A Course in Mathematical Statistics phần 10 pps

I.3 Basic Definitions About Matrices 507
and
rank rank rank
22
AA IAA
11
++−−
()
= n.
iv) If A
j
, j = 1, . . . , m are symmetric idempotent matrices of the same order
and Σ
m
j =1
A
j
is also idempotent, the A
i
A
j
= 0 for 1 ≤ i < j ≤ m.
The proof of the theorems formulated in this appendix may be found in
most books of linear algebra. For example, see Birkhoff and MacLane, A
Survey of Modern Algebra, 3d ed., MacMillan, 1965; S. Lang, Linear Algebra,
Addison-Wesley, 1968; D. C. Murdoch, Linear Algebra for Undergraduates,
Wiley, 1957; S. Perlis, Theory of Matrices, Addison-Wesley, 1952. For a brief
exposition of most results from linear algebra employed in statistics, see also
C. R. Rao, Linear Statistical Inference and Its Applications, Chapter 1, Wiley,
1965; H. Scheffé, The Analysis of Variance, Appendices I and II, Wiley, 1959;
and F. A. Graybill, An Introduction to Linear Statistical Models, Vol. I, Chap-

ter 1, McGraw-Hill, 1961.
I.4 Some Theorems About Matrices and Quadratic Forms 507
508 Appendix II Noncentral
t
,
χχ
χχ
χ
2
and
F
Distributions
508
II.1 Noncentral
t
-Distribution
It was seen in Chapter 9, Application 2, that if the independent r.v.’s X and Y
were distributed as N(0, 1) and
χ
2
r
, respectively, then the distribution of the r.v.
TX Yr=
()
was the (Student’s) t-distribution with r d.f. Now let X and
Y be independent r.v.’s distributed as N(
δ
, 1) and
χ
2

r
, respectively, and set
′
=
()
TXYr
. The r.v. T ′ is said to have the noncentral t-distribution with r
d.f. and noncentrality parameter
δ
. This distribution, as well as an r.v. having
this distribution, if often denoted by t′
r:
δ
. Using the definition of a t′
r:
δ
r.v., it can
be found by well known methods that its p.d.f. is given by

ft
rr
x
xt
x
r
dx t
t
r
r
r

′
+
()
−
()
∞
()
=
()
×−+ −
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎧
⎨
⎪
⎩

⎪
⎫
⎬
⎪
⎭
⎪
∈
∫
;
;
,.
δ
δ
π
δ
exp
1
22
1
2
12
12
0
2
Γ
ޒ
II.2 Noncentral
χχ
χχ
χ

2
-Distribution
It was seen in Chapter 7 (see corollary to Theorem 5) that if X
1
, , X
r
were
independent normally distributed r.v.’s with variance 1 and mean 0, then the
r.v. X =Σ
r
j =1
X
2
j
was distributed as
χ
2
r
. Let now the r.v.’s X
1
, , X
r
be indepen-
dent normally distributed with variance 1 but means
μ
1
, ,
μ
r
, respectively.

Then the distribution of the r.v. X* =Σ
r
j =1
X
2
j
is said to be the noncentral
Appendix II
Noncentral
t
,
χχ
χχ
χ
2
and
F
Distributions
II.3 Noncentral
F
-Distribution 509
chi-square distribution with r d.f. and noncentrality parameter
δ
, where
δ
2
=
Σ
r
j=1

μ
2
j
. This distribution, and also an r.v. having this distribution, is often
denoted by
χ
′
2
r:
δ
. Using the definition of a
χ
′
2
r;
δ
r.v., one can find its p.d.f. but it
does not have any simple closed form. It can be seen that this p.d.f. is a mixture
of
χ
2
-distributions with Poisson weights. More precisely, one has
fx Pf xx
r
j
j
rj
′
=
∞

