m ean
th a t
in
re m ain
c e rta in ch a p te r
b ia s e s th e y
s u s p ic io n b ia s
be
la te r
in e ffic ie n t,
n a tio n a l 32.
ra n g in g
can
sooner o r
or
but
in te rn a tio n a l
S till
la rg e r
b ia s e s
between
10 a n d
20
e a s ily
up
co rrected .
to
e s p e c ia lly
%)
(we are
5 %, when
th e y
S ta n d ard s, had
a
v a rio u s
e a s ily
dete c
375
Our
c o n c lu s io n
concern b ia s you
sh o u ld
be
(ch a p te rs destroy
s ib le o n ly
you
tio n ,
e x tra c tio n
few
anyway
w ith
p ie c e s
try
and
money
or
be
a d v ic e
to
is
th e
be
no
lo n g
c u ra tiv e
co m m ercia l of
p o s s ib le
e ra d ic a tio n .
you.
and
If
There
money
c h a ra c te riz e d
reduced
s a m p lin g ,
s a m p lin g
when
by
b ia s e s ,
is
is
th e
fir s t
p re p a ra tio n
com prom ise w i t h
d e stro y
random e r r o r s
o n ly
le ft, th e
e ith e r
one
then
pos but
v a ria n c e
s u p p re s s in g
th e
σ 2( Τ Ε )
d e lim ita
b ia s e s .
p e rm is s io n ,
buyers
s p e n t on sources
run
s ig n ific a n tly
p re p a ra tio n
re a d e r's
to
p o s s ib le
There in
reduce
to
is
a ll
27). w ill
and
lik e ly
the
of
th e y
p re v e n tiv e
can
is
i f
18 a n d
or
:
w hich
Now,
th a t
e lim in a te
17,
them
s tra te g y th e n
is to
and
we w o u l d
s e lle rs
of
lik e
to
m in e ra l
c lo s e
th is
ch a p te r w ith
co m m od ities
a ll
over
a
th e
w o rld . 1)
Never
hammer a n d w ith
re ly shovel
a c o rre c t,
2)
on
W h e th e r
a n o n -p ro b a b ilis tic
m e th od .
u nb ia sed
s e lle r
v e rn in g
sa m ple s
are
c e rta in
th ro ug h
an
or
v a lu a tio n
fo rg e t
th a t
th e re
d e v ic e s
a n d we
fo re ig n
C o u n try
fo r
fe g u ard
a g a in s t
b ia s .
The 1) or
fa c t no
o n ly to
way
never
by to
of
th e
3)
to
agree
-
p lin g check
syste m s
of
a n a ly s is ,
the
sure
th a t
agreem ent p a rty ,
th e
s a m p lin g
your
th a t
is
a b ig
th a n
fo r you
a c c o rd in g u n le s s
in sta n ce w ou ld
th e
Company
are
of
rig h t
sw orn
to
you
th e
spend
not
u s in g to
be
w hich
are
fa c ilitie s
in te re s ts
i n c o r r e c t ways
your
and
expert
agreem ents
in te re s ts
s a m p lin g
and
th e
to
be
used
g o in g
to
be
co rrect
is
to
im pose
s a fe ly
p rotecte d
are
je o
s a m p lin g in
d e fin ite ly
try in g
go
a b s o lu te ly
re p re se n te d ,
s a m p le r
are
fa c ilitie s ,
co rrected
w here y o u r
th e
th e
as
o th e r of
p a rty
one
of
e s tim a te d
d iffe re n c e
i f
rig h ts
as
e s tim a te d th e
c ritic a l
d iffe re n c e s sa m p lin g
its
a
no
sa
c o n d i
to
have
th e s e
is
:
d esig n ed
necessary,
are
on
not
s tric tly
between
e q u iv a le n t
to
between
:
o b ta in e d
o b ta in e d
between
e s tim a te d
gross
d iffe re n c e
fro m
a u n iq u e
p rim a ry
assays.
sa m ple s
a s s a y in g .
s y s te m a tic
m a g n itu d e s
tw in -s a m p le s
on
as
a c c e p ta b le
fo llo w in g
d iffe re n c e s
component
and
an
between
of
check
on
th e
:
assay
money
as
scheme.
o th e r
a c c re d ite d
p a rty
content
of
more
such
p a rty,
w ith
of
are
m ethod
e ith e r.
a s a m p lin g
content
c ritic a l
o th e r
th a t
th a t
an
an
th e
e v id e n c e by
own w e i g h i n g
s ig n
check
w e ig h t
th e
and y o u rs
c ritic a l :
in s ta n c e ,
make
o th e r
e s tim a te s
sam ple
to
have y o u r
th o s e
h is
th a t
s u ffic ie n t
sa fe g u a rd
e v a lu a te d 2)
have
and
much
sa m p lin g
s ig n
by
out
Never
is
never
o b ta in e d
la id
p a rd iz e d .
The
b uyer,
th o se
and
d e s ig n e d
3)
lo o s e
p ro b a b ilis tic
e x p e rt's
c o rre c tly
tio n s
You m i g h t
s a m p lin g
and
ta re
fro m
sa m ple s
by
both
and
tw o
p a rtie s
w e ig h ts ,
independent
sam
assays. in d e p e n d e n tly
m o is tu re
s a m p lin g
:
and
376
-
s e ttle m e n t
c lu d in g
A number o f out
of
p a rtie s
both
to
our 4)
-
w he th e r th a t, (see
5)
case
o n ly
fo r
of
our
d ete ct
are
o n ly
m in o r
th e
s ta tis tic a l
d iffe re n c e and
v a lu e
i f
th e
s y s te m a tic
upon
s o me
or
reason
and
i f
the
a rb itra tio n
are
w hich
la rg e r
E Q UI T Y
U n lik e ly c le v e r
model
b o n n a ire , 814
-
of
th e
in
of
order
d is a rm
s m a lle r
between
th e
in
assays
la b o ra to rie s
d e fin ite ly
in s u ffi
w h ic h ,
a c c o rd in g
d is c re p a n c ie s .
ch ap ter
than
both
is
31
to
d e c id e
on
a
e s tim a te s ,
s m a lle r
the
a rith m e tic w e ig h t
is
th e
la rg e r
of
at
th e
le a s t
com ple te
agreem ent
th a n
th e
m aximum a g r e e d
m aximum a g r e e d
m ean
of
c ritic a l
both
upon,
e s tim a te s
component
but
th e
w id e ly
p ric e
o f your
to
take
court
by
one
c o n tra c t w ritte n
s a tis fa c tio n be
reached
in te re s ts
re le v a n t
one
of
or
d e c id e as
th e
s e ttle m e n t
p ro d u c tio n
:
th e
or
th e
in
both
such
te rm s
p a rtie s . s e lle r
con
expe rt
If
and ask
fo r
buyer, fo r
P a ris .
expenses
to
upon,
in d e p e n d e n t
je o p a rd iz e d ,
in
lo s s e s
case
an
between
are
Commerce
im p o rta n t
fro m
m aximum a g r e e d
of
Chamber o f
much m o re
th e
out
can
th a t your
c o s tly
lia b le
than
a rb itra tio n
c e rta in
v a rie s
are
must
th e
be
cost
n e xt,
is
co nsu m p tio n
c o n s id e re d of
such
not
a
lik e ly
whereas
m uc h
p la c e .
