lSupported by the U.S. Army Research Office, Durham, under Contract No.
DAHC04-71-C-0042.
2Supported by the Air Force Office of Scientific Research under Grant
AFOSR-72-2386.
EXTENDED OCCUPANCY PROBABILITY DISTRIBUTION
CRITICAL POINTS
by
N.L. Johnson l
Department of Statistics
University of North Carolina at Chapel Hill
S. Kotz 2 and R. Srinivasan
Department of Mathematics
Temple University~ Philadelphia
Institute of Statistics Mimeo Series No. 934
June, 1974
EXTENDED OCCUPANCY PROBABILITY DISTRIBUTION
CRITICAL POliHS
Johnson, University of Horth Carol ina at Chapel Hill,
S. Kotz and R. Srinivasan, Temple University, Philadelphia
By .J.L.
1.
Introduction
In many instances of application of statistics, the underlying pro-
babilistic problem can be described in terms of the following urn model:
N balls are randomly allocated to
bability
K cells (so that each ball has pro-
11K of falling into the ith cell,
ball has probability
p
i
= 1,
of staying in its cell and
2, ..• , K).
1 - P of I1falling
through l1 (and hence not being available to 'fill' a cell).
variable of interest here is the number
equivalently the number
K- x
x
Each
The random
of 'occupied' cells, or
of 'empty' cells.
Several examples of
situations leading to this urn model in biology, nuclear phyics and other
fields are given in Feller (1968) and Harkness (1969).
Our purpose here is to tabulate critical points of the distribution
for various values of
N, K and
p.
The case when
p
=1
has already
been treated at length by Nicholson (1961), and we shall therefore not
include it here.
tion.
A description of our tables is given in the next sec-
They are set out in a similar form to Nicholson's tahle.
Critical roints corresponding to some of the values of
Nand
Knot
covered in our tables can be approximated using certain limit· theorems.
This
is discussed in Section 3.
The classical occupancy distribution (corresponding to
the probability of exactly
x
occupied cells as
p
= 1)
gives
2
(1)
x
L (-l)(x-i)
=
Hl(x;K,N)
.t rC~\
.: /
· N
i=O
=
J.....
(x
For general
p,
= 0,1, •.• , K)
the number of balls not falling through is distributed
binomially with parameters
N,p
and so the extended occupancy distribution
is the corresponding mixture of classical occupancy distributions.
probability,
tlp(x;K,N),
expected value of
meters
H,p.
of exactly
HI (x;K,J)
with
J
x
occupied cells is then the
distributed binomially with para-
Hence
H (x; K,N)
(2)
P
=
These probabilities satisfy the recursion relation
KHp(x,K,N)
(3)
=
P(K+I-x)hp(x-I;K,N-l)
+
{px
The r-th factorial moment is
(4)
ll(r)
=
=
=
=
where
t(r)
=
t(t-l) ... (t-r+l).
+
The
(l-p)K} H (x;K,N-I)
P
3
From (4) we obtain
~pressions
for the expected value:
N
(5.1)
K{l - (l-p/K) }
and the variance
.
K(K-l) {l-2 (l-p/K)
(5.2)
=
i(
{lo:. (l_p/K)N} ~
N
0+
N
(1-2p/K) }
+
N
K{l- (l-p/K) }
(l_p/K)N _ K(K-l) {(l-p/K) 2N _ (1_2p/K)N}
The r-th factorial moment of the number of empty cells has the simpler
form
K(r)(l _ pr/K)N
(6)
(Harkness (1969)).
2.
Description of Our Tables
K, X, P and lower tail area
Using (3), we can find, for given
a,
X
the smallest
N,
say
N ,
I
such that
These are presented in Table I for
0.99, and
I
for
8, N, 0.99]
N
>
17).
>
~
X
= 2 (1) 20,
N
l
L H (x,K,N)
x=O p
0.75,0.80,0.90, 0.95,
K, Nl,p]
P
=
=
given as
a + .
a.
up to 60, along with the
0.025
a
I
8, 17, 0.99]
17,
and that
Pr[x ~ 5
for
N
<
= 0.019,
Pr[x ~ 5
There is one exception to this system.
I
8, N, 0.99]
If there is a
N such that Pr[X ~ x I K, NI , p] lies between a and a + 0.0005
1
this is the value given in the tables. The corresponding value of a l is
value
~
= P [x ~ X I K,N 1 ,p]. (For example, the
l
corresponding to p = .99, K = 8, X = 5 and
N = 17, a = 0.019,
I
l
a = 0.025 in Table I means that
entry
0.019
K
a = 0.01, 0.025, 0.05, 0.10, and
the actual tail probabilities
Pr[x ~ 5
Pr[x
<
4
Similarly, using the recursion relations; for the same values of
K, X, P
and
we have tabulated the upper tail critical values
Ct
and the corresponding probabilities
K, X, P and
Pr[x
~
X
the entry
K, N, il]
~
= 12,
Ct
N2
2
10, N, 0.80]
0.021
for
existence of
if there is an
N >
N
and
Ct,
in Table II means that
;>
N
2
Ct
= .021,
p~[x ~ 9
<
0.025
= Pr[x
I
~
for
Ct <
X
such that
I
10, 12, 0.80]
N
K, X and
such that
N
Here, for given
K, N , p] . (For example,
2
corresponding to p = 0.80, K = 10, X = 9
2
$'
Dash marks
12,.)
for given
2
in Table II.
2
is the largest value of
N
Z
Ct,
Ct
N
2
P [x
r
is the value given in the tables, with
Ct
Pr[x
~
12,
and that
(-)
in table II indicate the non-
9Il0,N.0.80]
Here again, as an exception
Ct.
~
= 0.021,
X
2
I
K, N, p]
given as
In both lower and upper tail tables the entry
< Ct + 0.0005
Ct
this
+
> 60 is used when the
exact value is beyond the range of calculations we have carried out.
3.
Approximations
Harkness (1969) has suggested that a binomial with parameters
1 - (l_p/K)N
K,
would give a good approximation to the distribution (2).
This approximation gives the correct mean (see (5.1)) and is exact for
K = 1.
It would be expected to give good results for
the mixing distribution of
by a Poisson (with parameter
J
p
small, since
(see Section 2) may then be approximated
Np), and as Harkness shows a mixture of
classical occupancy distributions with such a Poisson mixing distribution is, in fact a Binomial distribution.
5
If the binomial approximation is good, then Molenaar's (1970) normal
approximation to the binomial may be used to get approximate expressions for
N
l
and N2 •
Writing
(7)
=
weN)
1 - (l_p/K)N
we have from Molenaar's 'quick work' approximation (1970, p. 110) for
a.. ::; 0.05
J
(8.1)
(3.2)
where
•
(I;f.;)-
1 fOO
2
V exp( - ~ ) du
=
a..
a.
T.~e
equations for
a.'. > 0.05
J
are of similar form.
In order to solve equations of form
=
AIl-w(N) - B/w(N)
we write
weN)
=
. 2
SIn
e
V
= sin
rp
and get
where
(9)
weN)
T
(when
A2+B 2
about
4K while hhe largest
<
V2 this equation is not applicable, but since
little restriction.)
2
V
we need is
5.4,
2
A2+B
is
this imposes
6
Once
w(N.)
J
has been found we can calculate the corresponding
N.
J
from (7).
Some trial calculations with
K = 20
indicated that when
p
~
0.75
this approximation gives values of
N which are too big by some 2-4
l
which are too small by some 1-2 units, at any rate for
units and of N
2
a = 0.025 - 0.05.
Further asymptotic formulae will be found in Samuel-Cahn, (1974).
7
REFERENCES
1.
F.N. David, (1950). Two combinatorial tests of whether a sample has
come from a given population, Biometrika, 37, 97-110.
2.
W. Feller, (1968). An Introduction to Probability Theory and its
Applications s New York: John Wiley &Sons, Inc.
3.
W.L. Harkness, (1969).
4.
W. Molenaar, (1970). Approximations to the Poisson, Binomial and Hypergeometric Distribution Functions, Mathematical Centre Tracts No. 31,
Math. Centrum, Amsterdam.
5.
N.L. Nicholson, (1961). Occupancy distribution critical points, Biometrika, 48, 175-180.
6.
Ester Samue1-Cahn, (1974). Asymptotic distributions for occupancy and
waiting time problems with positive probability of falling through
the cells, Annals of Probability, 2, 515-521.
The cZassical occupancy problem revisited,
Pennsylvania State University, Dept. Statistics, Technical Report
No. 11.
co
Lower tail critical points
..
-_._-_._-_.,~_.------
p
= .75
--._ '..
.....
----------_._._-~------------~---._
p
----
Lower tail probability level a.
_-.
----~-
= .75
--'--~_._".'-~~
t~vei C~
Lmrer tail probability
,'---"---.-~--
K
0.025
0.01
X
N
1
2
3
4
1
12
N1
0.
1
10
.007
__
0.05
~_.
0.
1
.018
N
1
8
0.
1
.047
..
~-~,",----
. -_...
0.10
-._~._._
N
1
7
..
~
.. _.
-
N
1
1
.074
9
20
.006
.010
-
17
.023
.023
6
15
.046
.040
5
12
.092
.094
1
2
3
8
14
29
.005
.008
.010-
7
12
25
.012
.021
.022
6
11
22
.027
.033
.041
5
9
18
.061
.083
.094
8
6
7
1
2
3
4
7
12
20
38
.008
.007
.008+
.010
6
10
17
33
.020
.024
.022
.023
5
9
15
29
.047
.042
.045
.045
5
8
13
24
.047
.074
.089
.099
1
2
3
4
5
7
11
16
26
48
.006
.007+
.010
.008
.010
6
10
14
22
41
.015
.013+
. 02 5+
.025+
.025
5
8
13
20
36
.040
.049
.039
.044
.049
4
7
11
17
31
.099
.093
.092_
.100
.094
1
2
3
7
10
15
.005
.009
.007
.009
.009_
.010
6
9
13
19
28
50
.013
.018
.020
.020
.023
.024
5
8
12
17
25
44
.035
.036
.034
.041
.046
.047
4
7
10
15
22
38
.090
.073
.093
.082
.091
.092
4
5
6
21
32
58
1
2
'1
4
5
6
7
}
5
0.01
X
0.
1
2
7
K
10
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
7
10
14
19
26
38
>60
6
10
13
,..,
..J..,
23
31
45
>60
0.025
0.
1
.004
.006
.007
.008
.009+
.010
-
+
.010
.005
.008+
.010
.009
.010
.009
.-
-
6 .010
9 .009
13 .006+
16 .010
21 .010
28 .068
27 .008
52 .009
-
N
1
..
a
1
_._.~-+---~.
6
9
12
17
23
34
59
.011
.013
.022
.019
.024
.023
.024
6
8
12
16
21
28
40
>60
.010
.024
.015
.017
.020
.022
.023
6
8
11
15
19
25
33
46
.010
.020
.021
.018
.022
.021
.022
.024
.~
0.10,
0.05
lIT
1
__.'--_.,--.. .~
5
fJ,J.
._._~,
..
.(32
.029
8
11
15
21
30
52
.038
.045
.044
.050'.047
5
8
11
14
19
25
36
60
.029
.024
"028
. Ol~3
.040
0049
"01;6
.048
5
7
10
14
17
.027
.045
.041
030i.050
.039
.044
. 041~
23
30
42
o
N
1
0.
1
4
7
10
14
19
27
45
.084
.060
.067
.068
.080
.089
.093
4
23
32
52
.080
.051
.C92
.069
.081
.082
.089
.095
4
6
9
12
16
20
27
37
.07h
.100.076
.081
.074
.097
.086
.092
...,
I
9
1~
1'7
.
0'\
Lo~ tail
_...--.
critical E.oints
'--_..........--.-.._-----_._-_ _-------_.
..
p = .75
p = .75
Lower tail probability level
K
0.01
X
Lower tail
C't
0.10
K
0.01
X
---~--
C't
1
N
1
C't
1
N
1
C't
1
H
1
C't
1
-_ _._--_.
0.025
-----------"---
.
N
1
pro~ability
-------_.,
0.05
0.025
N
1
<Xl
11
1
9
12
15
18
23
28
34
42
54
>60
>60
.006
.005
.007
.010
.007
.007
.008
.009
.009
C't
1
....
10
9
>60
-
>60
-
>60
-
59
.097
11
1
2
..,
4
5
6
7
8
9
10
6
9
12
16
20
25
32
42
59
>60
.009
.007
.008
.007
.008
.009
.01.01.009
6
8
11
14
18
23
29
38
52
>60
.009
.018
.017
.021
.021
.020
.023
.022
.025 -
5
7
10
13
17
21
27
35
47
>60
.026
.041
.034
.037
.033
.042
.040
.042
.049
4
6
.q
12
15
19
24
31
42
>60
.074
.091
.065
.064
.078
.082
.089
.092
.096
1
2
3
4
5
6
7
8
9
10
11
6
9
12
15
19
24
30
37
48
>60
>60
.008
.006
.007
.009
.008
.007
.007
.009
.009
5
8
11
14
17
21
27
34
43
59
>60
5
7
10
13
16
20
25
31
39
53
>60
.025
.037
.028
.029
.037
.038
.037
.042
.049
.049
4
6
9
11
14
18
22
28
35
48
>60
.071
.085
.057
.089
.092
.080
.093
.085
.100.089
.:>
12
13
1
6
-
-
.008
5
-
+
.025
.016
.014
.016
.023+
.025
.020
.020
.023
.023
-
.024
5
-
~.
.024
._.,-...- --
4
_.
.070
13
2
3
4
5
6
7
8
9
In
11
12
14
1
2
3
4
5
6
7
8
9
10
11
12
13
6
a-'
12
15
18
22
26
31
38
47
59
>60
>60
~
.~
.008
.005
.005
.005
.007
.007
.009+
.010
.009
.009 ..010
-
8
10
13
17
2l
25
31
38
48
>60
>60
5
8
10
13
16
20
24
29
35
43
54
>60
>60
.014
.025.023
.016
.016
.022
.020
.023
.025~
-
.-
.023
.013
.021
.019
.021
.018
.019
.020
.021
.021
.022
--
-
..
level
,_....-
C't
0.10
_-_ 0.05
.
iII
C't
C't
1
N"1
7
10
12
15
19
23
28
35
44
59
>60
.035
.025
.043
.046
.039
.044
.047
.044
.048
.049
6
8
11
14
17
21
26
32
40
53
>60
.080
.100'.076
.074
.086
.085
.081
.085
.092
.095
5
7
9
12
15
18
22
26
32
39
49
>60
>60
.023
.032
.046
.036
.036
.043
.042+
.050
.045
.048
.048
4
6
8
11
13
17
20
24
29
36
45
59
>60
.068
.076
.092
.066
.099
.067
.085
.091
.094
.086
.088
.092
---_.
~
-
-
1
0
.....
Lower tail critical J2..oints
._----
----"
p
_ . _ .... _ _• __
=
0.025
0.01
X
K
.L
0.1
11
'I
....... _ . _ _ _ _ _ _ _ _ _ _ _ • _ _ _ _ _ _ _ .... __
..
.75
p
~
0.05
N
1
1).1
5
7
.023
.030
.042
.031
.030
.033
.044
.046
.044
.045
.043
.049
= .75
Lower tail probability level
---------_.-_.