L O U I S - L E - D E B O N N A I R E 1S S P L I T T I N G METHOD i t
may a p p e a r ,
s p littin g
T h is a
la rg e
h is
draw
m a le the th e
e q u ity
son
was
co n c e iv e d
re s tle s s
between it
th e
We k n o w
a g a in s t
1 % of
b ia s e s
an
w ith
may seem
o b v io u s ly
840.
c o v e rin g
is
In te rn a tio n a l
p o lic y
of
L o u is
to
is
by
d is c re p a n c ie s
s a m p lin g
d e s c rib e d
e x is ts ,
co nte nt,
a m ic a b le
p ra c tic a lly
though
re ig n ,
to
no
C h a rle m a g n e 's to
fo r
p a rtie s .
o th e r
th a n
e m pire
L o th a ire
d iffe re n c e sam ple
a proce d u re
a ss a y in g
e x is ts
adopt
d iffe re n c e
fir s t
a few y e a r s
a procedure
im p o rta n t
3 0.6 .
from
both
in s u ra n c e
c o n tro l,
fa c to rs
ma y b e .
p ro v id e by
to
c ritic a l
case
fo r
be
the
th e
and w o r k i n g
more
of
as
s h o u ld
an
re le v a n t
3 above).
s y s te m a tic
S uch
m e th od
d iffe re n c e
d iffe re n c e
you
p rim a ry
com pared w i t h
d iffe re n c e
a s y s te m a tic
the
agreed
Such lo t
a ll
im p u ritie s .
a c c e p ta b le
a same
lis t) . by
between
p e n a liz e d
a c e rta in
as
the
6)
and
fro m
lo t
i f
tra c t
d iffe re n c e s
o b ta in e d
s p lit
p ric e
to
th e
p ro v id e
a s y s te m a tic
i f
p o in t
g o v e rn in g
as
can
of
com ponents
:
upon
(firs t
i t
im ple m en t
b a sis
check
tw in -s a m p le s
e x p e rie n c e , to
sound
to
on
because
:
v a lu a b le
c o n tra cts
c a rrie d
c ie n t
p ric e
secondary
and
of
je a lo u s y
n in th
more
h o ly ,
:
of
ce ntu ry
a c c u ra te ly
ru le d
to d a y 's
p ro je c t
o ffs p rin g s
lin e
or
h e ir,
n e ith e r
p art th e
the
over
nor
roman
w estern
and m i s t r u s t s p lit
sons,
e m p ire ,
p ro v id e s
"o v e re q u ity ".
Towards
and th e
"H o ly
th e
between
a very
L o u is -le -D e Roman
n e v e rth e le s s
e m pire
e ld e s t,
h is
th e
h is to ry
s o -c a lle d
b u t was
h is
th e
of
of
Europe.
s p littin g
L o th a ire ,
t h a t w ou ld
th e
French
end two
E m p ire "
a m ig h ty of of
h is th e
C h a rle s -le -C h a u v e . o ld
kn ow ing
Em peror th a t
most In
in v ite d
C ha rles
w ou ld
377
th e n
have
It
does
m e rc ia l of
tia l
of
not
tin g
th e
th e re
fo r
s tre tc h
and
th e
from
th e
a tte m p ts rem ains
a 50
in
th e
b ia s
in
L o u is -le -D e b o n n a ire 's be
ment o f
th e
in
the
fro m
a m oral
tra d e
of
th e
transpose p a rtie s
th is
to
th e
model
to
com
c o n tra ct,
each
s p littin g
o p e ra tio n s
th e
p riv ile g e
of
in
c h o o s in g
th e
th e
p re
poten
sa m p le . th e
:
fo r
approach, and
to
th e
idea
suppose
sam ple
e x p e rie n c e d
ch ea t w h ic h ,
a b o u t m o ra ls
have
24,
% p ro b a b ility
to
On e o f
perform
but
th e
p a rtie s
cares
im a g in a tio n
a c tu a l
ch ap ter
w hereas
e q u a lly
p o rtio n .
w ou ld
s e le c tio n
to
of
w o u ld
la tte r
become out
h is
co m m o d itie s.
in s ta n c e ,
p o in te d
o p e ra tio n
c h o o s in g
a huge
t h a t w o u ld
a lre a d y
of
o f m in e ra l
o th e r
o p e ra to r
ours,
need
tu rn s
sam ple
As
p riv ile g e
s a m p lin g
them by
sence
th e
th e
in th e
now t h a t
to
fo r
se pa ra te
some
a p ro b a b ilis tic b ia s
a ssum ing
c le v e r,
th e
s ta n d p o in t,
m ine ra l
rem ains
to th e
b ia s is
w ork
re aso n
in
fa v o u r
?
o th e r
such of
re p re s e n ta tiv e s o n ly
s p lit
or
approach
can w o rk
h ig h ly
co m m o d itie s
th e
to
as
th e of
both
th e
s a tis fa c to ry .
cheat
d e tri
B u t who
CHAPTER 30 N O TIO N
3 0.1 .
OF
E Q U IT Y
IN TR OD UCT ION
Th e
n o tio n
of
-
e q u ity
portan ce .
The
sh ip m e n ts
fro m
verned
lo n g -te rm
lo ts .
by
D EFINITIO N
tra d e
of
a same
A c c o rd in g
to
producer
agreem ents th e s e
is
d e te rm in e d
in
In
com m ercial
s a m p lin g ,
te rm s
com ponent w hich of
th e
th e
co m m e rcia l
to
a same
p ro v id in g
agreem ents,
of
dry
we
ta ken
assume
w e ig h in g ,
o n ly
th is
in
ta re
re s id u a l
u n d o u b te d ly pe o f
is
in
co m m o d itie s
th e
w e ig h ts
can
in to
a
account
in
w here
a lw a ys
is
of
e x tre m e
c o n s is ts
of
s e rie s
consum er.
Such
fo r
hundreds
te n s ,
purchase
and
d e fin e
s a m p lin g
n e a rly
c ritic a l
th e
s h ip m e n ts
p ric e
"c ritic a l
i t
of
or each
co nte nts c o nte nt"
purchase
are
even
u s u a lly
lo t
p ric e
th e
of
or
of
s h ip m e n t on
sam ples.
p ro p o rtio n
th e
go
th o u s a n d s
e v a lu a te d as
im of
m a te ria l
of
a
o b je c t
co ntract.
We s h a l l gross
a ris e s m in e ra l
very
th e
fo llo w in g
w e ig h in g ,
errors
b e in g
o p tim is tic
book.
It
w ou ld
s e c tio n s
m o is tu re th e
s a m p lin g
sa m p lin g
h ypotheses
anyway
th a t
but
re q u ire
re le v a n t and
e rrors th e ir
m easurem ents
a n a ly s is ,
in
th e
a ss a yin g
are
sense.
These
w id e r
d is c u s s io n
a m u lti-v o lu m e
such
fa lls
te x tb o o k
o u ts id e s till
to
as e xact, are
th e
sco
be w r i t
te n . We s h a l l in s ta n c e , chase ly
riu m -
-
iro n
in
such m a in
T h is
as,
e tc
p e n a liz e d
fo r
in
p e n a liz e d e tc
c ritic a l
com ponent,
a s in g le
te rm
on
ju s tifie d In
in s ta n c e component
a b a rite
v a lu a b le
by
th e
a g iv e n
fa c t
of
th a t
co m m od ity,
th e
m ain
th e
fo rm u la
a ll
te rm s
several
of
v a lu a b le
component
d e te rm in in g th e
th e
fo rm u la
com ponents
are
fo r
pur
are
u s u a l
u s u a lly
c ri
: :
iro n
in
co nce n tra te ,
co m ponent(s)
:
an
iro n
e tc
ore,
z in c
in
a z in c
c o n c e n tra te ,
ba
...
g o ld ,
s ilv e r,
in
a
lead
or
copper
con
... im p u ritie s
c o n c e n tra te ,
ore,
is
a s in g le w ords
same w a y .
v a lu a b le
secondary
on
o th e r
th e
s u lp h a te
ce ntrate ,
rite
in
p ric e .
b u ilt
tic a l
re aso n
or
e tc
o ff-s iz e ...
:
a rs e n ic
or
z in c
in
a
lea d
co nc e n tra te ,
o v e rs iz e
or
u n d e rs iz e
s ilic a
in
a ba
... m a te ria l
:
to
a g iv e n
mesh
in
an
368
We s h a l l
:
c a ll
p^n
:
th e
tru e
p^n
:
th e
s e ttle m e n t
te n t
m arket
a ^ n as
p ric e p ric e
of
a g iv e n
of
as
lo t
w ith
c a lc u la te d
a random e s t i m a t o r
of
a ^ .
c ritic a l
when
Th e
u s in g
p ric e
content
th e
p^n
a^n
sam ple
is
c ritic a l
a random
con
e s tim a to r
of PLn:
VEn
th e
"re la tiv e
%
“ d
VEn
is
the
cu m u la tiv e
^
s e ttle m e n t
,E
:
e rror"
“ Sn -
.
"
:
p. y l tj
a random
(V
fo r
v a lu e ),
re la tiv e
d iffe re n c e
between
p Ln
'Ln
v a ria b le . tru e
m arket
p ric e
of
a s e rie s
of
N lo ts
L
= 1,
(n
n
2,
...
N)
ρίΣ =Σ ρίη n
ΡςΣ :
the
cu m u la tiv e
s e ttle m e n t
:
th e
cu m u la tiv e
of
th e
s e ttle m e n t vc
error
d e fin e d
PS Z - P LJ:
?
is
th e
v a lu e s , is
w e ig h te d
we
equal
s h a ll
to
th e
me an
a d m it
:
E q u ity
:
3 0.2.
:
as
1 =—
E
:
by
J
th e
s h ip m e n ts
p ra c tic a l
u s u a lly
purposes
have
th is
a<*n ,
v a lu a b le
at
Fo rm u la s
pressed
p ric e
le a s t
by
lin k in g
or
fro m
th e
ores p^n
p^n up
one
most
and
is
w ith in
com ponents,
c om m odity
me an
:
n is
s a id
to
d e fin itio n w hich
be of
"e q u ita b le " VE
n
:
in v o lv e s
when
th e
me an
of
th e
: m(Psn)
= p Ln
FUNCTION OF THE
CONTENT
m e ta l-b e a rin g
s e ttle m e n t
c o m p a ra b le
w e ig h te d
ve
proce d u re zero
m e an
n
N
As a ll
PR OPER TI ES OF THE S ETTLEMENT P R IC E ASSUMED TO BE A L I N E A R
W ith th e
is
n
fo r
m(VEn ) = 0
CRITICAL
te n t
VE
VEn ·
a rith m e tic a l
A s a m p lin g
v a ria b le
of
th a t
Σ
random
:
N lo ts
ξ ρ ίη
ve
D e fin itio n
of
PL n VEn
Σ ν Ε Σ
s e rie s
" Σ pSn
p Sz VE
p ric e
n e a rly
th e is
th e
a lw a y s
lim its
p o s itiv e . p ric e
co ntract
general
c o n ce n tra te s
to
of
fo rm u la ,
lin e a r
a g iv e n
W ith
p^n to th e
a
as w e l l
th e
as w i t h fu n c tio n
range
p e n a liz e d c ritic a l
fix e d
in d u s tria l
m in e ra ls ,
of
th e
c ritic a l
by
th e
co n tra ct.
com ponents,
p^n
co n te n t
may v a r y
next
but
as
a general
v a lid
fo r
a s in g le
a ^ ru le ,
component
is
th e y
:
con W ith
n e g a tiv e .
can
fro m be
one ex
:
369
Psn
= ML n [ ( a Sn
ML n
:
dry w weight of tthe lot ei* 9 ^ lie
pQ
:
p ric e
of
'
one
p e n a liz e d by
W
to n
0
Pln
and
s m e ltin g
of
m e ta l,
a c ritic a l
a n > t>n ,
* 1 "
and
pure
m e ta l
m in e ra l
:
'
w ith
P2n
:
*-n '
co nte nt
p 2n
^
b n ) ( p On ’
or
a^
m in e ra l
(p o s itiv e )
(n e g a tiv e )
in
or
p e n a lty
a m a te ria l
per
to n
of
c h a ra c te riz e d
,
p ara m e te rs
p ro c e s s in g
or
m a te ria l
a c c o u n tin g
charges,
e tc
fo r
...
s m e ltin g
Some
of
or
p ro c e s s in g
lo s s e s ,
these
p a ra m e te rs
can
be
zero. W h ether th e
m e ta l
producer th is
is
by
th e
beyond
We c a n
PL n
th e se
th e
P in )
as
th is
procedure
is
th e
th e y
same
Pln >
.
,
( a Sn -
aLn}
s e c tio n
th e
e x a c tly
‘
p ro p o s itio n are
w ork
u s u a lly in
a lto g e th e r im po se d
fa vo u r
of
:
upon
th e
in th e
buyer
but
fo rm u la
:
p 2n] I
,
(lin e a r
m(VEn )
= 0
th e
e q u ita b le very
re s u lt
N becomes
of
h y p o th e s is ),
reduces
e q u ita b le
fo r
he
has
what
bn ,
its e lf
p Qn a n d
p ln
to
:
is
u n b ia se d .
re m ain
s a m p lin g
is
of
as
o n ly it
i f
it
u s u a lly
does· w i t h
s ta tis tic s
th a t
a
VE
When
th e
lo n g -ra n g e te n d s
tow ards
: :
VΕ Σ
re c e iv e d ,
in te re s ts
and
la rg e ,
in fin ite
is
The
i f
m a th e m a tic a l
sa m p lin g
s h ip p e d . i f
is
becomes
a known
as
both
in e q u ita b le
te n d s
th e
s e lle r
p a rtie s :
tow ards has
have
VE^, t e n d s
m(VEn ) been
been
= 0.
p a id
e q u a lly
to w a rd s
m(VEn )
The
buyer
e x a c tly
fo r
has what
re sp e cte d .