1).1
. _ . _ •• _ _ _
------._~ -~--
Lower tail probability level
K
R~ _ _ .. _ _ . - . . _
1IJ
'-1
0.10
K
0.025
0.01
X
N
1
1).1
12
13
14
15
57
>60
>60
>60
.009
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
r;
9
11
14
17
20
23
27
37
44
52
>60
>60
>60
>60
.007
.004
.007
.005
.006
.007 _.
.010
.010
.008
.010.008
.009
1
2
3
4
6
8
11
14
1).1
N
1
al
52
>60
>60
>60
.024
5
8
10
12
15
18
21
25
29
34
40
48
57
>60
>60
>60
.022
.011
.016
.024
.021
.021
.025 .022
.024
,024
.024
.021
.023
5
8
10
12
0
0.10
0.05
N
1
0.
1
N
1
a 1
- - - -.•_ _ _ _ _.,_._ _ _ _ _ _ _ _ _. _ _._ _ _ _ -....... _ _ _ _ _...._ _ _ .......__._ _ _ _ _ _. ___ ..._ _ _-_~_ _ _ _ _ _ _ _"'-- _ _ _ _ 4-_., _ _ _
15
16
1
2
3
4
5
6
7
8
9
10
11
12
13
14
6
11
14
17
21
25
30
36
43
52
>60
>60
>60
1
2
3
4
5
6
7
8
9
10
11
6
9
11
14
17
20
24
29
34
40
47
0
./
.007
.005
.009
.008
.009
.008
.008
.008
.007
.008
.009
5
8
10
13
16
19
23
27
32
-
39
47
59
>60
>60
.007
.004
.008
.007
.007
.009
.009
.007
.007
.008
.009
5
8
10
13
15
19
22
26
31
36
43
-
-
.023
.012
.020
.016
.017
.021
.019
.023 _.
.025
.022
.024
.024
.022
.Oll
.018
.014
.025 .016
.021
.022
.020
.024
.023
q
12
15
18
2l
25
30
36
44
51~
>60
>60
5
7
0
,
12
14
17
20
24
29
34
40
-
4
6
8
11
13
16
19
23
27
33
40
49
>60
>60
.067
.073
.086
.059
.085
.083
.093
.087
.099
.088
.089
.098
_.
.022
.029
.038
.028
.043
.043
.049
.045
.037
.041
.045
4
6
8
10
13
16
19
22
26
31
37
.066
.070
.080
.098
.074
.069
.074
.090
.091
.086
.084
16
17
-
18
....
')
.)~
=•
~--"
-
-
.,
.
~
.007+
.010
.007
.005
44
54
>60
>60
.092
.096
4
6
8
10
13
15
.065
.068
.076
.090
.066
.093
.090
.098
.089
.096
.097
.099
.097
.095
48
60
>60
>60
.047
.044
.022
.028
.036
.048
.037
.035
.039
.048
.048
.043
.049
.046
.045
.•.
-
5
7
9
11
14
17
20
23
27
32
37
44
53
>60
>60
>60
-
21
25
29
34
40
48
59
>60
>60
.021
.010
.015
.022
5
7
9
11
.021
.027
.033
.044
4
6
8
10
..
~~
"'
.
-
.-....
.-
.,
18
.064
.066
.072
.084
.....
.....
_-
tail ---critical - ,__.
- - . _.
... __.____.
' ___'" __._ . . __._ __.,_ -.0----_.--..--.---.-.. .-.... ____ _____ ___.. __'"-Lower
. _____ ._ . . _points
._._..____
~~
P
._~
~_._
~_~.
~
= .75
,_._------
Lower tail probability level a
K
0.01
X
K
-~_.,----
a
N
1
1
a
1
N1
a
1
N
1
--_....
0.01
X
-~-_._----
HI
a
-. ....
a
~J
"'1
1
.006
.009
.006
.010.008
.008
.009
.008
.008
.009
.010
.010
5
8
10
12
15
17
20
24
27
32
36
42
.021
.009
.014
.020
.016+
. 02 5+
.025
.019
.025 .019
.024
.021
12
13
14
15
16
17
.009
.010.007
.007
.008
.008
.010 .009
.008
1
2
3
4
5
6
7
8
9
10
II
12
6
8
11
13
16
19
22
26
30
34
39
45
11
19
.018
.017
.019
. 024
.023
.020
.022
.024
.023
-
15
18
21
24
28
33
38
44
52
>60
>60
>60
>60
16
19
23
27
31
36
41
48
57
>60
>60
>60
>60
5
6
7
8
9
10
-
..
-
-
~
14
16
19
23
26
31
35
41
48
57
>60
>60
>60
.033
.050.049
.037
.048
.038
.049
.047
.048
.050-
5
7
9
11
14
16
19
22
26
29
34
39
.059
.081
.075
.079
.093
.091
.081
.089
.098
.088
-
13
15
18
21
24
28
33
38
44
53
>60
>60
>60
.021
.026
.032
.040
.029
.043
.041
.043
.036
.050 .042
.044
4
6
8
10
12
15
17
20
2'.j.
27
31
36
.063
.064
.069
.078
.094
.072
.099
.096
.074
.091
.094
.089
-
..
.~
-
19
~
..
...
~
_----~.---
1
20
+
52
>60
>60
>(.')
>60
>60
.010
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
8
11
13
16
19
22
25
29
33
38
43
49
57
>60
>60
>60
>60
>60
.007
.009
.006
.009
.007
.007
.007
.009
.008
.009
.008
.009
.010 .009
.)
~
-
~"-'
~"
'"'~
.....
,
--_..._--,..
0.05
----_.__ ..._._,. __ _--------
N
1
a
48
.024 ...
.025
56
>60
>60
>60
>60
5
7
10
12
15
17
20
23
27
31
35
40
46
52
>60
>60
>60
>60
>60
0.10
..
HI
1
a
- __ .... _ - - - - _ . _ - - - - _ · ....._ . 0 .
13
14
15
16
17
18
.
_-~_.;------
0.025
_......
-----,-----~---_._--_._---
18
= .75
------------~~-,--_._-------,-,--_._.-
LmTer tail probability level a
~-'--
0.10
0.05
0.025
P
----~--
'"
_.
->
..
.~
.021
.025
.013
.018
.014
.022
.021
.023
.018
.018
.021
.021
.020+
.025
"'~
~
-.
"--
_.
1
45
52
>60
>60
>60
>60
.046+
.050
5
7
9
11
.021
.025
.030
.037
.048
.038
.034
.035
.041
.037
.039
.047
.041
.046
.045
13
16
19
22
25
29
33
37
43
49
57
>60
>60
>60
>60
..
~.
~
..•
-u
..
N
1
a
42
48
57
>60
>60
>60
.083
.098
.094
4
6
8
10
12
15
17
20
23
26
30
34
39
45
52
>60
>60
>60
>60
.062
.062
.066
.074
.087
.064
.087
.081
.084
.097
.091
.099
.098
.096
.099
1
N
~
Upper tail critical points.
p
-
= .75
0.01
X
:JII
3
4
5
6
7
8
9
3
4
0.025
2
1),2
-
-
..
...-.
N
2
-
1),2
•.
N
2
~
4
-
-
.-
--
5
5
.009
5
.009
6
U
5
6
'.
-
-
..
.022
.012
_.
5
6
7
5
6
7
8
5
6
7
8
9
6
-
6
8
--'"
.003
-.008
.004
'-
-
7
11
.003+
.010
-
.-,
7
9
13
.005
.005
.008
5
7
-
6
10
6
8
12
-
6
8
10
15
0.10
-
5
8
a
2
.030
J'T
T(
h
2
1),2
3
.094
5
4
7
.061
.077
-
4
6
10
.088
.077
.099
.022
.031
5
7
11
.036
.031
.045
5
8
12
.036
.075
.073
.014
.012
.020
5
6
9
13
.049
.014
.033
•034
5
7
10
15
.049
.053
. 067
.077
-
_.
5
7
9
13
18
.061
.076
,059
.092
.077
~
.008+
.025
....
.020
.023
.014
.025
6
8
11
16
10
6
7
8
9
10
11
5
6
7
8
9
10
11
12
.020
.023
.030
.040
5
6
7
8
13
2
a.
0.025
2
•.
5
~--,
7
0
11
15
<'-"
,-,,,
-
8
10
13
18
.,"
-
.008
.009
.005
.007
..
""-'
.003
.004
.005
.009
-
]\1
'2
.,....
".
.>
'T
9
12
17
-_.
7
9
11
15
20
..
..
.
8
-
7
9
11
14
17
23
_.
10
11
12
10
12
16
21
5
..
-
1),2
,.""
_.
.005
.006
.005+
.010+
.010
q
~
0.01
X
1'1
.093
-
.034
= .75
Upper tail probability level
I),
0.05
4
-
p
-
Upper tail probability level
K
-
-
.-_.
.008
.009
.013
.019
.011
. 015
.012
.022
.021
-
0.05
N
2
-
6
8
10
14
19
-
0.10
1),2
N
2
a
.-.
5
7
9
11
15
22
.072
.098
.087
.057
.073
.095
5
8
11
14
18
25
.082
.033
.048
.087
.088
.084
.087
5
6
8
10
13
16
21
29
.091
.040
.060
.059
.083
.074
.090
.097
.027
.035
.027
.046
.041
h.
6
8
10
12
16
22
'--
.033
.048
.042
.027
.037
.041
~"
.022
6
7
9
12
15
19
25
.040
.015
.022
.044
.046
.045
.040
-.
•.
..
~
.
.015
.022
.020+
.025
.c18
I),
6
c
./
2
.099
t-')
.-l
Y~er
tail critical
point~.
p = .75
Upper tail probability level
K
13
14
6
7
8
9
10
11
12
13
6
7
8
9
10
11
12
13
14
15
6
7
8
9
10
11
12
N
2
0.
.'
.•.
-
N
.2
2
-
8
10
12
14
18
23
.006
.010.009
.005
.009
.007
_.
-
..
8
9
11
14
16
20
26
-
-
8
9
11
13
16
~
~.
.008
.003
.004
.010.005
.007
.008
~
7
8
10
13
16
20
26
-
.010
.004
.006
.006
.010+
7
8
10
12
15
18
22
29
..
7
8
10
12
14
17
C't.
0.10
-_._-
0.05
0.025
0.01
X
2
N
2
C't.
2
"2
...
.018
.006
.010
.021
.023
.023
.022
6
7
9
11
14
17
21
29
.046
,018
.029
.029
.043
.039
.035
.048
6
8
10
12
15
19
24
32
.046
.073
.077
.064
.075
.091
.092
.087
6
.051
.085
. 094
.084
.064
.066
.074
.092
.092
0.
.
-
.022
.008
.014
.014
.021
.020
.018
.021
.025
.010
. 018
.019
.016
.020
+
p
--,-'
._--
..
7
9
11
13
16
19
24
32
.022
.037
.040
.033
.040
.033
.039
.044
...
..
~
7
9
10
13
15
18
.025
.045
.018
.045
.033
.036
liT
8
10
12
14
17
21
27
36
C't.
16
8
9
11
14
17
20
.056
.096
.oll5
.051
.086
.098
.087
N
2
C't.
13
14
19
23
Pi
29
.010.009
.008
20
2'"')
32
6
7
8
9
10
11
12
13
--
..
_.
11~
15
16
17
0.025
N
2
2
15
0.01
X
6
7
..
9
10
11
12
13
14
15
2
......
-
9
11
13
15
18
21
25
32
.
.-
-
.:'
9
10
12
14
17
20
23
.006
.002
.004
.004
.008
.009
.007
IJ'2
.017
.021
.020
22
27
36
<"'"
8
10
11
14
16
19
23
28
36
.012
.022
.008
.023
.017
.018
.022
.023
.024
_.
..
8
9
11
13
16
18
21
25
-
'~
.,
2
C't.
C't.
0.10
0.05
.-",-
.005
.008
.009
.007
.010
.008
.007
.008
(3
f)
Upper tail probability level
---K
= .75
..
.014
.006
.011
.013 ..
.025
.017
.016
.018
0.
2
N
2
0.
2
.042
.041
.049
24
30
~o
.083
.090
.095
-
-
7
8
10
12
15
17
2l
.029
.012
.022
.026
.047
.032 _.
.050
25
30
39
. 01.~2
.044
6
7
9
11
13
16
19
22
27
33
44
.061
.029
.052
.062
.059
.083
.088
.073
.089
.086
.096
6
7
.066
.032
.060
.073
.073
.062
.079
.077
.086
.092
0..'.
7
8
10
12
14
17
20
23
27
• 01~8
_.
.032
.014
.027
.033
.031
.0 47+
.050
.042
.039
C\
,/
11
13
15
18
21
25
30
'd"
..-4
Y~~~il
p
critical points
--------
= .75
-
Upper tail probability level a
K
N
2
17
18
16
17
28
35
6
-
7
8
19
0.025
0.01
X
~
2
N2
.009
.008
31
39
lX
~
_.
".-'0
--"'"
_.
8
9
10
11
12
13
14
15
16
17
18
9
10
12
14
16
19
22
26
31
39
.007
.003
.005
.006
.005
.008
.008 _.
.010
.010.010+
9
11
13
15
18
20
24
28
34
43
6
7
8
9
10
11
-
-
-
..
-
9
10
12
-
.008
.003
.006
~
8
9
11
13
lX
.024
.022
..""
.016
.007
.013
.017
.016+
.025
.016
.023
.022
.025 -.025 -
,.018
.008
.016
.021
N
2
33
43
~
lX
2
.042
.047
-
l\T
2
0.
6
.070
.035
.067
.084
.087
.079
.063
.072
.096
.095
.093
.093
.096
7
9
11
13
15
17
20
24
28
33
40
52
.-
-
6
7
0
J
11
12
15
X
2
.099
.097
.035
.016
.032
.040
.040
. 03!~
.01f5
.044
.035
.042
.041
.049
.038
,018
.036
.047
.050-
K
37
48
7
8
10
12
14
16
19
22
25
30
36
47
7
8
10
12
14
..
.074
.038
.074
.095
.047
.095
--
0.01
T,T
"2
19
20
0.
.021
.018
.024
.025 .019 ."'.
.025 ..•
.025
.022
_.
..
..
8
9
11
12
14
17
19
22
25
29
33
40
50
.020
.009
,019
.008
,OIl
.024
.019
.023
.022
. 021~
.019
.024
,024
28
.008+
18
34
42
.010..
.010
.....
~
'.~
0
/
10
12
13
15
18
20
23
27
31
36
45
.009
.004
.008
.003
.004
.009
.007
.007+
.010
.009
.008
,009
2
]IT
-'2
2
N2
.01f5
.035
.042
.039
.042
.045
.049
.044
17
20
23
26
30
36
43
56
0.
-
r
{
~
~
a
15
17
20
23
26
31
3
46
;;;:
0.10
0.05
N,.,
.008
.008
.006
.008
.007
~
= .75
2
14
16
It)
21
24
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
p
._---_._._----_._-_._---,.~.
0.025
12
13
14
15
16
17
19
__
Upper tail probability level a
---0.10
0.05
2
-_ _----,-_.
16
18
2l
24
28
33
40
50
-
7
8
10
11
13
15
18
20
23
27
31
36
43
54
.....