φ
0.
T wo
cases
m ay
: m(VEn )
is
v a lu e .
Th e
in te re s ts
tru e th e
to
'
c o n d itio n
s h ip m e n ts
it
Then,
i f
has
-
in
e q u ity
ru le ,
th e y
book.
b n ) ( p 0n
ML n
a n o th e r
= a Ln
m(VEn ) . 1)
assumed
th e
s a m p lin g
co n tra ct,
th e
th is
is
in d u s trie s ,
a general
n p Ln
number N o f
tru e
* 1 ’
bn)
The
-
W
-
m( a Sn>
a ris e
"e q u ita b le "
a c c o rd in g
:
When,
2)
'
p^n
As of
we d e d u c e
co nsta nt,
p a id
scope
c a lc u la te
n =
are
m e ta llu rg ic a l
w hich
vr
he
and
consum er.
= ML n [ ( a L n
fro m
fo rm u la s
m in in g
p o s itiv e
is
v a lu e .
The
in te re s ts
sa m p lin g
of
m(VEn )
:
th e
s a m p lin g th e
s e ttle m e n t
in e q u ity
buyer
n e g a tiv e
of
th e
:
are th e
in
s e ttle m e n t
are
w orks
is
fa v o u r
s ig n ific a n tly
s y s te m a tic a lly
h ig h e r
th a n
th e
of
In
long
run,
th e
s e lle r.
th e
je o p a rd iz e d .
p ric e in
is
fa vo u r
s ig n ific a n tly
in e q u ity
s e lle r
w orks
p ric e
s y s te m a tic a lly of
th e
buyer.
je o p a rd iz e d .
lo w e r In
th e
th a n
th e
lon g
run,
370
As
fa r
as
sa m p lin g
a~
is
are
u n b ia s e d .
3 0.3.
its e lf
is
concerned
e q u ita b le ,
the
and
ina sm u ch
s e ttle m e n t
is
as
th e
e q u ita b le
fo rm u la
i f
and
lin k in g
o n ly
i f
PROPER TI ES OF THE SETTLEMENT P R I C E ASSUMED TO BE A N O N - L I N E A R THE C R I T I C A L
Fo rm u la s tra c t.
up
th e se
m e te rs.
An e x a m p l e
c e rta in
iro n
ore
p^n
to
sam ples
FUNCTION OF
CONTENT
lin k in g
O u tsid e
up
th e
p~
w ill
is
to
lim its ,
a^
are
th e
illu s tra te
p a id
lin e a r
fo rm u la th is
as
fo llo w s
o n ly
rem ains p o in t.
fo r
one
w ith in lin e a r
A c c o rd in g
to n
of
d ry
lim its
d e fin e d
b u t w ith to
a c e rta in
ore
:
per
1
in
d iffe re n t
th e
con-
para
c o n tra ct,
a
-
N ormal
:
65
% Fe
> a^n >
62
% Fe
:
21
-
Low g r a d e
1
:
62
% Fe
> a^n >
61
%
Fe
:
p e n a lty
of
42
cents
per
1
% belo w
62
%
-
Low g r a d e
2
:
61
% Fe
> a^n >
60
% Fe
:
p e n a lty
of
70
cents
per
1
% below
61
%
T h is
13.2-
range
fo rm u la ,
p Sn
us
w hich
$ per
may seem c o m p l e x ,
is
cents
illu s tra te d
in
%
F ig .
Fe
30.1.
to n
-----------
13.1-
1 13.0-
A
I I1
12.9-
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a. Sn
---------- 1---------- 1---------- 1------% Fe 63
Flg. 30.1. Example of non-tineaA neZoutionshtp between the settlement pnlce of an Inon one and I t s c/UXlcal content Fe 1 .
371
L e t's the
suppose
broken
te n t
a^n
sae. mal
lin e ,
is ,
fo r
A s su m in g and
th a t
th e
apex
A fo r
th e
th e
sake
th e
both
c o n fid e n c e
re s p o n d in g
s e ttle m e n t
Due
change
to
ves, th e is
th e
th e
lo w e r
o rd in a te th e
m (pS n)
of
co u ld
and
a p p lie d
be
s h o u ld
o u r e x a m p le , be
th e
a lit tle
be
to th e
best
we
can
even
th e n
th e
w ill
ever
be
3 0.4 .
TE
η
:
:
in to
c a ll
as
is
by
fa c to r is
sa m p lin g
is
of
of
sam ple th e
p ra c tic a lly broken
th e
of
con
a b s c is nor
lin e th e
co r
o rd in a te s .
d is s y m m e tric a l The
th e
curve is
tru e
p ric e
h a l p^n
is
s e ttle m e n t
p ric e
w h ic h ,
to
due
its
:
< 0
of
is
In
b ia s
1.0 0 2
(p o in t
is
th a t
The
a ll
of
th e
a^n
is
d e p a rtu re
e xam ples
fig .
fa c to r
of
when
in
3 0 .1 .,
th is
c o n tra ct,
la rg e r
order near
of 62
th is
th a n
%.
1
%
0 .2 It
in
s h o u ld
B ).
concerned, and
in
an e q u i t a b l e
a c o rre c tiv e
th e
As
S in c e
lik e
buyer.
be
in e q u ita b le .
p ^ .
downwards
a correct
in e q u ita b le .
pS n ,
consequence
u s in g
%
61
of
one.
of
fre q u e n cy
s h o u ld
apexes
th e
th e
tw o
th e
As
near
of
The
th e
co nte nt
is
on
a x is
upper
m e an
th e
of
d is trib u tio n
th e made
th e
s e ttle m e n t
of
of
a x is
curve
arrow s
p ^ .
concave
a g a in s t
is
th e
G of
d e fin itio n
fa v o u r
a c h ie v e is
th e
The
fre q u e n c y
th e
one
can
we
th e re fo re
th in g s
seem v e r y
are,
do n o t h in g u nb ia sed
th e
about
s a m p lin g
chances
th a t
it
:
and
th is
p o in t
rem ote.
AND RANDOM ERROR I N
COMMERCIAL
S AM PL IN G
:
Tc
_ a Sn
'
error
c o m m itte d
on
lo t
L
n
.