. 04·1
.020
.041
.019
.026
.027
.047
.035
.038
.049
.048
.046
.047
.045
0.
2
.e80
.097
.096
.082
.078
.093
.087
.095
6 .078
7
9
10
12
14
17
lq
22
25
29
33
39
47
60
.041
.080
.041
.055
.059
.099
.079
.089
.085
.09 1f
.083
.091
.094
.093
l/)
...-t
Lower tail
--------- ..
_. _ _ _ _
. _ ~ . _ . _ . _
._ _
~"_
_~
___
~
... ""-"'._ _
~_~4··
_
~ritica!. p?ints
•. _ _ _ .• _. _• •_ ••_
•. ,_
~
~.
__ ,_
~
_ _ _ •.
~
..... _
....
_• •. - . • • _ _ _
._~.
_ _...... _
•. - . . , _ ._ _. _ .. _ _ _ _ ..... _ _ _ _ _ . - .•
_ _ _ _ _ _
-_.------~ ~
.. _ _ _ _ _ _ •
--~~
.
~
__
_ _ _ _ _ _ ...... _ _
A
6
2
1
11
u
1
.007
HI
9
u
1
.020
N
1
u1
N
1
._._.___. _.__._____
_~N.·_
:(
_ _
8
.034
u1
.~.
6
_.w_... ___.. __ ........
.093
3
1
2
8
19
.007
.008
7
16
.014
.021
6
14
.031
.039
5
11
.066
.098
4
1
2
3
7
13
27
.007
.008
.010 -
6
11
23
.016
.021
.024
5
10
20
.040
.035
.046
4
8
17
.098
.096
. 089
1
2
3
4
7
11
18
36
.004
.007 _.
.010
.009
6
10
16
31
.011
.014
.020
.022
t::
1
2
3
4
5
6
10
15
24
45
.008
.007
.009
.008
.010 -
5
9
13
21
39
1
2
3
4
5
6
6
9
14
20
30
54
.006
.009
.006
.007
.008
.010-
5
8
12
17
26
47
5
6
7
8""
14
27
.029
.049
.043
.045
4
7
12
23
.078
.091
.090
.089
.023
.015
.024
.021
.023
5
8
12
19
34
.023
.030
.039
.039
.046
4
7
10
16
29
.066
.061
.099
.094
.093
.020
.021
.020+
.025
.023
.023
5
7
11
16
23
41
.020
.045
.035
.037
.049
.048
4
6
10
14
21
35
.059
.097
.061
.079
.080
.098
_~.
__ .
_-_._~._.--.-------
___ .__.. _.. . _.Lower . _tail
·__.. . __.'.probability
. . . .___.. . . . __._ level u
__.. . -0.01
. __ ____,___.
X
~
N
1
. _ _ _ _ _ _ _ _ . _ _ . .... _ _ .• _ . _
p = .80
-_
• _ _ _ _ _ _ _ _ _ _
~+
--------_._..
Lower tail probability level u
.. ,.
._.. -- -- ._-0.10
0.025
0.05
0.01
X
.•_ " - , , , _ _ _ ._ .•
p = .80
-~---
K
~
___
8
9
10
0.05
0.10
. _ . _-..,_.______,
______ ____
0.025
_~_~_._._.4
N
N
~._.
_.~.
-.-~
u
TT
1 ..___"1
. _.. -... . . 1
. . . _.. . . '____'\
_ _ _ _1_.....___
.-.,.~_
1
2
3
4
5
6
7
,
~_
6
9
13
17
24
36
>60
2
3
4
5
6
7
8
6
9
12
16
21
29
42
>60
1
2
3
4
5
6
7
8
9
6
8
12
15
20
26
34
48
>60
.J..
~
.~,
.005
.006
.006+
.010
.010 .009
..
.005
.005
.007
.008,.,.
.010
.009
.009
-
.004
.010
-
.005
.009
.007
.008
.OOG
.010 u
~
N
1
ul
.017
.015
.022
.016
.019
.021
.024
5
7
10
14
20
28
48
.017
.036
.041
.042
.037
. 04 9+
.050
8
11
14
19
26
37
>60
.012
.015
.025 .023
.022
.024
4
7
10
13
18
24
34
56
.050
.029
.029
.041
.034
.040
.042
.048
5
7
10
.014
.025 ."
.022
.015
.018
.022
.020
.023
4
7
9
12
16
21
28
39
>60
.047
.025
.045
.049
.045
.044
.043
.045
5
8
11
16
22
32
55
1~
18
23
31
43
>60
.~
-,
-
..
'\
4
6
9
13
17
25
42
.054
.081
.075
.066
.097
.091
.093
4
6
9
12
16
21
30
48
.050
.070
.057
.069
.074
.091
.087+
.100
4
6
8
11
15
19
25
34
55
.047
.062
.090
.084
.069
.085
.089
.100.099
\0
...-i
&ovrer_
_ _ _ _ _ _
• _ _ _ _ • _ _ u_~ _ _ _
p
• _ _ _ _'~ _ _ _ _ ...... _ _ • •
A_._.. _.
= .80
0.01
X
N
1
a
1
N
1
~
rT
1
12
13
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
11
1
2
3
6
8
11
14
18
23
30
39
55
>60
6
8
11
14
18
22
27
34
44
>60
>60
6
8
11
.004
.008
.008
.010:
.010+
5
7
10
13
17
.010
2l
.009_
.010
.009
-
.004
.007
.006
.007
.006
.008
.010:
.010+
.010
-
.003
.006
.005
27
35
49
>60
.013
.022
.017
.019
.017
.023
.022
.024
.023
..
5 .013
7 .019
10 .014
13 .014
16 .019
20 .020
25 .020
31 .023
40 .024
55 .023
>60
5
7
10
.012
.018
.012
4
7
9
12
15
19
25
32
(44
>60
4
7
9
12
15
18
23
29
37
50
>60
4
6
9
__ . . __.__ . . . _._... ___._..________ .-. _ _._ _.__._.__.. ____ --......_..__________.. . . ._____._._,_....._-_
_~
l'
a
!III
1
.045
.022
.037
.036
.046
.049
.041
.048
.048
..
.043
,019
.031
.028
.032
.047
.040
.040
.043
.046
.042
.047
.027
-_
K
4
6
8
11
14
18
22
29
39
>60
4
6
8
10
13
17
2l
26
33
44
>60
4
6
8
0.01
X
a
1
"' _ _ _ _ _ . _ _ . • _ _ _ _ _ _ _
.045
.056
.077
.065
.074
.071
.097
.090
.099
-
:N
1
.~
......
~
_ _ _ _ • • ~ _ . _ ,_ _ _
13
14
.043
.051
.067
.096
.089
.070
.078
.087
.093
.099
a.
.......... , . . . _ . _ _
~_
1
IJ
'I
a
1
0.05
0.10
N
a
11
14
17
21
26
33
41
55
>60
.043
.041.-.
.050
.049
.047
.040
.049
.05
4
6
9
11
14
17
20
25
30
37
46
>60
>60
.040
.044
.023
.036
.032
.036
.048
.038
.042
.041
4
6
.039
.042
1
1
11
-1
a
10
13
16
19
24
30
37
50
>60
.081
.071
.077+
•let>
.085
.081
1
. - - . .. . . . . . _ _ _ _ . _ . . . . , ~ . _ .
14
17
21
25
31
39
50
>60
>60
.005
.007
.007+
.010
.010.009
.009
1
2
3
4
5
6
7
8
10
11
12
13
6
8
10
13
16
20
24
29
35
44
55
>60
>60
.003
.006
.010.009
.010
.008
.009
.009
.010
.008
.010
1
2
6
8
;;
15
0.025
4
5
6
7
8
9
10
11
12
f"I
-
.042
.047
.059
= .80
Lower tail probability level a
0.10
1; _ _ -
11
_.~.~
----
0.05
0.025
_....
--
Lower tail probability level a
K
critical points
~ail_
..___~.
~"
-
~
.003
.005
12
15
19
23
28
35
45
>60
>60
5
7
9
12
15
18
22
27
32
1~0
50
>60
>60
5
7
.022
.024
.020
.023
.025+
.025"'
.023
..
.-
.012
.016
.023
.018
.018
.022
.021
.019
.024
.021
.024
~
....
,-
.011
.015
~
.01~6
.~
-
.097
.090
4
6
8
10
12
15
19
23
27
33
42
55
>60
.040
.044
.054
.071
.099
.093
.071
.073
.094
.096
.088
.093
4
6
.039
.042
r--.
.-1
J:,owe~
_ _ ~ _ _ _ _ _ _ . _ _ _ _ _ _ _ • _ _ _ . . - _ _ _ _ _ _ _ ...
~
_ _ _ _ - - - - . . . . _ _ .•. ,
, _ _ ._ _ _ ~ ~ - ..
~
.._, _ _
••., ....• _ . _
•
p = .80
..
tail
___
K
X
~-_
..
_._--~_._-
...
W
l
_
.
_
.
~
_
.
15
16
N
l
~1
__._.---_.
3
4
5
6
7
8
9
10
11
12
13
14
10
13
16
19
23
28
33
40
48
>60
>60
>60
.009
.007
.007
.009
.008
.007
.008
.008
.010
1
2
3
4
6
8
10
13
16
19
22
26
31
37
44
53
>60
.003
.005
.008
.006
.005
.007
.009
.010.009
.008
.000
.010
5
6
7
8
9
10
11
12
13
Ct
1
~
...,,~
-
--
...
..... --
_--_._~-_.
N
1
~1
1'1
_ ...........
_~
...... _
........ _ _ _ _ • . . - ___
....
_ _ _ _. _ _ _ _ _ _ ~ _
.~~~_._._
_ . . . _ _ _ _ _ . ._ _ _ _ _ _ _ _~ _ _ e _ _ •
----~_.
__
.
p = .80
__ ....
~_,_~
K
-
4
6
8
10
12
15
17
21
24
29
34
41
51
.039
.040
.045
.055
.073
. 061
.095
.072
.095
.083
.093
.093
.089
16
17
1
.. ~.-...
....
14
15
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
.-
18
1
2
3
4
5
6
____
......__ " . _ • ......- '''h ._._ ...... _ _ _ _ _ _ _ _
0.01
X
1\1
__._. ----- ...- _._-_
.039
. 040
.045
.027
. 040
.036
.039
. 04 9+
.050
.048
.048
.045
.045
_ _ ~_._ _ _ _.....
- - _ ...._-,_._._--------
~
'1
_.-
4
6
8
11
13
16
19
22
26
31
37
45
56
:e,.oints_
~.-.-_.
Lower tail probability level
.049
.062
. 084
.074
.079
.098
.077
.082
.094
.094
.097
.011
.014
.019
.012
.021
.021
.024
.022
.018
.020
.024
.021
.025+
5
7
9
12
14
17
20
24
29
34
40
49
60
__
~--
8
10
12
15
18
21
26
31
37
46
60
>60
-
-
,._.
-._---
....
-
.049
.031
.047
.045
.034
.049
.042
.039
.043
.045
.021
.015
.014
.016
.022
.023
.023
.023
.023
.024
~-_
---,,------- ..
8
11
13
16
20
23
28
34
41
51
>60
>60
9
12
15
18
21
25
30
36
44
55
>60
>60
--
0.10
0.05
0.025
0.01
------_.- - - - - - ----------- _------ -
~ri.!ical
._._ .•.•.••
_------
Lower tail probability level a
------------------ .-...
~._
0.025
~1
N
l
----------~---~----'--
>60
>60
>60
>60
~
-
6 .003
8 .004
.10 .007
13 .005
15 .009
18 .009
22 .006
25 .010
30 .007
35 .008
41 .008_.
48 .010
58 .009
_.
>60
>60
>60
"
"0'
5
8
10
12
15
18
.010
.004
.006
.009
.007
.007
+
5
7
q
-
11
14
17
20
23
27
32
37
44
53
>60
>60
>60
5
7
9
11
14
16
.~,
~
..- . _..• _.._ ...._ _ _ _. _ ._ _ _ ......
0.05
II
I
__._--_
~l
N
>60
>60
.-
>60
>60
.038
.038
.042
.050
.034+
.050
.048
.036
.048
.039
.042
.046
.048
.-
4
6
8
10
13
15
18
22
25
30
35
41
49
>60
>60
>60
.-
17
20
23
27
32
38
45
55
>60
>60
.010
.012
.016
.021
.015
.024
4
6
8
10
13
15
.037
.036
.039
.045
.029
.043
4
5
7
9
12
14
~1
~
.
.011
.013
.017
.023
.017
.016
.018
.023
.023
.021
.025
.024
.024
-
~
"".,.
_ _._ _
0.10
-----
------_.
_.~.
.-
-
..
~
.
~1
1
..
4
6
7
10
12
ll~
.038
.038
.097
.050
.064
.086
.077
.080
.096
.095
.089
.086
.093
.096
.037+
.JOO
.092
.094
.057
.074
, --./
....
00
Lm~e!__t!lg__c}'_i~i~~
---,--_.
~_ . . _ _ _ ~_._ _ _ _ _ _ _ _
.__ . . . .
.~~ ___ ~L _ _ _ _ ._• •
_
p = .80
0.025
0.01
X
0.10
..
N
l
N
1
(11
--_._-----_.18
19
7
8
9
10
11
12
13
14
15
16
17
1
2
3
4
5
6
7
8
9
10
11
12
13
21
24
28
33
38
45
53
>60
>60
>60
>60
5
8
10
12
15
17
20
24
27
32
36
42
49
N
l
(11
~-~~_
...
~
-
+
.010
.004
.006
.008
.006+
.010+
.010
.007+
.010
.008+
.010
.009
.009
19
23
26
30
35
41
48
58
>60
>60
>60
5
7
9
11
14
16
19
22
25
29
34
39
45
.023
.017
.023
.025
.024
.023
.025
.022
18
21
~
"c'>
~.
-
-
.010
.012
.015
.019
.013
.020
.018
.020 _.
.025
.024
.020
.021
.023
24
28
33
38
45
54
>60
>60
>60
.039
.041
.050
.049
.043
. 049
.01..!-6
.044
M
,-
-
4 .03h
6
8
10
13
15
18
21
24
27
31
36
42
.035
.037
.042
.026
.036
.031
.032
.037
.048+
.050
.048
.045
point!§..
_ _,
_ . _ _ _ _ .. _.~ _
_
• _
_ _ ~,~,_ _ _ ~
_ _ _ _ " . _ . . . . . . .•. _ _ .• _ _ _ _ _ _ _ _ •. _ , '_ _ _ _. _ . _ . - ; , __ _ . . . _ ' , _ _ •. _ _ _ _ _ _ _
p = .80
---------._......_----------_._--0.01
0.025
0.05
------ ----,,'_.- - ---_........_- --- __ ..
,~._._-
_ ...
17
19
23
26
30
35
41
49
60
>60
>60
4
5
7
9
11
14
16
19
22
25
29
34
39
_-
K
X
_.. -
~
.. ~
.•
.~_
_ -
.063
.095
.072
.092
.097
.097
.098
.095
.093
..
..
.036
.097
.087
.088
.096
.065
.087
.078
.081
.00 4
.090
.079
.086
~_.~-_._._~-~
6\.T
(11
-1
- - - _ . . ._ . _ . _ . _ - - - .. -'> .....
.008+
.010+
.010
.009
.010
.008
.008
--..
_--,~-
1'1
(11
... _ '
----_.,~
Lower tail probability level (1
0.05
- - - - - - - - - - _ . - - - _ . _ - - , , - _ . _ - --~
~
_
----
Lower tail probability level (1
K
,~ . • •
1'1
-~.
III
(11
"'1
(11
-.-._--. _ _ . - - - - ~ - - - , ~ .
_ _ . _ ~•. _ - - ~ - - , ~ -
19
20
14
15
16
17
18
57
>60
>60
>-:;0
>61
.010
1
2
3
4
5
6
7
8
9
10
11
12
13
1)-!15
16
17
18
19
')
.010
.004
.005
.007
.005
.009
.008
.009
.007
.007
.009
.009
.009
.009
8
10
12
15
17
20
23
27
31
35
40
46
53
>60
>60
>60
>60
>60
53
>60
>60
>60
>60
-,
_..
.,
.....
.~
'""
0.10
-.~-_.,_
, -_._---
..
(11
N
1
49
58
>60
>60
>60
.046
.047
1!-5
53
>60
>60
>60
4
6
8
10
12
15
17
20
23
26
30
35
40
46
53
>60
>60
>60
>60
.036
.034
.035
.038
.046
.031
.045
.041
.043+
.050
.047
.040
.041
.042
.047
.,
liT
-'1
_._---_._
.....
"-.,"
,-
-
.,.,_
._--_._-
~
5
7
9
11
13
16
19
21
25
28
32
37
42
49
57
>60
>60
>60
>60
.021
,-
-,
. ""
,.010
.011
.014
.017
.023
.017
.01 5+
.025
.018
.024
.024
.022 ..025
.022
.023
" ...
_.
~
.-
<-
.,.,-.'
..
~
",-,
4
5
7
9
11
14
16
19
21
25
28
32
37
42
49
57
>60
>60
>60
(11
.09~·
.097
.036
.095
.084
.083
.089
.057
• 0~r5
.065
.096
.072
.088
.090
.084
.095
.091 +
.100
0)
..-l
Upper tail critical points
p = .80
Upper tail probability level (l
Upper tail probability level (l
K
0.01
X
4
5
6
7
8
9
4
4
5
5
6
5
6
7
N
2
<X
-
_.
_.
-
-
..
~
..013
-
-
"
.004
7
.017
-
-
-
..
-
8
.006
-
-
5
6
-
-
-
7
9
12
.,.
_.
•008
.007
.006
..
..
-
6
9
..
.Oll
.017
-
6
8
11
.020
.018
.015
-
-
'.
7
10
14
--,
7
JlT
K
2
(l2
N2
(l2
4
.038
4
.038
CO-"
6
~,
.045
4
7
10
.079
.098
".-.
.008
.021
.023
-
.013
.030
.042
5
9
.030
.079
5 .049
5
7
12
.049
.044
.096
5
7
10
14
.067
.075
.092
.073
5
6
9
12
17
.084
.030
.083
.080
. 078
5
8
7
10
.044
.035
6
9
13
00'-
.020
.047
.048
..
..
6
8
11
15
.030
.034
.045
.037
-
6
7
..
.040
.011
5
6
8
.099
.040
.052
8
6
'7
I
12
"-
13
0.01
X
9
10
11
..
6
.004
.007
5
6
7
2
5
7
10
8
-9
~
0.10
0.05
-
5
6
7
8
'r
10
0.025
(l2
N
2
p = .80
-
~T2
(l2
N
2
8
11
15
.003
.009
.010+
9
12
16
.-~.
~
...
8
9
10
11
8
10
13
17
6
7
8
9
10
11
12
.'
6
7
8
9
10
11
12
0.025
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8
9
12
15
19
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.,
<"-'
9
11
14
17
..
.~
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.006
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7
9
11
14
19
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•.
2
1\1"2
(l2
N
2
(l2
.015
10
13
18
.040
• Ol~O
11
14
20
.083
.068
.079
6
8
10
13
17
24
.049
.070
.063 .
.077
.085
.097
6
8
10
12
15
20
27
.058
.089
.088
.068
.070
.096
.092
(,
.067
.029
.046
.096
.067
.093
.077
0.
.021
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.018
. 024
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.019
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6
7
9
12
15
21
-
-
.~
.008
.002
.008
.009
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7
8
10
13
16
22
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-
.'.
"'-
'I.",
'"'
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.005
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.
8
10
12
15
19
0.10
0.05
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.016
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.024
7
9
11
14
18
24
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7
9
11
13
16
20
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.031
.040
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.046
.035
.037
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7
9
12
14
18
22
o
N
Upper:. tail c_ritical poit:lts
p
-
= .80
p
Upper tail probability level a
Upper tail probability level a
K
0.01
X
N
2
13
14
15
13
6
7
8
9
10
11
12
13
14
6
7
8
9
10
11
12
13
14
15
16
6
7
22
2
N
2
(12
.008
25
.025
0.
..
-
-
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~
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-
9
11
13
16
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25
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9
10
12
15
18
22
28
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-
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8
10
12
14
17
21
28
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-,
-
~
-
. 006
.002
.003
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...
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8
9
11
13
16
19
24
31
..
.•
N
2
+
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-
.014
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-
-
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-
..
0.10
0.05
0.025
27
2
N
-2
.044
30
.-
-
7
8
10
12
15
18
23
30
.034
.014
.023
.023
.036
.033
.044
.043
..
7
8
10
12
14
17
2l
26
34
...
7
K
.0.
_.
.040
.017
.030
.033
.027
.034
.045
. 04 9+
.050
"'....
.O~5
= .80
6
7
9
11
14
16
20
25
34
0.
',r
L
2
.086
.075
.034
.058
.062
.098
.064
.079
.082
.094
6
7
9
11
13
16
19
23
28
37
•083
. 040
.070
.079
.072
.095
.091
.093
.084
.086
6
7
.090
.045
0.01
X
16
17
8
9
10
11
12
13
14
15
16
6
7
8
()
10
11
12
13
14
15
16
17
18
7
8
2
•. ,
9
0.025
0.
2
N
2
a"c:.
8
.008
Q
10
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1~
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11
13
15
18
22
26
34
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.025
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.,
.
8
9
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14
17
20
24
30
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9
10
12
14
16
19
22
26
33
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-
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11
13
15
17
20
24
29
37
-
0.10
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.017
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.022
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N
'2
0.
8
10
12
14
16
23
28
37
.020
.037
.043
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.032
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.047
..
7
8
10
11
13
16
18
22
26
31
40
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.024
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1(1
....
./
,,-.
..
..
8
2
1\T
u'2
0.
9
11
13
15
18
21
25
31
41
.081
.096
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.077
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.078
.085
.092
6
7
9
10
12
15
17
20
24
28
34
45
,097
.050
.092
.045
.055 ..
.100
.077
.082
.098
.086
.084
.096
----
+
7
.027
8
2
.055
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.....
N
Upper tail critical
p
POi~1tS
= .80
Upper tail probability level a
K
Upper tail probability level a
0.10
K
18
19
9
10
11
12
13
14
15
16
17
18
7
8
9
10
11
12
13
14
...."
10
12
13
15
18
2l
24
29
36
~
-
10
11
13
15
17
a
2
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-
~
.006
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N
2
a2
9
11
12
14
17
22
26
32
40
.012
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.012
.023
.016
.016
.018
.024
.022
..
-
.......
10
9
10
12
14
16
19
0-
.014
.006
.012
.016
.015
.025 -
N'
2
9
11
13
15
18
21
21+
28
34
44
-
8
9
11
13
15
17
20
a
11
2
a2
.012
.024
.030
.029
.044
.048
.041
.038
.042
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10
12
14
16
19
22
26
31
38
49
.053
.066
.067
.059
.075
.075
.081.j.
.091
.100
.098
-
7
8
10
12
14
16
18
21
.060
.030
.060
.078
.082
.075
.061
.071
2
.030
.014
.029
.037
.038
.033
.044
20
-
15
16
17
18
19
7
8
9
10
11
12
13
14
15
16
17
18
19
'?O
0.025
0.10
0.05
.
2
a2
N
2
a2
20
23
21
32
3°
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21
25
29
35
43
.016
.023
.022+
.025
.021
1\T
19
--_.-
-
0.01
X
----.-
N
2
= .80
._.--,,-..<-_._-.---,
0.05
0.025
0.01
X
p
._-------
-
-
~.
..
0.-.00
10
11
13
15
17
19
22
25
29
34
43
<.-
.007
.003
.006
.008
. 008
.006
.008
.008
.008
.008
.010+
_.
9
10
12
14
16
18
21
24
27
31
37
47
-~
.016
.007
.015
.020
.021
.018
. 02 5:
.025
.020
.018
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a
-2
a2
23
26
31
37
47
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.043
.042
.044
25
29
34
41
53
.096
.096
.095
.096
.099
-
7
8
10
12
14
16
18
21
24
27
31
37
44
57
.064
.033
.068
.090
.098
.093
.080
.098
.098
.084
.081
.097
.091 ..100
}IT
~
8
a
11
13
15
17
19
22
25
29
34
40
51
./
.033
.016
.034
.045
.048
.044
.035
.042
.040
.044
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.042
.047
lJ
2
2
N
N
..
__
...
._~-..--._--_._-------
tai..J.:. critic:tl_ points
_-----_._'--_._.._----~ow~
..... _---_.- .'-._..... --_._......... _---.-,~.-
~
...
- .... ..
-
~_._._----,
..
-..... _.. _--_.------_._ ..._.__.--_
.. ---_._---_.~..-._.-------
p = .90
P = .90
_.------~------_._-------
Lower tail probability level a
._-------
K
• •_
___
•• . -
_
._.... ' _ .•
_.~_.
0.05
0.025
._-----_._-'---_._,.
.
0.01
X
_ _ _ _ _• _ _ _ _
--"-~-~-_
.'-----
N
1
0.
1
N
1
0.
1
N
1
_ _, _ _ _ _ _
..
__
_..
•
Lower tail probability level a
M · __
.~
__ •
0.10
-_ ... _.....
---'
N
0.
1
1
_ _ .......
~-~._..__.
~.
_. __.....
9
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8
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7
.030
6
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3
1
2
7
16
.005 =
.010
6
14
.012
.020
5
12
.031
.041
4
10
.077
•084
4
1
2
3
6
11
24
.005
.008
.009
5
10
20
.014
.015
.024
4
8
18
.044
.049
.041
4
7
15
. 01~4
.088
.087
5
1
2
3
4
5
9
16
32
.009
.009
.008
.009
5
8
14
27
.009
.019
.019
.023
4
7
12
24
.030
.042
.045
.042
4
6
11
20
.030
.088
.070
.093
6
1
2
3
4
5
5
8
13
21
40
.006
.009
.008
.008
.009
4
7
12
18
34
.023
.023
.015
.024
.023
4
7
10
16
30
.023
.023
.046
.047
.045
3
6
9
14
26
.089
.056
.080
.093
.086
1
2
3
4
5
6
5
8
12
17
26
48
.004
.005
•006
.008
.009
.009
4
7
10
15
23
41
.019
.014
.023
.021
.022
.025 -
4
6
.019
.039
.045
.034
.038
.049
3
5
8
12
18
31
.078+
.100
.087
.083
.089
.096
14
21
36
_ _ _ _ _ ~_
_ _ .... _.• _ _ _ _ _ . _ _
.~
.•. '_ __________ ._ _ _ _ _ _...._ _ _ _ •
._,~
0.025
0.10
0.05
•. _ _ _ • _ _. . . . _ · . - . · _ _ _ _ _ · _ _ _ _ e. _ _ _ _ • _ _ _ _ • _ _
H
1
.~
~_~
0.01
X
ell
"""'- __'.O'-·._d __.• _._ •.·____
1
Q
K
_--
2
7
_ _ .__ .. _ .. _.,'W _
_ _ _
0.
N
1
1
N
ell
'1
_ _ _,_ _ _ _ _ _ _ _ _ _ •_ _ _ _ _ _ _ _ _
a
1
H
1
ell
3
5
8
11
15
22
3,'
.070
.081
.058
.074
.091
.090
.094
~
8
':J
10
1
2
3
4
5
6
7
5
7
11
15
21
31
56
.003
.010 '"
.006
.008
.009+
.010
.010 -
4
7
10
13
19
28
49
.016
.010
.013 ..
.025
.020
.021
.023
12
17
25
43
.016
.029
.027
.043
.043
.044
.047
1
2
3
4
5
6
7
8
5
7
10
14
19
25
37
>60
.003
.007
.008
.007
.007+
.010
.009
4
6
9
12
17
23
33
56
.014
.022
.018
.024
.018
.020
.022
.025 -
4
6
8
11
15
21
29
50
.014
.022
.041
.045
.045
O-~9
• J
+
.050
.046
3
5
7
10
14
19
26
43
.064
.068
.093
.082
.070
.074
.094
.095
1
2
3
4
5
6
7
5
7
10
13
17
22
30
42
>60
.002
.005
.005
.007
.008
.010
.009+
.010
4
6
9
12
16
20
27
38
>60
.012
.013
.012
.014
.014
.022
.021
.022
4
6
8
12
14
18
24
34
56
.012
.018
.031
.014
.040
.049
.050
.048+
.050
3
5
7
10
13
17
22
30
49
.060
.058
.074
.057
.067
.071
.087
.099
.096
8
9
~,
K;o •
"':-
.-
4
6
9
.~
t<")
N
-,-
-----_.
.~-----.--~----_._---
__._-----p
Lower tail critical points
... _--
--~_.
-~-
-
- _.- _._ .......-
-........ ~.-
= .90
.---..,. ..
.-.-'-
--~- .----~--
.-
.... __ ._~._.
._-~-
...
----_ _._..
..
'..-...._~
•.._.
_._._p
--
-~
....
= .90
_-_._---.__.--------
L01-rer tail probability level a
Lower tail probability level a
--~-_._--
K
0.10
0.05
0.025
0.01
X
.......
---------------._._'--~-~----_._
"~
N
1
a
1
N
1
(l1
N
1
(l1
F'I
_ _ _ _ _ . ___ __ . _
11
12
13
1
2
3
4
5
6
7
8
9
10
34
48
>60
1
2
3
4
5
6
7
8
9
10
11
4
7
9
12
15
19
24
30
39
54
>60
1
4
7
9
2
3
5
7
9
12
16
20
26
.002
.004
.009
.009
.007
.009
.009+
.010
.010-
+
.010
.003
.007
.006
.008
.008
.007
.009
.009
.010-
.009
.003
.005
4
6
8
11
14
18
24
31
43
>60
4
6
8
11
14
17
22
27
35
48
>60
4
6
8
.011
.015
.024
.020
.024
.025 .019
.022
.024
-
.010
.013
.019
.014
.015
.022
•018
.024
.024
.025 --
-
.009
.011
.015
4
6
8
10
13
17
22
28
39
>60
.011
.015
.024
.041
.043
.039
.038
.0~8
.046
_.