By
d e fin itio n
:
aLn
a,L n
be
lo ts
of
th e
regarded
as
v a ria n c e c u m u la tiv e
T|-
asI
Σ As
than
th e
IMPORTANCE OF B I A S
s a m p lin g
th e
g ra v ity
a lo n g
a lo n g
than
but
m(VEn )
of
in
The
curve
la rg e r
c o n s id e ra tio n
to ta l
tio n a ry ς
to
th e
When
Τ Ε
fa r
is
ta ke n
" can
a^n
s e ttle m e n t
RELATIVE
We s h a l l
when
do
of
lin e
sam ple
th e
d o lla rs ),
s m a lle r
p ro v id e d
as
A,
w ith
d is trib u tio n
fre q u e n c y Fe.
re p re se n te d
u n b ia s e d ,
c o rre c tiv e
la rg e r
is
o b v io u s ly
w orks
th e
U n fo rtu n a te ly , th e
broken
a lw a y s
%
± 2 σ ( a^n ).
in v o lv e s is
th e
a ^ n = 62
p o in t
(1 3.02
a consequence
e q u ity
to
at
ce ntre
s a m p lin g
is
p o s s e s s io n
la c k
th e
w h ic h
th e
e q u ity
A
corresponds fre q u e n c y
u n b ia s e d , of
p Sn
n e c e s s a rily
< PL n
Though
our
of
is
be
in te rv a l
b e in g
p o in t
o rd in a te
d is s ym m e try ,
fro m
h a lf
of
to
s lo p e
a Ln The
c o n v e n ie n c e ,re p re s e n te d
sid e s
p ric e
of
co n te n t
in s ta n c e .
of
s a m p lin g
s y m m e tric a l,
d e lim it
tru e
" a
th e
s a me m a t e r i a l a random
are
v a ria b le
sa m ple d w ith
under
ro u tin e
a s ta tio n a ry
m e an
c o n d itio n s , m(TE)
and
TEn
a s ta
σ 2 (ΤΕ). s a m p lin g
e rro r
co m m itte d
on
th e
set
of
N lo ts
Ln
(n
= 1,
2,
aLZ
LZ
w e ig h ts
ML n
of
th e
lo ts
are
very
o fte n
c o m p a ra b le ,
we m ay w r i t e
:
..N )
372 1 m (TE_)
= m( T E )
and
σ 2(ΤΕ_ )
Σ
= - σ 2(ΤΕ) Ν
Σ
The
% p ro b a b ility
95
For
a s in g le
lo t
Ln
For
a set
N lo ts
c o n fid e n c e
in te rv a ls
: Τ Ε ρ
= m(TE)
± 2 σ( ΤΕ)
: TE
= m (TE)
± —
are
:
2
In
th e
m itte d nary
of
lin e a r
on
means
and
m(VEn )
= m(TE)
m( VE
= m( VE
τ
)
(see
lo t
L
on
3 0 .2 .)
a set
expressed
= m (TE)
% p ro b a b ility
95
s e c tio n
and
n
v a ria n c e s
)
η
The
range
a s in g le
σ( ΤΕ)
as
of
th e
s e ttle m e n ts
N lo ts
fo llo w s
are
a 2 (VEn )
= σ 2(ΤΕ)
and
a2 (V E J σ
1 = — a2 (V E J N η
in te rv a ls
For
a s in g le
lo t
:
VEn
= m (TE)
± 2 σ( ΤΕ)
For
a set
N lo ts
:
VE„
= m(TE)
± —
VEn a n d
v a ria b le s
V EZ c o m
w ith
s ta tio -
:
and
c o n fid e n c e
errors
random
are
1 =— σ 2(ΤΕ) N
th e re fo re
:
2 of
Expressed hand,
th o se
These very
re la tiv e TEn a n d
fo rm u la s
la rg e ,
standard the
in of
d e v ia tio n
b ia s ,
im p o rta n t
p o rta n t
can
become
3 0 .4 .1 .
R e la tiv e
s h a ll
random e r r o r s ,
present
buyer.
know n
to
We s h a l l rie n c e ) is the
assume th a t
σ 2(ΤΕ)
re a d e r must
e xam p le
s ix th a t
th e
and
s a me o r d e r
of
w ith
v a lu e s
s m a lle r w h ils t
(s e c tio n
VΕ Σ
on
th e
one
3 0 .4 .1 .)
and
o f N lik e ly
to
be
c h a ra c te riz e d
by
th e
th e
unchanged.
of
und e rsta n d
σ 2(ΤΕ).
u ran ium
th e lb s
te rm
in v o lv in g
We s h a l l on
show
a no the r
how
on one unim
th o se
order
to
of
100
lo ts
c a lc in e s ,
was
s ign e d
c o n ta in s
pound
In
co m m ercia l
an
average
(Decem ber
errors
1978).
of
of
c a lc in e d between
lo t
(h y p o th e s is
lo ts
have
is
v a ria n c e equal
L e t's
n il
th e
is
b ia s
m a g n itu d e ).
: me an
th a t
th is
zero
p o in t,
we
uran ium
o x id e ,
each
a s s o c ia te d
s ta tis tic a l
assume
now t h a t
a g iv e n
% U^Og.
93
Each
th e
s a m p lin g
w ith
illu s tra te
to
s a m p lin g
sa m p lin g
c o n c e n tra te s .
s h ip m e n t of
in
d is trib u te d
th e
VEn a n d
random e r r o r ,
tw in -s a m p le s
to ta l
th a t
of
id e n tic a l.
rem ains
random e r r o r s
v a ria n c e
US $ p e r of
of
th e
The m a t e r i a l
s e rie s
th e
and
m(TE)
b ia s
in te rv a ls
are
c o n tra cts,
s m a lle r
th e
c o n c e rn in g
be 4 3
te rm
in v o lv in g
me an
th e
hand
random e r r o r .
a b o u t 20,000
g iv e n
s in g le
by
th e
A co n tra ct c o n ta in in g
th e
u n im p o rta n c e
ch a ra c te riz e d
lo n g
te rm
be
c o n fid e n c e
o th e r
becomes
by
can
th e
th e
th e
on
th e
σ( Τ Ε ) ,
c h a ra c te riz e d
how
By
th a t
m ore,
e xam ple
are
v a lu e ,
TE^, o n
show
100 o r
σ( ΤΕ)
S
Σ
The to
p a rtie s
w ith
w e ig h ts th e
p ric e
be or
w e ll each
of
kept
in
U^Og
th e
a is
by
a
re serve .