K
-------.'.,,--• __
..-
, __
•_ _ _ _4
3
5
7
9
12
15
20
25
35
55
.056
.051
.061
.085
.076
.094
.075+
.100
.089
.098
3
5
7
9
11
14
18
23
29
39
>60
.053
.045
.051
.067
.100.099
.087
.081
.093
.097
14
4
5
8
10
13
16
20
25
32
44
>60
.010
.045
.019
.031
.029
.037
.040
.044
.048
.046
4
5
7
.009
.041
.043
-
3
5
7
-
.051
.041
.043
15
N
1
(l1
........
..... _ _ ~ _ _ _ _. _
13
-...---, ..,,_.------- ......
_ ---_._---'1
_,,-_
0.025
•.
--.-----_._~-_
1\T
(l1
. __
0.01
X
4
5
6
7
8
9
10
11
12
11
14
11:\
22
27
34
44
60
>60
1
2
3
4
5
6
7
8
9
10
11
12
13
4
6
9
25
31
38
49
>60
>60
1
2
4
6
11
1)~
17
2l
+
.010
..010
.007
.008
.109
.009
.J09
.010 --,.
.009
.010 .004
.008
.007
.008
.007 .
.010
.008
.010
.009
~
'r
-
.008
.009
--.~
a
1
..
.024
.020
.023
.022
.020
.022
.021
.023
4
6
8
10
13
16
19
23
28
35
.009
.010
.013
.019
.014
.015
.021
.022
.023
.021
.023 _.
.025
59
>60
4
6
N
1
_.---._-~._-_
10
13
16
20
25
31
41
54
>60
~.4
0.10
0.05
_--_.'.-._---- .._'.... _.-
""""
-
.008
.009
..
(l1
_--_.-
10
12
15
19
23
29
36
49
>60
3
5
7
9
12
15
18
21
26
32
41
54
>60
3
5
.024
.040
.041
.034
.042
.039
.049
.046
'-.'
.049
.037
.038
.045
.030
.028
.034+
.050
.045
.046
.042
.047
-'
.047
.035
N
1
(l1
9 .055
11 .077
14 .070
17 .083
21 .084
26 .088
33 .089
44 -. 092
>60
3
5
7
9
11
13
16
20
24
29
37
48
>60
3
5
.049
.037
.038
.045
.061
.090
.089
.074
.084
.096
.088+
.100
.047
.035
'<:t
N
r...o~_e!_j;_a...i_1_.~!_~t.i.caJ.
..
p
.1'oint ~
_-_._--_._--------- ------
= .90
P
= .90
_ . _ _ _ _ _ _ 4 _ _ _ _ . _ . . _ _ _ _ _ _- _ , . , . _ _ _ _ _ _
Lower tail probability level a
..
K
0.05
0.025
0.01
X
0.10
K
._-----_ -
X
1
(Xl
'CH1
tIT
a
N
1
1
a
1
a
N
1
..
18
19
-
17
20
23
27
31
36
42
51
>60
>60
>60
.014
.015
.018
.017
.020
.022+
.025
.023
.007
.006
.006
.008
.010 .006
.005+
.010
.007
.010.008
.008
.008
4
6
7
9
11
14
16
19
22
25
29
34
39
.007
.006
.022
.022
.024
.014
.021
.019
.020+
.025
.025
.022+
.025
7
8
9
10
11
12
13
14
15
16
17
18
21
25
29
33
39
46
55
>60
>60
>60
.007
.008
.007
.007 ..
.010
.009
.009+
.010
1
2
3
4
5
6
7
8
9
10
4
6
8
10
12
15
18
20
24
27
32
37
43
II
12
13
-
-
16
18
21
25
29
34
40
47
58
>60
>60
.027
.044
.046
.038
.041
.040
.041
.048
.047
3
5
7
9
11
13
15
18
2l
24
28
32
37
0.01
._---------~--,
.____..-.__.___._ ,-....-...-...... ___ _'-'- "._c . . _,_.•_.,,-, ___ ___.____._____
~
..
!'I
Lower tail probability level a
..._-------0.025
0.10
0.05
_--------_.. __
1
_--_
-
14
17
20
23
27
31
36
43
53
>60
>60
.091
.074
.072
.082
.078
.090
.100
.098
.092
.043
.027
.022
.022
.024
.030
.041
.033
.033
.039
.036
.041
.043
3
5
6
8
10
12
14
16
19
22
26
29
34
.043
.027
.072
.059
.058
.063
.076
.098
.085
.087
.072
.099
.091
-
a
50
60
>60
.010
.009
1
1
N
1
a
46
55
>60
>60
>60
.024
.024
4
6
7
9
11
13
16
19
21
25
28
32
37
43
50
59
>60
>60
>60
.007
.006
.020
.019
.021+
.025
.017
.014
.024
.017
.023
.024
.023
.022
.023 ..."
.025
1
H
1
(Xl
N
1
(Xl
43
51
>60
>60
>60
.046
.048
40
47
57
>60
>60
.087
.093
.095
3
5
7
9
11
13
15
17
20
23
26
30
35
40
47
55
>60
>60
>60
.042
.026
.020
.019
.021
.025
.033
.046
.040
.042
.050
.048
.041
.046
. 01~3
.048
3
5
6
8
10
12
14
16
19
21
25
28
32
37
43
51
>60
>60
>60
.042
.026
.068
.054
.051
.055
.064
.081
.067
.098
.072
.090
.095
.093
.094
.090
..
19
-
IJ
w_c_·_ _ _ _...._ _• _ _ _ _ _ _
20
14
15
16
17
18
>60
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
4
6
8
10
12
15
17
20
23
26
30
35
40
46
54
>60
>60
>60
>60
>~O
.-
-..
..
",-0
.007
.006
.006
.006
.008
.005
.008
.007
.008+
.010
.010
.008
.009
.010
.010 ~
~
~.
-
..
.-
...
'-~
"'-~
-
..
_.
~',",
..,
-
.~
~
L!'l
N
-'-~--
--_.--#---_._-._.. . .
..-.,--..
_--~-_._.-
p
= .90
.
L~T_e!__ :t~.i_1__c:rJ::t.i~a1_.:e9ints
_._.- ---,_.•
~.,.,
0.01
X
0.025
..
_- --....
._~_._----_._-,~,
HI
15
16
Ct
1
3
4
5
6
7
8
9
10
11
12
13
14
9
11
14
16
20
24
29
35
42
53
>60
>60
.003
.006
.005+
.010
.007
.008
.007
.008+
.010+
.010
1
2
3
4
5
6
7
8
9
10
11
12
13
4
6
8
.008
.008
.009
.005
.008
.007
.008
.007
.008
.009
.010
.009+
.010
11
13
16
19
23
27
32
38
47
58
.-
-
-
TiT
'1
_~_
....
0.
1
'---_.~.
.011
.015
.023
.020
.023
.020
.024
.020
.021
.022
4
6
8
10
12
15
18
21
25
29
35
43
53
40 _ _ _ •• _ _ _ _ _ _ ~ _ ~
_ . ~_ _
..... _
._.......
N
1
._.~
Ct.
1
.033
.038 .•.
.050
.038
.008
.008
.009
.012
.018
.015
.015
.019
.019
.025
.024
.021
.024
3
5
7
9
11
14
16
20
23
27
33
40
49
.046
.032
.029
.032
.041
.029
.048
.032
.044
.049
.041
.041
.047
.038
.048
.049
.049
.043
.044
.049
~
N
1
..
. - . . ~ . ~_ _ _
..... _
..-_•• _
-_.__
_ _ _
_• _ _ •
_ _- " ' _ _ _ _ • _ _ _ _ _ • _ _ _ • _ _ _
p
= .90
.,.--
.
Lower tail probability level
0.
,'
7
9
11
14
17
20
24
29
36
45
59
>60
~
_._.>
_.~
------0.10
0.05
.._.---_._-----_..-._-. _.--,"-
_ _ _ _ ._ _ _ _ _ _ _ _ _ _ •._.... _~~ _ _ _ _ _ _
8
10
12
15
18
22
26
32
39
49
>60
>60
....
~
--------- _._-
Lower tail probability level
K
.. '-
-
K
X
'
_._
0.01
..
,-
-_
N
l
0.
1
0,025
.. ~----
._-~.,-,,-
0.
1
..
~
0.
0.05
.--------._...'------.--.--- _.--'._- _-..._--lJ
0.
N
(\
HI
·'1
1
1
0.10
Ct.
1
_ _ _ u._.~_~_#,_.·._~"'.<___.... _ _ _ _ _ _ _ _ _ _ _ _ _ _ .__ _._ ________._ _ _ _ _ _ . _ _._ _ _ .'.•_ _ _•._______,.......___,. _ ..._
.~_.
6
8
11
13
16
19
22
27
33
41
53
>60
.096
.091
.050
.070
.064
.073
.098
.086
.085
.087
.097
3
5
6
8
10
13
15
18
21
25
30
36
45
.046
.032
.088
.081
.087
.056
.081
.080
.093
.093
.090
.094
.088
16
17
-
14
15
>60
>60
1
2
3
4
5
6
7
8
,~
10
11
12
13
14
15
16
6
8
10
13
16
18
22
26
30
35
42
51
>60
>60
>60
1
2
3
4
5
6
4
6
8
10
13
15
0
,/
18
>60
>60
c~
..003
.007
.008+
.010
.006
.005+
.010
.007
.007
.009+
.010
.010
.009
4
6
_.
8
10
12
14
17
20
24
28
33
39
47
58
>60
>60
.007
.007
.007
.009
.005
.008
4
6
7
9
12
14
~
-
~~
~
.008
.007
.008
.010
.015
.022
.020
.021
.018
.020
.020
.021
.022
.023
-_.
.007
.007
.024
.024
.012
.018
>60
>60
-
58
>60
.096
3
5
7
9
11
13
16
19
22
26
31
36
43
53
>60
>60
,045
.030
.026
.028
.034
.045
.036
.036
.043
.041
.037
.046
. 04 9+
.050
3
5
6
8
10
12
15
17
20
24
28
33
40
49
>60
>60
.045
.030
.082
.072
.075
.087
.064
.094
.097
.083
.090
.094
.087
.090
3
5I'
.044
.028
.076
.065
.065
.074
3
5
rr
9
11
13
..
.~
.044
.028
.024
.024
.029
.037
0
8
10
12
\0
N
Upper tail
p
cr~tica1
points
= .90
p ::;: .90
._-~----_.
Upper tail probability level
Upper tail probability level
~
_.
.-
..
K
:N
2
4
4
-
5
5
6
5
6
-
7
8
9
10
5
6
7
6
7
8
6
7
8
9
6
'7
I
8
9
10
~2
&".
.062
.023
6
.075
5
..
.008
-
7
.003
.009
.005
6
8
_.
.023
.013
7
-
6
9
_.
-
7
10
.009
.014
6
8
11
-
.-
-
~2
r
4
-
~
t!
..
..
..
N
5
6
~2
.032
8
.055
.072
-
5
7
10
.089
. 080
.064
.023
.033
.041
.037
.031
6
9
13
.041
.088
.086
.060
.069
.087+
.100
.080
.029
.083
.081
.080
11
.004
.006
7
9
12
.018
.017
.014
7
10
14
.018
.044
.046
-
-
.
-
-
..
.029
.034
.045
.038
6
7
10
13
18
-8
-
8
10
13
-
~
.008
.007
.006
-
8
11
15
.008
.021
.024
7
9
12
16
r.'
".
0.01
X
N,.,
c_
11
6
7,.,
..
~
12
~2
~.
..
_.
9
12
15
.003 ..
.010
.006
7
•.
.-
-
..""
.006
.007
9
10
11
12
14
0.025
9
10
11
Ij
13
6
8
11
16
~
I(
.023
-
_.
N
2
5
.008
7
9
• r
~2
..
6
-
N
2
0.10
0.05
0.025
0.01
X
7
8
9
10
11
12
13
7
8
9
10
9
11
13
18
.OO~'+
.010
. ...
..
,
..."..
~
,,- ..
9
10
12
15
20
~
•.
~.
10
.009
.003
,003
.005
.009
"""
~.~
..004
JlT
~2
11'2
-
-
..
7
8
11
14
19
o •
~
r'
.013
.016
.023
.020
•.
8
9
12
15
19
.020
.006
.022
.023
.017
-
-
9
11
14
17
22
.009
.013+
.025
.021
.021
..
-
9
11
.014
.021
-
-
0.10
0.05
2
8
10
13
17
~
~"
8
10
13
16
21
~
8
10
12
15
18
24
~
8
9
11
~2
N
2
~2
.
6
7
9
12
15
.100
.041
.055
.089
.073
.086
"
.041
.013
.043
.0)+4
.046
2l
.-
7
.020
.028
.048
.040
.037
9
11
14
18
24
.053
.078
.072
.088
.097
.088
-
7
8
10
13
16
20
27
.066
.027
.042
.081
.084
.083
.089
7
8
10
12
.078
.035
.057
.059
.027
.042
.038
.049
.036
.041
~
.035
.014
.021
~---
l"N
Upper tail critical
p
point~
= .90
p
------
Upper tail probability level u
K
14
15
16
11
12
13
14
u2
N
2
u
2
N
2
u2
12
l1.t
17
22
.006
.005
.005
.007
13
15
19
25
.020
.013
.020
.024
14
17
21
27
.045
.048
.049
.044
-
-
.-
-
..
10
9
11
12
15
17
21
27
-
-
~.
...
11
11
12
13
14
15
14
16
20
25
.007
.002
.010 -.006+
.010
.009
8
9
10
11
12
-
.-
14
15
16
17
N
2
7
8
9
10
13
8
0.10
0.05
0.025
0.01
X
-
10
11
13
15
18
22
27
.009
.003
.005
.005
.007
.009
.008
9
10
12
1).j.
16
19
24
30
-
_.
..
..
-
-8
-
.044
.018
.031
.032
.049
.046
. 042
.045
.018
.031
.011
.024
.014
.018
.019
11
13
16
19
23
30
-
-
.013
9
11
13
15
18
21
25
33
.023
.042
.047
.040
.050.042
.036
.045
-
..
.023
.009
.016
.017
.014
.024
.021
-
9
~
N
2
K
Ct
15
18
23
30
.085
.079
.097
.088
7
8
10
12
14
17
20
25
33
.091
.044
.074
.082
.070
.085
.072
.082
.085
8
10
.052
.092
.042
.099
.077
.082
.093
.090
.098
11
14
16
19
23
28
37
8
l'J
2
.060
-2
17
18
19
9
10
11
12
13
14
15
16
17
8
9
10
11
12
13
14
15
16
17
18
Upper tail probability level a
0.025
0.05
0.01
X
..
u
2
N
2
.005
.008
.009
.006
.007
.008
.009
10
12
13
16
18
22
26
33
'"
~.
11
13
IS
17
2J
24
30
."
= ·90
.-
--
""'~
.~
_.
u
2
,.,.'~
.012
.023
.008
.022
.015
.024
.021
.022
_.-
2
u2
N
2
u2
9
10
12
14
17
20
23
28
36
.029
.012
.023
.026
.045
.049
.039
.043
.045
9
11
13
15
18
21
25
31
40
.029
.053
.063
.059
.080
.078
.082
.096
.093
liT
<-.'.
.03!t
.016
.030
.036
.034
.048
.047
.051
.048
.044
-
8
9
11
13
15
17
20
23
28
34
43
.069
.034
.066
.081
.080
.067
.080
.073
.097
.100.088
8
9
11
13
.077
.040
.079+
.100
.006
.002
.004
.004
.008
.007
.007
..010
10
11
13
15
18
20
24
28
36
.016
.006
.012
.014+
.025
.016
.022
.018
.023
a.