su pp o rte d of
and
re p re s e n te d
(p h y s ic a l
tru e
s e lle r
by expe
tw in -s a m p le s w e ig h ts
(unknown)
of
c ritic a l
373
co nte nt p^n
a^n o f
= 2 0,000
th e
to
02 ( a σ2 (ΤΕ)
u n b ia s e d , lo t
Ln
to
be
= --------------=
tru e
$ o r,
in
we
kn ow
(unknow n) round
th e
about ± 0 .4
95
%
% U^Og,
m arket
fig u re s
0 .8
p ro b a b ility fro m
w hich
p ric e
of
x
$
10^
is
c o n fid e n c e we
can
in te rv a l
e s tim a te
4.62 X
10
σ(ΤΕ)
and
= 2 .1 5
x
σ(ΤΕ) have
th e
we
can
= 0 .8
x
assumed
chances
com pute
:
106 x 2 .1 5
th e
are,
x
sa m p lin g
as w e l l
1 0 ' 3 = 1720
to
fo r
be
th e
$
c o rre c tly s e lle r
or
c a rrie d th e
out
buyer,
and
and
th e re fo re
fo r
a s in g le
:
%
to
g a in
or
lo o s e
more
5160
$
2 .1 5
%
to
g a in
or
lo o s e
between
2 and
3 σ(ρ^ η )
or
:
between
3440
and
5160
$
13.59
%
to
g a in
or
lo o s e
between
1 and
2 σ(ρ^ η )
or
: between
1720
and
3440
$
34.13
% to
g a in
or
lo o s e
between
0
1 σ(ρ^ η )
or
:
0 and
1720
$
T h is
c a lls
As of
tiv e
v a lu e ,
2)
th e
s o me
S ince
th e
fo r
lon g
5 ,000
dard
th e
s a m p lin g
1
The
pLz
at
now c o n s i d e r
1
= —
a(VE
a(VE
& to ta l
or
v a lu e
seem t o is
be
very
u n b ia s e d ,
: m ore
th a n
between
of
800,000
e x c e s s iv e . s m a ll
th e
maxim um p o s s i b l e
re pre se n ts
%
0.6 5
in
re la
in d e e d .
i t
is
th e re fo re
or
lo o s e
co n tra ct
of
100
to
$,
It
w in
lik e ly
an
e q u ita b le
w hich
a p p re c ia b le
means
th a t
p ro p o rtio n
of
sta k e .
of
w h o le the
cu m ula te d
1
= 2 .1 5
x
error
10
lo ts
(N
co m m itte d
= 100). on
th e
The
set
of
is
:
re la tiv e 100
lo ts
s ta n is
:
-4
*
10
p ric e
of
= 80
a b s o lu te
is is
= ------ σ(ΤΕ)
)
and
co m m e rcia l not
th e
)
n
= 100 p Ln The
nobody
3 σ(ρ^ η )
:
p ro b a b ility
money
d e v ia tio n
σ ( VE )
th e
$ does
its
run,
th a n
re m arks
and
amount o f L e t's
tw o
com pared w it h
loss
in
:
10 J
0 .1 3
1)
:
(93)2
= σ (TE)
= PL n we
= 799,800
The
(0.2)2
)
a Ln As cr ( VEn )
S in ce
% ^O g.
e x p e rie n c e ,
co nte nt
= ----------—
σ ( ρ 5 η)
93
% U308
0.2
=
is
x 43
past
c ritic a l
a (aSn)
Ln
x 0.9 3
A c c o rd in g of
lo t
x
th e
100
lo ts
is
of
d e v ia tio n
of
th e
th e
order
of
:
1C)6 *
sta n d a rd
c u m u la tiv e
p ric e
p Lj;
1 σ ( pS z } Th e s in g le cept or
=
pL e
g(VE
re la tiv e lo t,
%of
as
= 100
s ta n d ard
w h ils t
3 σ(ρ^ Σ )
0.0 6 5
}
th e
th e
th e
p Ln
d e v ia tio n
a b s o lu te
maxim um
to ta l
Χ ^
σ( ΤΕ) is
te n
sta n d a rd
p o s s ib le
co m m e rcia l
lo s s
v a lu e
= 10 ö ( p S n ^ tim e s
s m a lle r
d e v ia tio n of of
e ith e r th e
= 1 7 ’ 200
is
fo r
te n
p a rty,
*
100
tim e s it
co n c e n tra te s
lo ts
th a n
la rg e r.
am ounts o b je c t
of
fo r
I f
to
a
we a c
5 1,600
th e
$
c o n tra ct.
374
Such
a maxim um
u n re a lis tic neous
m a te ria ls
a q u ite
b a b ility
fo r
th e w ho le
such
pass
d u stry
is
What
is
w ith
th e
is out
to
mes
b ia s
s a m p lin g
o fte n
succeed.
3 0.5 .
CONCLUSIONS
Who c a n th e of
a ffo rd
in tro d u c tio n a much m o r e
o n ly
co rrect p lin g
and a p t
th a n
u s u a lly but
in
to o
s a m p lin g
to
d e liv e r
te c h n ic a l
s m a ll
data
is
of
th e
la rg e
to
enough.
in
h is
n a tu re
ch ap ters
17,
to
s a me
th e are
g a in
th e
th e
very
le t's
is
o n ly
any
n ow
loss
w o u ld
% pro
a 0 .1 3
F u rth e rm o re ,
amount o f
on
homoge
assume
th e re
is
money s i n c e
ra n g in g fa c t
th a t
u n lik e ly or
lo o s e
s a m p lin g
fro m
is
1 to
th e
to
in
observed).
% as
0.0 6 5
2 % c o u ld
u ran ium
be
b ia s e d
com pared
and
i f
p o s s ib ility
your
and
fa v o u r.
The
tru th
th e
is
of
th e
tin
m a tte r
m ine
co m m ercia l A c c o rd in g
fa c ilitie s
be
can
perhaps
number o f fro m
in
can
in s p e c tio n
th e
th e lon g
d e te ct of
s till
th e s e
m e n tio n e d
to
re ga rd e d le s s
our as
in
in e v ita b le run.
th e se
a
i f
th e
sa m p lin g
and
c h e ckin g
w h e th e r
re s p e cte d ,
e x p e rie n c e
1 to
sam
2 % b ia s
s ta tis tic a l
th e
number o f
p re p a ra tio n
rem ains
is
random e r r o r s
a s p e c ia lis t a c tu a lly
in
b ia s
a b s o lu te ly
A c a re fu l
b ia s e s
th e y
co m m e rcia l
in tro d u ce
a
s o m e ti th a t
s a m p lin g ,
random e r r o r s .
31)
are
lik e
YOU ? I n
d e trim e n ta l
27
th ro u g h
c o n d itio n s
sa m ple s,
by
money
sa m p lin g
d is tin g u is h e d
out
lo o s in g
o wn
A fa ir
chapter
of
th e ir
p ro d u c tio n
or
u n b ia s e d
s a m p lin g .