10
12
14
16
19
22
26
31
39
8
..
..
-
-_.
..
.,
-
.0
9
10
12
.040
.019
.038
. .•
-
..
11
12
14
16
19
22
26
33
/
10
11
.11
~
<---,
.008
10
11
'"
.019
.008
0.10
~
00
N
UFper tail critical points
p
= .90
p
-
Upper tail probability level a
= ·90
Upper tail probability level
Ct
~-£--._---
K
19
20
0.01
X
N
2
Ct
12
13
14
15
16
17
18
19
12
14
16
18
24
29
35
.003
.00G
.007
.006
.008
.007
.010.008
8
9
-
-
21
...
2
-
0.10
0.05
0.025
K
N
2
Ct
2
N
2
a
2
N
2
Ct
13
15
17
19
22
26
31
39
.016
.020
.019
.014
.016
.020
.022
.023
14
16
18
21
24
28
33
1.J.2
.048
.048
.039
.049
.044
.045
.040
.043
14
17
19
22
26
30
36
47
.048
.092
.072
.078
.095
.084
.083
.096
-
..
8
9
.085
.046
--.
..
~,
""
9
.,
.046
X
0.01
~J
J'-2
2
20
10
11
12
13
14
15
16
17
18
19
20
0.025
--11
12
14
15
18
20
23
26
31
38
Ct
2
.010+
.004
.009
.003+
.010
.007
.009
.007
.009
.009
0.10
0.05
N
2
Ct
10
11
13
14
16
19
21
24
28
34
42
.023
.010
.022
.009
.011
.023
.016
.016
.018
.025 .023
2
N
Ct
10
12
13
15
17
20
23
26
30
36
.023
.047
.022
.028
.027
.043
.048
.041
.039
.043
.050--
2
1~6
2
N
2
Ct
11
12
14
16
19
21
24
28
33
39
.092
.047
.061
.064
.099
.074
.075
.085
.094
.085
.090
50
2
0'1
N
-_._~-_.~-,----,-_.
__
LmTex:
._------_.---'-~--~--.--
p
._-_ ... _.-.- .. - ...
= .95
__
~ail_ E..r.J:~_ical poiI~_i?.
..
_._~
....
-.-,.-
.,- .-.
-_._.-
.. ,~.
__ .-...
,'_.~
...
_---
------~_
...
-
--_._ _-_ _... --._
..
..•,..-..........-- -
(Xl
HI
•
-
-...- - . _ ..• _ - - - .• _ - -
2
1
9
.006
7
.022
6
.042
5
.080
3
1
2
6
15
.007
.010-
5
13
.020
.021
5
11
.020
.045
4
9
.054
.097
1
2
3
5
10
22
.008
.010
.010~
5
9
19
.008
.018
.023
4
8
17
.027
.034
.040
3
7
14
.095
.065
.089
1
2
3
4
5
9
15
30
.004
.005
.008
.009
4
8
13
26
.017
.011
.020
.021
4
7
11
22
.017
.026+
.050
.048
3
6
10
19
.069
.060
.080
.090
4
_ ....... -.....
8
';)
5
6
7
1
2
3
4
5
5
8
12
20
38
.002
.005
.009
.007
.009
4
7
11
17
32
.011
.013
.016
.022
.024
4
6
10
15
28
.011
.034
.030
.047
.048
13
24
.054
.090
.055
.097
.094
1
2
3
4
5
6
4
7
11
16
24
45
.008
.007
.006
.008+
.010
.010-
4
6
10
14
22
39
.008
.022
.013
.021
.019
.024
3
6
9
13
19
34
.044
.022
.027
.035
.048
.049
3
5
8
11
17
29
.044
.064
.057
.093
.087 _.
.100
3
5
9
10
.----._.~.
.....
°1
1
- _ . _ - 'c.
_-~
= .95
._-----_
...• ~--------
Lower tail probability level °
._ .... - - - - - - _....... _---. '*_.- ,_.....
0.10
0.05
0.025
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'
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~
- - - - - - - - - - - - - . _ - - _ . _ - _ .•_
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A
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0.01
v
T'
1'-
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.
0.10
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1
.
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-
N
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-
0.025
0.01
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..
~
X
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-
K
..
~
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Lower tail probability level a
._ .....
...
p
-
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......
__
-,~----
--,--_....
N
1
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_.
__
....
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N
-1
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(Xl
N
1
(Xl
_ _ _ _ _ _._ _ _ _ _ _ ._ _• _ _ _ _ _..•___ ...._. _ _ _
1
2
3
4
5
6
7
4
7
10
14
20
29
53
.006
.004
.006
.008
.008+
.010 -.'
.010
4
6
9
13
18
26
46
.006
.015
.015
.014
.018
.023
.024
3
5
8
11
16
23
40
.038
.048
.035
.048
.042+
.050+
.050
3
5
7
10
21
35
.038
.048
.080
.085
.093
.084
.094
1
2
3
4
5
6
7
8
4
7
9
13
17
24
35
>60
.005
.003
.009
.006 .....
.010
.008
.009
4
6
8
12
16
21
31
53
.005
.011
.023
.013
.016
.025
.022
.024
3
5
8
11
14
19
28
47
.033
.038
.023
.026
.045
.050
.043
.047
3
5
7
9
13
17
25
41
.033
.038
.057
.100
.073
.098
.084
.091
1
2
3
4
5
6
7
8
9
4
6
4
6
8
11
14
19
25
36
60
.004
.008
.016
.015
.023
.019
.023
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.029
.031
.043
.033
.041
.046
.043
.048
.050
3
5
7
9
12
16
21
29
46
.029
.031
.043
.070
.072
.069
. 07.~J
.086
.099
9
12
16
21
28
40
>60
,.
.004
.008
.006
.007
.007
.008
.009
.009
.-
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3
5
7
10
13
17
23
32
53
~
~
l~l
0
~
--_
..-_.
__ ......-,._._._.
Lower tail critic'u E.oints
p
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.
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n.
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= .95
P
K
0.025
0.01
X
N
1
II
12
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
11
13
1
2
3
4
6
9
11
15
19
24
32
45
>60
4
6
8
11
14
18
22
28
37
51
>60
4
6
8
a
l
.004
.006
.004
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.010
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-
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-
.003
.004
.006
JIT
1
a
_--_ .. -- _-..... --- .. ----
l
4
5
8
10
13
17
22
29
41
>60
.004+
.025
.011
.022
.024
.022
.023
.023
.022
3
5
8
10
13
16
20
26
33
46
>60
.024
.022
.008
.015
.015
.020
.023
.019
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3
5
7
.023
.019
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-
-
-.---_._---_._.N
l
a
0.10
3
5
7
9
12
16
20
27
37
60
.027
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.050
.046
.037
.048
.040
.045
.048
3
5
7
9
12
15
19
24
30
41
>60
.024
.022
.026
.037
.030
.036
.035
.037
.050 .050 -
3
5
7
_
..
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N
1
1
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l
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11
14
18
24
33
52
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.093
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.085
.096
.099
.089
.090
.098
3
4
6
8
11
14
17
21
27
37
58
.024
.091
.078
.088
.060
.061
.082
.098+
.100
. 094
.097
3
.023
.083
.067
3
J:;
....
6
9
4
6
= .95
--_._----~._---------_
Lower tail probability level a
-.- ........
~
..... __--,--'..-----_ ......- ---------Lower tail probability level a
--~----_._-
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~
. _
----~~
13
14
15
....
........ - .•
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0.01
X
K
~
---_.-...
.....
a
1
l
_~~.--~._---
4
5
6
7
8
9
10
11
12
11
13
17
21
26
32
41
57
>60
1
2
3
4
5
6
7
8
9
10
11
12
13
4
6
3
10
13
16
19
2)+
29
35
46
>60
>60
1
2
6
4
.004
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.006
.007
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;.. ..
._-~A
___ .___...
~
..._ _ .....
0.025
__ u _ • • • _ _ _ _ . _ . _.. _
1'T
____
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.010
.007
.009
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..
_.
.002
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_~~
_ _ _ _ _ _.._ _ _ _ _
0.05
0.10
.• _ _ _ _ - _ _ _ • _ _ _ _ _ _ _ ~ _ _ ~ _ _ _ . . . _ _ _ _ _ _ _ _ _ _ _ _
';IT
..
_.~---
..
-1
a
1
'~T
1.
Cl.
N
1
a
8
10
13
16
20
24
31
41
>60
.071
.090
.073
.079
.075 ..
.100
.091
.099
3
4
6
.021
.076
.058
.059
.071
.097
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.100+
.100
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.086
.092
1
l
_-._- -._-.,----_.-..._ _---_.-_.
...
, _ _ _•
l
..
10
12
15
19
23
29
37
51
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.011
.021
.022
.019
.024
.023 " ....
.025
.023
3
5
7
9
12
15
18
22
27
33
42
56
>60
.021
.016
.017
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.014
.014
.018
.018
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3
5
.020
.014
-
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9
11
14
17
22
27
34
46
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.029
.044
.040
.050
.035
.042
• 01~8
.048
3
5
7
9
11
14
17
20
25
30
38
51
>60
.021
.016
.017
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.033
.027
.031
.044
.037
.047
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.047
3
5
.020
.014
,,,-
~,
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3
10
12
15
18
22
28
35
46
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3
4
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r-l
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Lower_
_ _ • _ _ . . . . -_ _ _ _ •
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~
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K
_ _ _. _ _ _ _ " _ _ _ _ _ _
_ . _ . _____ b'_
------~---_
a
1
N
1
(l1
N
1
3
4
5
6
7
8
9
10
11
12
13
14
16
1
2
3
4
5
6
7
8
9
10
11
12
13
8
10
12
15
19
22
27
32
40
50
>60
>60
.004
.006+
.010
.009
.006
.009
.008+
.010
.009+
.010
-
25
30
36
46
>60
>60
4
6
8
10
12
15
18
21
25
30
36
44
55
.012
.002
.003
.005
.008
.006
.007
.009
.009
.009
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.010 -
3
5
7
9
11
14
16
20
23
28
33
40
50
7
9
11
14
17
2l
.014
.018
.025 .019
.020
.016
.018
.020+
.025
.023
7
8
11
-
13
16
19
23
28
34
42
56
>60
.019
.013
.012
.014
.019
.013
.024
.016
.023
.019
.023
.024
.024
3
5
6
8
10
13
15
18
22
26
31
37
46
-
.__ ..
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____
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~
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.039
.041
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..
_ _ _ _ . ' •.
_.-0
6
8
10
12
15
18
2l
25
31
38
50
>60
3
4
6
8
10
12
14
17
20
24
28
34
42
X
.~_~
__
---- .... __ ..
_.~._._
..
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0.01
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16
17
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TIT
-1
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,~._~_"_~
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0.10
0.05
-
>60
>60
..
1
4
6
7
10
12
14
17
20
24
28
33
40
48
60
>60
>60
.002
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.010
.004
.006
.010
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.010
.008
.009
.010.008
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.018
.012
.010
.012
.015
.021
.017
.017
.022
.022
.020
.019
.023
.022
-
3
5
7
9
11
13
16
19
22
26
31
37
44
55
>60
>60
4
.002
.002
.009
.010 .004
3
'5
7
9
11
.017
.011
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.012
1
2
3
4
5
6
7
9
12
~
~
.-
1'1
1
a
l'J
a
55
>60
.093
1
1
1
_ _ _ _ _ _.... _ _ _ _ _ _ _ _ _ _...._
>60
>60
3
4
5
6
7
8
9
10
11
12
13
14
15
16
18
~_."
14
15
2
~
.019
.066
.045
.042
.046
.059
.081
.074
.083
.078
.095
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.....
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.049
.057
.075
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.086
.097
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.098
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......_-----""-_.
Lower tail probability level a
----- -'_.'-"---,"-'_.,-, - _... •.-..._._---._---._.-
(l1
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K
_._---l'J
(l1
_ _ _ .• , _ .. _ _ _ _ _ _. _ . _ •. ,_ _ w _ _ _ . _ _ _ _ _ •. - _ . _ _ _ ...... _ _ _ _ _ _ _ · ......... _ _ ...._. _ _
15
_
0.10
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N
1
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0.05
0.025
0.01
X
.... _
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. ._'.____.,_____,__.. _. . .___ ,_._.
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_
-.~_
Lower tail probability level a
,~
~._._
p = .95
p = .95
_ _ _ _.... _ . _ _ _ _ _ ..
. . ______.. . . -. . _
~
...""
>60
>60
.-
3
5
6
8
10
12
15
18
21
24
29
34
41
50
>60
>60
.018
.012
3
5
6
8
10
..
~.
3
4
6
.oln
.036
.038
.047
.033
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.050
.039
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~
. O!~l~
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,"....
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.017
.011
.037
.031
.032
~
+
8
9
11
14
16
19
23
26
31
37
46
>60
>60
3
4
6
7
9
.018
.062
.041
.036
.092
.097
.063
.090
.087
.070
.100
.096+
.100
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a<"
.017
.058
.037
.093
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N
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_ _ •_ _ _ _ _ _ _ _
---~----_.-
p
~
- - _ . _ - _ . ~ . " --_.~
N
1
(l1
18
19
6
7
8
9
10
11
12
13
14
15
16
17
14
17
20
23
27
31
37
43
52
>60
>60
>60
1
2
3
4
5
6
7
8
9
10
11
12
13
4 .002
5 .010
7 .008
9 .008
11 .01014 .005
16 .009
19 .008
22 .009
26 .007
30 .001
35 .007
•40 .009
.007
.006
.006
.008
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.010.008+
.010
.010-
-
-
-
-
N
l
(l1
_
(l1
•• _ _ _ _ _ , _ _ • _ _ - . ,
13
15
18
21
25
29
34
40
48
59
>60
>60
.017
.025
.021
.023
.019
.021
.021
.022
.023
.024
3
5
7
9
11
13
15
18
21
24
28
32
37
.017
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~
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___ ... _ ....___ ..- •. _ ••, '.,.. ___ .____ • ___ ...... __ .. _ ,__..•
p
~
12
14
17
20
23
27
32
37
45
55
>60
>60
3
5
6
8
11
12
14
18
19
22
26
30
35
--,
_._
1\1
1
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.039
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X
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_ _ _ _ ~ _ _ ~ _ _ _ _ ..o O - _ . •,'
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.098
.090
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3
4
6
7
9
11
13
15
18
21
24
28
32
.017
.055
.033
.084
_.,
-
.070
.070
.078
.095
.076
. 074
.084
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.091
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0.01
........... ,_
-_.
19
20
•...__ _..
14
::"5
16
17
18
1
2
3
4
5
6
7
8
0
.;
10
11
12
13
14
15
16
17
18
19
~_
......
----~
N
1
..._ ......_
_ _ _ _ _ ~.____=,
11
13
16
18
22
25
29
34
41
50
>60
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_~
= .95
__.... _ _ _
. ___. _ _
_.
. __
- _ . _ - ~ - _ . _
rr
0.10
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N
l
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.
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0.05
0.025
0.01
~
= .95
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X
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....... _-_ ........ ---_.- -- _....... _----
Lower tail
K
Lower
tail critical points
----_._---_..