18 a nd
th e re
b ia s e s
b ia s e s
? Can
th a n
d e v ic e s
c a rrie d
but
exceeded.
e m p h a s iz e
2 % i f
to
p art
A c ritic a l
when
and
th e
th a t
ris k
im pose
9 % of
(see
g a in
fo u n d e d
are
Th e m aximum p o s s i b l e
s lig h tly
to
such
1 or
o wn
c le a rly
re p ro d u c ib le
v a lu e
not
e q u ita b le .
th e ir
s ig n ific a n tly re s u lts
m ent and m e th o d s , s ta te d
be
very
is
c o n ce n tra te s
s id e .
to
e ig h th
of
n e v e rth e le s s
a n a ly s is
in
tre a ch e ro u s
a m in o rity
lo o se
try
lo o s e th is
or
we s h o u l d
w rong
c o n c lu s io n
in te rv a l.
o cca sio n s
s e n s itiz e d
th e y
to to
be
one w he re
to
so
th a t
th e
to
Our
u ran ium
to ta l
a sym m e trica l
th e
s a m p lin g
to
is
chance)
v a rio u s
of
b ia s e d
th e
reached
(or
(th o u g h
on
are
be
o n ly
ris k
be
Some C o m p a n i e s
to
on
th e
s y s te m a tic
th a t
sa m p lin g
assumed
im p o rta n c e
happens
triv ia l.
happens
%of
0 .1 6
ris k
u nd e te cte d
th e
so
th e ir
a lo s s
p ro b a b ly
a b s o lu te ly i t
1 % U^Og c o n f i d e n c e
an e q u a l
p o in te d
e a s ily
:
$ or
procedure
N o w , we
is
th a t ±
129,000
h y p o th e s is
p a rty
and
im p ro b a b le
n o t exceed
by
ris k
h y p o th e s is
th e
th e
e q u ip
c o n d itio n s
best
b ia s
de
te c to r. L arger
b ia s e s
e x p e rim e n ta l are
im p le m e n te d
p o i n t w hich
in
w ill
o p p o rtu n itie s te d w hich
are
lik e ly
m e thods
does
to
be
of
arouse fo r
accordance w ith d e a lt w ith
c ome
not
to
c h e ck in g
across
m ean
th a t
in
re m ain
c e rta in ch a p te r
b ia s e s th e y
s u s p ic io n b ia s
be
la te r
in e ffic ie n t,
n a tio n a l 32.
ra n g in g
can
sooner o r
or
but
in te rn a tio n a l
S till
la rg e r
b ia s e s
between
10 a n d
20
e a s ily
up
co rrected .
to
e s p e c ia lly
%)
(we are
5 %, when
th e y
S ta n d ard s, had
a
v a rio u s
e a s ily
dete c
375
Our
c o n c lu s io n
concern b ia s you
sh o u ld
be
(ch a p te rs destroy
s ib le o n ly
you
tio n ,
e x tra c tio n
few
anyway
w ith
p ie c e s
try
and
money
or
be
a d v ic e
to
is
th e
be
no
lo n g
c u ra tiv e
co m m ercia l of
p o s s ib le
e ra d ic a tio n .
you.
and
If
There
money
c h a ra c te riz e d
reduced
s a m p lin g ,
s a m p lin g
when
by
b ia s e s ,
is
is
th e
fir s t
p re p a ra tio n
com prom ise w i t h
d e stro y
random e r r o r s
o n ly
le ft, th e
e ith e r
one
then
pos but
v a ria n c e
s u p p re s s in g
th e
σ 2( Τ Ε )
d e lim ita
b ia s e s .
p e rm is s io n ,
buyers
s p e n t on sources
run
s ig n ific a n tly
p re p a ra tio n
re a d e r's
to
p o s s ib le
There in
reduce
to
is
a ll
27). w ill
and
lik e ly
the
of
th e y
p re v e n tiv e
can
is
i f
18 a n d
or
:
w hich
Now,
th a t
e lim in a te
17,
them
s tra te g y th e n
is to
and
we w o u l d
s e lle rs
of
lik e
to
m in e ra l
c lo s e
th is
ch a p te r w ith
co m m od ities
a ll
over
a
th e
w o rld . 1)
Never
hammer a n d w ith
re ly shovel
a c o rre c t,
2)
on
W h e th e r
a n o n -p ro b a b ilis tic
m e th od .
u nb ia sed
s e lle r
v e rn in g
sa m ple s
are
c e rta in
th ro ug h
an
or
v a lu a tio n
fo rg e t
th a t
th e re
d e v ic e s
a n d we
fo re ig n
C o u n try
fo r
fe g u ard
a g a in s t
b ia s .
The 1) or
fa c t no
o n ly to
way
never
by to
of
th e
3)
to
agree
-
p lin g check
syste m s
of
a n a ly s is ,
the
sure
th a t
agreem ent p a rty ,
th e
s a m p lin g
your
th a t
is
a b ig
th a n
fo r you
a c c o rd in g u n le s s
in sta n ce w ou ld
th e
Company
are
of
rig h t
sw orn
to
you
th e
spend
not
u s in g to
be
w hich
are
fa c ilitie s
in te re s ts
i n c o r r e c t ways
your
and
expert
agreem ents
in te re s ts
s a m p lin g
and
th e
to
be
used
g o in g
to
be
co rrect
is
to
im pose
s a fe ly
p rotecte d
are
je o
s a m p lin g in
d e fin ite ly
try in g
go
a b s o lu te ly
re p re se n te d ,
s a m p le r
are
fa c ilitie s ,
co rrected
w here y o u r
th e
th e
as
o th e r of
p a rty
one
of
e s tim a te d
d iffe re n c e
i f
rig h ts
as
e s tim a te d th e
c ritic a l
d iffe re n c e s sa m p lin g
its
a
no
sa
c o n d i
to
have
th e s e
is
:
d esig n ed
necessary,
are
on
not
s tric tly
between
e q u iv a le n t
to
between
:
o b ta in e d
o b ta in e d
between
e s tim a te d
gross
d iffe re n c e
fro m
a u n iq u e
p rim a ry
assays.
sa m ple s
a s s a y in g .
s y s te m a tic
m a g n itu d e s
tw in -s a m p le s
on
as
a c c e p ta b le
fo llo w in g
d iffe re n c e s
component
and
an
between
of
check
on
th e
:
assay
money
as
scheme.
o th e r
a c c re d ite d
p a rty
content
of
more
such
p a rty,
w ith
of
are
m ethod
e ith e r.
a s a m p lin g
content
c ritic a l
o th e r
th a t
th a t
an
an
th e
e v id e n c e by
own w e i g h i n g
s ig n
check
w e ig h t
th e
and y o u rs
c ritic a l :
in s ta n c e ,
make
o th e r
e s tim a te s
sam ple
to
have y o u r
th o s e
h is
th a t
s u ffic ie n t
sa fe g u a rd
e v a lu a te d 2)
have
and
much
sa m p lin g
s ig n
by
out
Never
is
never
o b ta in e d
la id
p a rd iz e d .