--
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_._~_
-
.... ....
...
56
>60
>60
>60
4
5
7
9
11
13
16
19
21
25
28
33
38
44
51
>60
>60
>60
>60
..
_--------_._.'-~._-----_.,---
0.025
1
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0.05
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.
N
1
a
1+1
48
59
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.041
.049
.047
3
5
6
8
10
12
14
16
19
22
25
29
33
38
44
52
>60
>60
>60
,016
.009
,031
.024
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1
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47
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Lower tail probability level a
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- --.._._--_.. .... _---- ......_---_. __
a
~
~
44
52
>60
>60
>60
.020
:023
3
5
7
8
10
12
15
17
20
23
26
30
35
40
47
56
>60
>60
>60
.016
.009
.007
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1
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44
54
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.093
3
4
6
7
9
11
13
15
17
20
23
26
30
35
41
48
58
>60
>60
.016
.052
.031
.077
.062
.059
.065
.077
.099
.085
.086+
.100
.098
.090
.086
.092
.094
---_ .._-
tI')
tI')
Upper tail critical points
p
= .95
Upper tail probability level a
K
0.01
X
0.025
N
2
a
4
4
-
-
5
5
..
..
6
5
6
-
7
8
9
6
7
6
7
8
6
7
8
9
10
11
7
8
9
10
7
8
~
7
..
2
..
..006
.009
-
-
-
--
.002
.010-
N
2
~
5
8
.072
.092
.011
-
7
6
0
.-
-,
--
-
..
-
9
12
.024
.021
7
9
13
7
9
11
15
8
.076
-
7
11
-
4
-
.013
.020
."
-
..
.012
.011
.019
..
.021
-
-
8
.042
.031
.044
..
.013
.042
6
10
6
8
12
.057
.052
.072
.026
.024
.038
6
8
10
15
. 084
.094
.061
.092
.042
.049
.031
.033
7
9
12
17
.042
.049
.064
.077
-
7
9
.060
.079
~
.021
lL
9
10
11
12
7
8
9
10
11
12
.031
.083
13
1 11
0.01
X
2
_.
-
~
a
N
.030
7
10
~~~
'"2
a2
5
..
8
10
14
K
.030
.018
-
0.10
0.05
5
8
8
11
-
6
Upper tail probability level a
-
.004
..
9
13
-
•.
.-
..
..
~
a2
..
9
-
·2
~
.008
-
TeT
= .95
p
--,----
7
8
9
10
11
12
13
8
!J
'2
a
2
N
2
C!2
a.11
15
.005
.005
.009
10
12
16
.024
.015
.017
"
.
...,,,
'J
10
13
17
..
~.
~
10
12
15
19
",-'
.-0
10
11
12
13
14
0.025
O'
=,-,
.-
-
.010
.012
.017
.025
-
-_.
-
.005
.006
.008
.008
9
11
13
16
21
.015
.022
.017
.018
.021
..
'--.,
_.
.008
.002
.008
.009
.007
N
"2
a2
10
13
18
.024
.034
.044
11
14
20
.063
.062
.086
.8
_.
7
.078
.031
.042
.070
.091
.093
~
2
10
12
15
20
.'
8
9
11
11~
17
23
....
..
...
0
.022
.008
.011
.021
.018+
.025
9
11
13
16
20
26
~
0'
10
11
14
17
21
o.
9
11
14
19
2
JlT
~
.010.002
.007
.009
10
12
15
18
24
0.10
0.05
lX
.031
.042
.033
.034
.037
~
.042
.015
.022
.039
.033
.043
-
.022
.034
.032
.042
.049
.048
8
10
13
17
23
26
.096
.042
.063
.058
.073
.081
.096
8
10
12
14
17
21
29
.054
.086
.088
.069
.072
.072
.097
7
8
10
12
15
1~
~
f")
Upper tail critica:J._ ;ps>ints
p
= .95
p
= .95
-~--_._.
Upper tail probability level ex
K
0.025
0.01
X
Upper tail probability level a
----_._---_.
0.10
0.05
K
0.01
X
0.025
0.10
0.05
----~-
15
16
8
9
10
11
12
13
14
15
8
9
10
11
12
13
14
15
16
17
8
9
10
11
12
13
14
N
2
a
-
--
.-
-
11
13
15
19
24
2
.-
.004
.005
.OO~.
.009 =.
.010
_.
-
10
12
14
17
20
26
-
..
..
11
13
15
17
21
26
-
.006
.009
.009
.005
.009
.008
10
11
13
16
19
23
29
-
-
-
-
11
12
14
16
-
N
2
v·
..008
.003
.005
.004
.
.."
-
-
10
11
13
15
18
a2
N
2
-
..
.011
.018
.017
.023
.017
.021
~
-
.016
.006
.009
.021
.024+
.025
.024
..
-
.021
.0oB
.015
.015
.025 -
9
11
12
15
18
22
28
9
10
12
14
17
20
24
31
9
10
12
14
16
19
a
2
~
.030
.049
.018
.039
.042
.043
.039
..038
.016
.027
.028
.044
.041
.038
. 041
-
.047
.021
.038
.043
.036
.01~6
J'i
2
8
9
11
13
16
19
a
32
.067
.030
.049
.051
.075
.069
.087
.097
8
9
11
13
15
18
22
27
35
.080
.038
.066
.073
.063
.077
.096
.099
.095
2)~
8
9
11
13
15
17
20
~!2
2
.093
.047
.084
.098
.091
.071
.077
0.
--17
18
19
15
16
17
19
23
29
2
.006
.009+
.010
.-
-2
a
2
n2
21
25
31
.024
.023
.020
22
27
34
l\T
9
10
11
12
13
14
15
16
17
18
12
14
16
18
21
25
31
.004
.008
.008
.006
.007
.008
.008
11
13
15
17
19
23
27
34
9
_.
-_.
...
..
.006
.002
.004
.004
.007
.007
.007
.010
11
12
14
16
19
21
25
30
37
10
11
12
13
14
15
16
17
18
19
..
&-...,
~.
_.
12
13
15
17
20
23
27
34
~
•.
r,.'
~
~
-
.-
.011
.021
.024
.020
.014
.023
.020
.021
-
.015
.006
.011
.013
.025 .015
.022+
.025
.023
-
10
12
13
15
18
21
24
29
37
..
10
11
13
15
17
20
23
27
32
40
0.
2
.039
.047
• 01~3
-
.0 27+
.050
.021
.024
.043
.047
.037
.041
.044
-
.033
.015
.029
.035
.033
.047
.046
.050+
.047
.044
N
2
a
24
20
38
.087
.085
.093
9
10
12
14
16
19
22
26
32
41
.056
.027
.050
.059
.055
.076
.075
.079
.093
.091
9
.065
.033
.063
.078
.078
.065
.078
.071
.095
.099
;
10
12
14
16
18
21
24
29
35
44
2
.088
Lfl
rJ")
Upper tail critical
p
poip~
= .95
Upper tail probability level a
K
20
9
10
11
12
13
14
15
16
17
18
19
20
0.•.025
0.01
X
N
2
0.
.-
.-
2
-_.
-
12
13
15
17
19
22
25
30
36
.008
.003
.006
.007
.006
.008
.007+
.010
.008
-
N
2
~
-
11
12
14
16
18
20
23
27
32
40
0.
0.10
0.05
2
.019
.008
.016
.020
.019
.014
.016
.020
.022
.024
N
2
-
10
11
13
15
17
19
22
25
29
34
43
0.
2
....,~
.039
.019
.'138
.0\+8
.047
.039
.049
.045
• 01~5
• ol~O
.044
N;:>
0.2
9
.075
.039
.078
.099
.048
.092
.072
.0713
.095
.084
.084
.097
10
12
14
15
18
20
23
27
31
37
48
\0
I"')
._-_._
...
_----_._--------~-
..
~---~--_._,._-_.~.
=
p
__ . __
-_-----_------_._._._,.--_.__
Lower tail ..critical points
..
.~
-,---.~
..
-
.~-_._.-
·99
P
• _ _ _ .. _ _ _ _ _
Lower tail probability level
_ _ R
. _ . ________ . . _ _ _
- - . _• • _ _ • _ _ •
_ _ • . _ _ .,' _ _
. _ • • . •_
~
.••
~-_.-
K
X
_-_ _. __
...
2
1
0.05
---_._~-
N
.
0.025
_. 0.01
.
1
8
~'
7 .. 017
6
15
.004
.007
5
12
.014
.025
4
1
2
3
5
10
21
.005
.006+
.010
4
8
18
5
1
2
3
4
4
8
14
28
.009
.007
.009+
.010
1
2
3
4
5
4
7
12
19
36
1
2
3
4
7
10
15
23
43
6
7
4
5
6
.._.
__
6
1
2
3
N
l
~1
._--_._----~._--_._._------
.008
.. __ -_.
_.,.~
N
l
.
~1
.~._-----
.033
-
-
0.10
..
I(
... -_._._ ...... ,_
5
..
_-_
.066
.040
.037
4
9
.040
.081
.018+
.025
.024
4
7
16
.018
.049
.042
3
6
13
.068
.097
.098
4
7
12
24
.009
.018
.023+
.025
3
6
11
21
.045
.044
.038
.048
3
6
10
18
.045
.044
.062
.093
.006
.008
.005
.007
.009
4
6
10
16
31
.006
.022
.021
.024
.022
3
6
9
15
27
.032
.022
.040
.036
.046
3
5
8
13
23
.032
.064
.077
.077
.093
.004
.004
.008
.008
. 010.010-
3
6
9
13
21
37
.024
.013
.018
+
.025
.019
.025
3
5
8
12
18
33
.024
.042
.040
.042+
.050
.045
3
5
7
11
16
28
.024
.042
.087
.070
.093
.100-
-
v
f.
..
~1
_. __ .-- ....
= .99
". _ _ • _ _ . . _ . _ _ _ _ _ _ J_. _ _ _ _ _
.. _ _ _ _ _ ."'- _ _ _ _ _ _
_ _... _.,_ ....._·_·_..·....._._________ ·____ . . ___·_·___ .w._· ___ .
~-
N
l
_
_-------'--~~.
Lower tail probability level
,._-
4
11
~
..
._---_._--._..-....._-----~------~--
__ 0.01
..... --
"'r
·1
..
9
10
--- ._----_._--_..
N
~1
1
2
3
4
5
6
7
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
9
4
6
2'i
51
.003
.008
.009
.009
.008
.010
.009
)+
.002
()
-'
13
19
~
58
'-I
.001
8
11
i5
20
27
38
>60
.ooh
.ooa
.009
.008
.008
.008
.010
~
~1
(1,1
N
l
C1.
'
.019
.030
.022
.033
.046
"""
.050
.046
3
5
7
10
14
20
33
.019
.030
.055
.061
.070
.085
.100-
3
''''1
..
.019
<•
1
.
8
12
17
25
44
.022
.017
.019
.022
.024
3
5
8
11
15
22
39
3
5
8
11
15
20
30
51
.016
.022
.013
.016
.018 ,.025
.020
.023
3
5
7
10
14
19
27
45
.016
.022
.037
.034
.031
.036
.041
.047
9
12
17
24
39
.016
.089
.098
.071
.087
.074
.082
.093
3
5
8
10
14
18
24
34
58
.013
.016
.008
.020
.014
.020
.022
.023
.024
3
5
7
9
12
16
22
31
51
.013
.016
.026
.046
.050
.04<:,
.042
.044
.049
3
4
6
9
11
15
20
27
44
.013
.073
.075
.046
.089
.076
.080
.099 ...
.100
6 .008
.005
.008+
.010
.008
.010:
.010
6
1
3
6 .005
9
12
16
23
33
...
'J
-_._-_.. ---..'------_._._--_
----- ...._------_......
_------_.~--
8
..........
~
__0.05._------, _0.10
---
0.025
.~._-~.~._
~,_
4
6
",--~
~-
1"'l")
-_._,,-----'-- ------_ . --- -_.
~wer ~ail
---,-----~_.---._-
._.. ---_._----
critic.§.l E..oints
_._-_.--._-~
----~
... -----
-.-----_._---_.-..-
= .99
p
. -..
...... - _._ ...
_,~_._--
-----..
= .99
-p
--.... - -_.'----'_-.'--------- --..- ...
Lower tail ~robability level a
- -- '--_ _..-- ._._----.---,'--------_._ __.._---_.-...-.
.
Lower tail probability level
-----_._-------_ --.... _--- -_...- __.. _---_._
0.05
0.025
0.01
X
..
._-_._-----_
...
N
1
ot
......
11
12
13
1
2
3
4
5
6
7
8
9
10
4
6
8
11
14
18
23
31
43
>60
.001
.002
.005
.005
.007
.008
.010.009+
.010
1
2
3
4
5
6
7
8
9
10
11
3
5
8
10
13
17
21
27
35
49
>60
.009+
.010
.004
.008
.008
.006
.009
.008
.009
.009
1
2
3
3
5
7
.008
.008+
.010
-
-
.. _-~.-
..
-_._._.__ a
HI
I'll·
1
1
------_.__._--_ _. __
K
at
ot
1
..
- ---0.10
ot
.011
.013
.018
.013
.014
.024
.022
.021
.022
3
5
7
9
12
15
19
25
35
57
.011
.013
.018
.031
.029
.040
.049
.050.048+
.050
3
4
6
8
11
14
18
23
31
50
.011
.062
.059
.075
.058
.068
.072
.087
.099
.096
.009
5 .010
7 .014
9 .022
12 .018
15 .022
19 .023
24 .025
32 .021
44 .022
>60
-
3
5
6
9
11
14
18
22
29
40
>60
.009
.010
.047
.022
.038
.040
.036+
.050
.045
.044
3
-
4
6
8
10
13
16
20
26
35
56
.009
.053
.047
.056
.080
.072
.089
.099
.095+
.100
.094
.008
.008
.010
3
4
6
.008
.046
.039
2
4
6
.095
.046
.039
~
.,
.)
-
3
5
7
K
X
..
...
0.01
0.025
._---N
1
1
._--------_._----~_.--._----_
3
5
7
10
13
16
21
28
39
>60
.........
_--_._._N
1
---------~--
ot
1
:N
'I
0.05
ot
1
lT
1
a
8
11
13
17
21
26
33
44
>60
.043
.026
.047
.032
.034
.039
.043
.048
3
4
6
8
10
.007
.041
.032
.034
.044
.032
.033
.044
.049
.043
.041
.046
_,_.---_._.---_.-------_-."'---_._ _-----_.
13
14
15
0.10
1
~T
"I
ot
1
..
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
13
1·
2
1)
13
1~
20
24
30
39
54
>60
3
5
7
10
12
15
18
22
28
34
44
60
>60
3
5
.005
.005
.006
.'106
.0l0~
.01)+
.010
.010 ~.
.007
.007
.008
.004
.007
.007
.010:
.010
.007 .,
.010
.009
.009
-,
.007
.006
0-'
12
14
18
22
28
36
49
>60
3
5
7
9
11
14
17
21
25
31
40
54
>60
3
5
.015
.011
.025 .019
.023
.020
.021
.022
,-
.007
.007
.008
.011
.018
.015
.018
.017
.023
.024
.022
.022
..
.007
.006
13
16
19
23
29
37
49
>60
3
4
~
.007
.036
8
10
12
15
19
23
30
40
>60
.043
.059
.088
.088
.077
.097
.084
.088
2
4
6
7
9
12
.090
.041
.032
.097+
.100
.063
.099
.069 ..
.100
.098
.094
.092
Ih
18
2l
26
33
44
>60
2
4
.085
.036
00
l")
______
__
___
M_~
_ _ _ _ '
_ _ _ _ _ _ _ _ • _____
p
- __
-
___
•
~
"'"-
__
~~
'"
_.__.
Lower
._-_.- tail critical points
--__ M
= .S9
---~'-'-----
K
X
N
l
_ _ _ . _ _.•••
~_
a
l
HI
~
a
l
N
l
16
1
2
3
if
5
6
7
8
9
10
11
7
9
12
14
18
21
26
31
38
48
>60
>60
.006
.008
.005
.010.005
.009
.007
.009
.009
.010 -
-
+
4 .000
5 .005
. 7 .005
9 .006
11 .009
14 .007
17 .006
20 .008
24 .008
29 .007
34 .010-
7
9
11
13
16
19
23
28
35
44
59
>60
4
5
6
8
11
13
16
19
22
26
32
.006
.008
.013
.022
.021+
. 02 5+
.025
.025 .021
.022
.023
~
+
.000
.005
.023
.022
.009
.015
.013
.015
.021
.024
.019
6
8
10
12
15
18
22
26
32
40
54
>60
4
4
6
8
10
12
15
17
21
25
29
".-
._-~--
al
n_
N
1
~_.~_._
.027
.027
.033
.046
.039
.043
.038
.048
.047
.049
.044
+
.000
.032
.023
.022
.026
.034
.026
.047
.034
.036
.049
a
./
11
14
17
20
24
30
37
48
>60
2
4
5
7
9
11
13
16
19
23
27
1C
• • .,. _ _
_---
.-..~-~-,,_._---_.-.---_._._---._--_
__._ _ _ _._ -__ .... __ .....
~
=
,---_.- ......... _. --.
0.0l
. J.
"
'
_
_
·
.
~
4
_
·
_
....
_.. _.
oq
"/..-.
••____ .._,____'. _ _ _ _ _ _ "_ ........._'__._._ _ ·_ _
tail probability
level. a
'.
..
....
_-------_..-""0.025
0,05
0.10
~,--~
_._.~~.
.
~------_
..
.. ,,-----~ ...-_._.,_.....
a
1
HI
,009
.009
38
48
>60
>60
a
lJ
1
1
a
1
N
1
a
1
..... _ ' - _ _ _ _ _ _ ~ _ _ ~ _ _ _ _ . _ · _ ._ _ o _ _ ~ _ _ _ _, _ . . _ _ • _ _. . _ . . _ _ _ • _ _ •• _ _ _ _ _ _ _ _ _ .,•
16
17
~
.081
.032
.096
.070
.066
.075
.096
.080
,084
.075
.088
~--
Lo~er
~
--- _.1\T
l
.027
.082
.081
,095
.071
.070
.086
.092
.078
.087
.096
.......... - .....
--------~~---_._
-- "---,.....-.' ..•
•• __. . . . . . . . . . _ _• . _ _ . _ •. _ _. . . . . . . . . . . . . _ _ _ _ _ _ •• _
6
7
-
p
V
a
-
- - _ . _ - ....
-----
_
~.
• _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ , _ _....
3
4
5
6
7
8
9
10
11
12
13
14
--~
0.10
0.05
0.025
0.01
____ ___._ _._ __ . _._,__ . ._.&.______ __._.-...__.__.. . . ___ .. _.._.. __
~
15
-
-------
Lower tail probability level a
_ _ _ _ _ _- . _ ........ _ _ _ _.• _ _ _ _ _ _ _v
-
12
13
14
15
42
5'3
>60
>60
1
2
-3
4
5
6
7
8
9
10
11
12
3 ,005
5 ,004
7 ,004
9 .005
11 .007
14 .004
16 ,009
19 .009
23 .007
27 .008
32 .008
38 .009
46 ,009+
57 .010
>60
..
>60
13
14
15
16
18
1
2
~
.,.,.,
~
3
5
.005
.003
.025
,023
......,
...,..,
_.
3 .005
5 ,004
6 .019
8 .018
10 .020
13 ,OIl
15 .018
18 ,017
21 .020
25 .019
29 ,024
35 .021
42 .023
"'
52 .025
>60
>60
'""-.
~
3
5
,005
.003
36
41~
58
>60
3
if
6
8
10
12
14
17
20
23
27
32
39
48
>60
>60
3
4
.041
.049
.049
33
40
53
>60
,005
.029
,019
.018
.020
.026
.037
.032
.034
.045
.048+
.050
.046
.049
2 .078
4 .029
5 .087
7 ,060
9 .055
11 .060
13 ,074
15 .100 18 .090
21 .100
25 .094
30 .086
36 .088
44 .094
57 ,095
>60
..
,-
_.
.005
.027
2
4
.083
.098
.090
.074
.027
0'1
t')
---_...
--_._-_.
_._----
Lower
tail critical --_.noints
---- ._-
..
_,~---------_.
-- '._- --.-
...........
-----------~
.. .
~,-
---,~-
-
~
p = .99
.. _._--
__ ._---
Lower tail probability level a
--_.
K
N
1
a
1
-_.__._--_._---_.
18
19
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
1
2
3
4
5
6
7
8
7
9
11
13
16
19
22
25
30
35
41
50
>60
>60
>60
3
5
7
9
11
13
15
18
N
1
a
0.10
a
1
N
1
-
6
8
9
11
14
16
19
22
26
30
36
43
52
>60
>60
.017
.015
.046
.048
.027
.041
.038
.043
.039
.046
.040
.042
.049
.005
.024
.015
.012
.013
.015
.021
.015
3
4
6
7
9
11
13
16
-
.017
.015
.016
.020
.013
.022
.022
.017
.018
.024
.024
.022
.022
a
.005
.003
.003
.003
•004
.006
.009
.007
3
4
6
8
10
12
14
17
X
.079
.052
.046
.048
.058
.076
.0fA
.067
.083
.086
.086
.094
.091
.-
.005
.024
.015
.046
.038
.039
.045
.030
2
4
5
7
9
10
12
15
.071
.024
.073
.046
.038
.094
.096
.058
"".'
-_.__
0.01
1-J
1
. ._ " _ , . . . . . _~
5
7
9
11
13
15
18
21
24
28
33
3q
48
>60
>60
-
K
--
_ _ _ . _ _ _ _ • _ _ .. ~ _ _ _ _ • • 0' _ _
6
8
10
12
15
17
20
24
28
32
38
46
57
>60
>60
.003
. 001~
.005
.008
.006
.006
.007+
.010
.008
.009+
.010
.009
N
1
.,-.-_.---~--_._-
Lower tail probability level a
--_.- ---
1
---_ .... _ . _ - .-.-#-._...._._---_
..__
..
--_._----_.
0.05
._---_.-----_._----_.
----_.. _------
._._._-_.~.------
0.05
0.025
0.01
X
.~,,'-
__._-_ ----_.-.._---_._--_.--
_.
P = .99
••...
~'1
~~
_ . . . . . . . , ._ _ _
19
r-!1
1
1
N
1
a
.01)+
18
.017
.023..025 .
.025
.020
.021
2l
.046
.045
.035
.036
.045
.043
.048+
.050
a
1
0.10
N
1
(Xl
17
20
23
26
31
36
43
52
>60
>60
.080
.073
.079
.099
.081
.087
.084
.089
2
4
5
7
8
10
12
14
17
19
22
25
29
.069
.022
.067
.040
.097
.081
.079
.088
.061
.086
.083
.093
.088
v. _ _ _ _ _ _ _ _ _ _, _ - . . . _ _ _ _ _ ~_. _ _ . . . _ _. . . . _ _ _ . _
Q
10
11
12
13
14
15
16
17
18
20
a
0.025
1
2
3
4
5
6
7
8
9
10
11
12
13
21
24
28
33
38
45
54
>60
>60
>60
.008
.010
.009
.008
.010
.009
.009
3
5
7
8
10
13
15
18
20
23
27
31
36
.004
.003
.002+
.010,or
.010
.004
.007
~
~
~
-
.005
.009+
.010
.008
.009
.008
20
23
26
30
35
42
50
>60
>60
>60
3
4
6
8
10
12
14
16
19
22
25
29
33
';
-,
..
.004
.022
.013
.010
.010
.012
.016
.022
.018
.018
.022
.020
.024
25
29
33
39
46
56
>60
>60
3
4
6
7
9
11
13
15
18
21
24
27
31
.004
.022
.013
.040
.033
.032
.036
. 045
.033
.031
.034
.044
.047
o
-.:t
________=L=o~w=e=r:..:t=a=i=l:...=.c=r=i=t=i=C::al=_:~:o..::i:nt=s_:p
_
= ·99
__
Lower tail probability level a
..
..
-------._--------_.-- _.•._._0.01
K
x
0.025
--- -----.--_.... _---_. __ --_._--_.,,---
-_._-~--_._
.....,.
20
14
15
16
17
18
19
~-
N
1
(11
N
1
(11
42
40
58
>60
>60
>60
.008
.009 ..
.010
39
45
54
>60
>60
>60
.019
.023
.021
./
.,
-
-
..
-
-
._.~
~-~-
._---0.10
0.05
.
---_._-----~._-----
~~
.J..
al.
36 .• 045
It?
50
>60
>60
>60
.046
.045
-
N'
1
a
1
34 . ~078
39 .089
46 .091
56 .088
>60
>60
.....
"'"
Upper
p
N2
4
4
-
5
5
-
5
6
..
6
7
8
9
6
7
6
7
8
7
8
.'0
10
11
~
7
-_.
,--Co"
8
10
-'
-
9
10
9
12
7
8
9
...
~.
-'
9
""
~2
-="
."
-
5
"'
-
.015
-
.006
-
- +
.008
.004
.003
.006
~
~
.008
N
2
..
6
~
Upper tail probability level
0.10
0.05
-
_.
..
9 .010
8
7
~2
NC)
~2
N
2
K
.-
4
.090
.037
5
.037
6
.015
5
7
. 088
.051
8
..
.023
6
8
.040
.023
6
9
.040
.055
•.
-
-
.
.018
.026
7
-
8
11
-
8
10
13
...-..
.,
9
.018
.010
-
.008
.012
,.."
.017
.015
.013
..
-
.008
~
7
10
7
9
12
-
8
11
15
-
8
10
'
.035
.032
.027
•.
.017
.041
.043
-.
.029
.033
6
8
11
7
10
14
X
~2
-'
9
point~
11 = .99
------.
0.025
0.01
X
cri.:tical
= .99
Upper tail probability level
K
ta~l
11
10
11
12
8
9
10
11
12
.072
.066
.053
14,
8
9
10
11
12
13
7
8
11
.056
.064
.082
.095
.080
.029
.083
10
11
12
13
14
15
8
9
0.025
~2
N
2
~2
12
16
.021
.023
13
17
...
.-
..
c,.
Q
.014
.017
.024
.021
S
q
~2
11
14
.007
.006
",,--.
~
10
1]
16
-
10
12
14
18
,~
.11
13
16
21
-
~-
.003
.010+
.007
-
"
11
14
18
.-.,
.-
q
.007
.008
.005
.006
10
13
16
20
.-
-
.-"
10
12
.003
.004
.006+
.010
','
..
c.. ..,
.... ,
....
~
.022
.007
.024+
.025
.019
~
-
18
23
.011
.016
.013+
.025
.024
_.
..
11~
.~
--,0.10
0.05
N
2
'-
8
q
7
9
12
17
.
IT,.,
13
.035
.078
.077
0.01
~
-
12
15
20
_.
Q
-'
11
13
17
22
.,",
9
11
13
15
19
25
.. ...
-
9
riT
'2
~2
.045
.038
14
19
.081
.080
.043
.014
.046
.046
.048
8
10
13
16
22
.043
.057
.093
.077
.089
-
8
10
12
15
18
25
.059
.086
.079
.097
.071
.096
8
9
14
17
21
28
.076
.032
.049
.093
.097
.094
.099
8
9
.093
.043
.022
.031
.021i
.045
.041
0-
.032
.049
.045
.030
.01~2
.047
......
.043
11
('I
-.:t
~r
p
p~int~
tail critical
= ·99
Upper tail probability level a
K
15
16
17
0.01
X
10
11
12
13
14
15
9
10
11
12
13
14
15
16
9
10
11
12
13
14
15
16
17
0.025
N
2
a
-,
-
11
13
15
18
23
2
.006
.008
.006
.007
.009
a
10
11
14
16
20
25
.017
.006 •.
.025
.016
.024
.021
2
-
..
_.
11
12
14
17
20
25
.009
.003
.004
.008
.007
.008
10
11
13
15
18
22
28
.024
.009
.014
.013
.018
.023
.024
•.
-..
..
-
11
13
15
17
20
24
30
..
..
.013
.022
.023
.018
.020
.021
.020
-
-
12
14
16
19
22
28
-
.005
.007
.007+
.010
.007
. 010+
-
0.10
N
2
a
10
12
14
17
21
27
.017
.027
.025
.033
.039
.039
,~
10
12
11~
16
19
23
30
. -"
10
11
13
15
18
2l
26
33
2
..
.024
.040
.041
.032
.035
.036
.043
.032
.013
.022
.023
.037
.035
.047
.046
lTpper tail probability level a
._-_.
0.05
·'2
TIT
.p = ·99
--_._-,..
11
2
11
13
15
19
23
30
K
a
.070
.071
.055
.094
.084
.083
.055
.093
.040
.088
.062
.089
.100. 084
9
10
12
14
17
19
23
28
36
.068
.032
.056
.062
.099
.066
.083
.087
.085
0.01
18
19
20
9
10
11
12
13
14
15
16
17
18
9
10
11
12
13
14
15
16
17
18
19
10
11
12
0.025
-_.
r··T
2
11
12
15
17
21
25
33
9
X
"'2
a
-.
..
-~
....'.,
-
-
,-
11
12
14
16
19
22
26
33
.018
.007
.012
.013
.021
.020
.019
.023
o.
12
13
15
18
20
24
30
<.
-
12
13
15
17
19
22
26
33
2
.
F."
.007
.002
.004
.010
.005
.007
.009
~
"2
..
.~
..
.010
.004
.006
.007
.005
.006
.007+
.010
0-
-.
'"
l\T
-
a
2
~
.Il
12
14
16
18
21
24
28
36
s·
..
12
..
,
.023
.010
.018
.021
.018
.024
.020
.018
.025
-
..
.013
N
·'2
-
.'eo
~_
0.10
0.05
~
10
11
13
15
17
20
23
28
36
....
10
11
13
14
17
19
22
26
31
39
..
11
12
a
2
N
2
a
.
.040
.018
.032
.036
.031
.040
.034
.042
.048
a
10
12
14
16
18
21
25
31
40
.081
.040
.073
.086
.080
.062
.067
.077
.098
.099
9
10
12
13
15
18
20
24
28
33
43
.095
.049
.092
.044
.053
.093
.068..100
.097
.085
.097
10
11
13
.059
.029
.057
J
_.
.049
.023
.044
.018
.04q
.038
.042
.049
.050
.049
0<.
.029
.013
-
2