The
b uyer,
th o se
and
d e s ig n e d
3)
lo o s e
p ro b a b ilis tic
e x p e rt's
c o rre c tly
tio n s
You m i g h t
s a m p lin g
and
ta re
fro m
sa m ple s
by
both
and
tw o
p a rtie s
w e ig h ts ,
independent
sam
assays. in d e p e n d e n tly
m o is tu re
s a m p lin g
:
and
376
-
s e ttle m e n t
c lu d in g
A number o f out
of
p a rtie s
both
to
our 4)
-
w he th e r th a t, (see
5)
case
o n ly
fo r
of
our
d ete ct
are
o n ly
m in o r
th e
s ta tis tic a l
d iffe re n c e and
v a lu e
i f
th e
s y s te m a tic
upon
s o me
or
reason
and
i f
the
a rb itra tio n
are
w hich
la rg e r
E Q UI T Y
U n lik e ly c le v e r
model
b o n n a ire , 814
-
of
th e
in
of
order
d is a rm
s m a lle r
between
th e
in
assays
la b o ra to rie s
d e fin ite ly
in s u ffi
w h ic h ,
a c c o rd in g
d is c re p a n c ie s .
ch ap ter
than
both
is
31
to
d e c id e
on
a
e s tim a te s ,
s m a lle r
the
a rith m e tic w e ig h t
is
th e
la rg e r
of
at
th e
le a s t
com ple te
agreem ent
th a n
th e
m aximum a g r e e d
m aximum a g r e e d
m ean
of
c ritic a l
both
upon,
e s tim a te s
component
but
th e
w id e ly
p ric e
o f your
to
take
court
by
one
c o n tra c t w ritte n
s a tis fa c tio n be
reached
in te re s ts
re le v a n t
one
of
or
d e c id e as
th e
s e ttle m e n t
p ro d u c tio n
:
th e
or
th e
in
both
such
te rm s
p a rtie s . s e lle r
con
expe rt
If
and ask
fo r
buyer, fo r
P a ris .
expenses
to
upon,
in d e p e n d e n t
je o p a rd iz e d ,
in
lo s s e s
case
an
between
are
Commerce
im p o rta n t
fro m
m aximum a g r e e d
of
Chamber o f
much m o re
th e
out
can
th a t your
c o s tly
lia b le
than
a rb itra tio n
c e rta in
v a rie s
are
must
th e
be
cost
n e xt,
is
co nsu m p tio n
c o n s id e re d of
such
not
a
lik e ly
whereas
m uc h
p la c e .
L O U I S - L E - D E B O N N A I R E 1S S P L I T T I N G METHOD i t
may a p p e a r ,
s p littin g
T h is a
la rg e
h is
draw
m a le the th e
e q u ity
son
was
co n c e iv e d
re s tle s s
between it
th e
We k n o w
a g a in s t
1 % of
b ia s e s
an
w ith
may seem
o b v io u s ly
840.
c o v e rin g
is
In te rn a tio n a l
p o lic y
of
L o u is
to
is
by
d is c re p a n c ie s
s a m p lin g
d e s c rib e d
e x is ts ,
co nte nt,
a m ic a b le
p ra c tic a lly
though
re ig n ,
to
no
C h a rle m a g n e 's to
fo r
p a rtie s .
o th e r
th a n
e m pire
L o th a ire
d iffe re n c e sam ple
a proce d u re
a ss a y in g
e x is ts
adopt
d iffe re n c e
fir s t
a few y e a r s
a procedure
im p o rta n t
3 0.6 .
from
both
in s u ra n c e
c o n tro l,
fa c to rs
ma y b e .
p ro v id e by
to
c ritic a l
case
fo r
be
the
th e
and w o r k i n g
more
of
as
s h o u ld
an
re le v a n t
3 above).
s y s te m a tic
S uch
m e th od
d iffe re n c e
d iffe re n c e
you
p rim a ry
com pared w i t h
d iffe re n c e
a s y s te m a tic
the
agreed
Such lo t
a ll
im p u ritie s .
a c c e p ta b le
a same
lis t) . by
between
p e n a liz e d
a c e rta in
as
the
6)
and
fro m
lo t
i f
tra c t
d iffe re n c e s
o b ta in e d
s p lit
p ric e
to
th e
p ro v id e
a s y s te m a tic
i f
p o in t
g o v e rn in g
as
can
of
com ponents
:
upon
(firs t
i t
im ple m en t
b a sis
check
tw in -s a m p le s
e x p e rie n c e , to
sound
to
on
because
:
v a lu a b le
c o n tra cts
c a rrie d
c ie n t
p ric e
secondary
and
of
je a lo u s y
n in th
more
h o ly ,
:
of
ce ntu ry
a c c u ra te ly
ru le d
to d a y 's
p ro je c t
o ffs p rin g s
lin e
or
h e ir,
n e ith e r
p art th e
the
over
nor
roman
w estern
and m i s t r u s t s p lit
sons,
e m p ire ,
p ro v id e s
"o v e re q u ity ".
Towards
and th e
"H o ly
th e
between
a very
L o u is -le -D e Roman
n e v e rth e le s s
e m pire
e ld e s t,
h is
th e
h is to ry
s o -c a lle d
b u t was
h is
th e
of
of
Europe.
s p littin g
L o th a ire ,
t h a t w ou ld
th e
French
end two
E m p ire "
a m ig h ty of of
h is th e
C h a rle s -le -C h a u v e . o ld
kn ow ing
Em peror th a t
most In
in v ite d
C ha rles
w ou ld
377
th e n
have
It
does
m e rc ia l of
tia l
of
not
tin g
th e
th e re
fo r
s tre tc h
and
th e
from
th e
a tte m p ts rem ains
a 50
in
th e
b ia s
in
L o u is -le -D e b o n n a ire 's be
ment o f
th e
in
the
fro m
a m oral
tra d e
of
th e
transpose p a rtie s
th is
to
th e
model
to
com
c o n tra ct,
each
s p littin g
o p e ra tio n s
th e
p riv ile g e
of
in
c h o o s in g
th e
th e
p re
poten
sa m p le . th e
:
fo r
approach, and
to
th e
idea
suppose
sam ple
e x p e rie n c e d
ch ea t w h ic h ,
a b o u t m o ra ls
have
24,
% p ro b a b ility
to
On e o f
perform
but
th e
p a rtie s
cares
im a g in a tio n
a c tu a l
ch ap ter
w hereas
e q u a lly
p o rtio n .
w ou ld
s e le c tio n
to
of
w o u ld
la tte r
become out
h is
co m m o d itie s.
in s ta n c e ,
p o in te d
o p e ra tio n
c h o o s in g
a huge
t h a t w o u ld
a lre a d y
of
o f m in e ra l
o th e r
o p e ra to r
ours,
need
tu rn s
sam ple
As
p riv ile g e
s a m p lin g
them by
sence
th e
th e
in th e
now t h a t
to
fo r
se pa ra te
some
a p ro b a b ilis tic b ia s
a ssum ing
c le v e r,
th e
s ta n d p o in t,
m ine ra l
rem ains
to th e
b ia s is
w ork
re aso n
in
fa v o u r
?
o th e r
such of
re p re s e n ta tiv e s o n ly
s p lit
or
approach
can w o rk
h ig h ly
co m m o d itie s
th e
to
as
th e of
both
th e
s a tis fa c to ry .
cheat
d e tri
B u t who