+
()
=
() ()
≥
∑
χ
δ
δδ
;
;,,
2
0
2
0
where
Pe
j
fj
j
j
rj rj
δ
δ
χ
δ
()
=
()
=

−
++
2
2
2
22
2
2
0
!
, , and is the p.d.f. of 1,
II.3 Noncentral
F
-Distribution
In Chapter 9, Application 2, the F-distribution with r
1
and r
2
d.f. was defined as
the distribution of the r.v.
F
Xr
Yr
=
1
2
,
where X and Y were independent r.v.’s distributed as
χ
2

r
1
and
χ
2
r
2
, respectively.
Suppose now that the r.v.’s X and Y are independent and distributed as
χ
′
2
r
1
:
δ
and
χ
2
r
2
, respectively, and set
′
=F
Xr
Yr
1
2
.
Then the distribution of F′ is said to be the noncentral F-distribution with r

1
and
r
2
d.f. and noncentrality parameter
δ
. This distribution, and also an r.v. having
this distribution, is often denoted by F′
r
1
,r
2
;
δ
, and its p.d.f., which does not have
any simple closed form, is given by the following expression:
ff e c
j
f
f
f
Fj
j
j
rj
rr j
rr
′
−
=

∞
−+
+
()
+
()
=
()
+
()
≥
∑
12
2
1
12
2
2
0
1
2
1
1
2
2
1
0
,:
;
!

,,
δ
δ
δ
δ
where
c
rr j
rj r
j
j
=
+
()
+
⎡
⎣
⎢
⎤
⎦
⎥
+
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝

⎜
⎞
⎠
⎟
=
Γ
ΓΓ
1
2
1
2
1
2
01
12
12
, , ,
510 Appendix II Noncentral
t
,
χχ
χχ
χ
2
and
F
Distributions
REMARKS
(i) By setting
δ

= 0 in the noncentral t,
χ
2
and F-distributions, we obtain the
t,
χ
2
and F-distributions, respectively. In view of this, the latter distribu-
tions may also be called central t,
χ
2
and F-distributions.
(ii) Tables for the noncentral t,
χ
2
and F-distributions are given in a reference
cited elsewhere, namely, Handbook of Statistical Tables by D. B. Owen.
Addison-Wesley, 1962.
Tables 511
511
Appendix III
Tables
Table 1 The Cumulative Binomial Distribution
The tabulated quantity is
j
k
j
nj
n
j

pp
=
−
∑
⎛
⎝
⎜
⎞
⎠
⎟
−
(
)
0
1.
p
nk1/16 2/16 3/16 4/16 5/16 6/16 7/16 8/16
2 0 0.8789 0.7656 0.6602 0.5625 0.4727 0.3906 0.3164 0.2500
1 0.9961 0.9844 0.9648 0.9375 0.9023 0.8594 0.8086 0.7500
2 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
3 0 0.8240 0.6699 0.5364 0.4219 0.3250 0.2441 0.1780 0.1250
1 0.9888 0.9570 0.9077 0.8437 0.7681 0.6836 0.5933 0.5000
2 0.9998 0.9980 0.9934 0.9844 0.9695 0.9473 0.9163 0.8750
3 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
4 0 0.7725 0.5862 0.4358 0.3164 0.2234 0.1526 0.1001 0.0625
1 0.9785 0.9211 0.8381 0.7383 0.6296 0.5188 0.4116 0.3125
2 0.9991 0.9929 0.9773 0.9492 0.9065 0.8484 0.7749 0.6875
3 1.0000 0.9998 0.9988 0.9961 0.9905 0.9802 0.9634 0.9375
4 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
5 0 0.7242 0.5129 0.3541 0.2373 0.1536 0.0954 0.0563 0.0312

1 0.9656 0.8793 0.7627 0.6328 0.5027 0.3815 0.2753 0.1875
2 0.9978 0.9839 0.9512 0.8965 0.8200 0.7248 0.6160 0.5000
3 0.9999 0.9989 0.9947 0.9844 0.9642 0.9308 0.8809 0.8125
4 1.0000 1.0000 0.9998 0.9990 0.9970 0.9926 0.9840 0.9687
5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
512 Appendix III Tables
Table 1
(continued)
p
nk1/16 2/16 3/16 4/16 5/16 6/16 7/16 8/16
6 0 0.6789 0.4488 0.2877 0.1780 0.1056 0.0596 0.0317 0.0156
1 0.9505 0.8335 0.6861 0.5339 0.3936 0.2742 0.1795 0.1094
2 0.9958 0.9709 0.9159 0.8306 0.7208 0.5960 0.4669 0.3437
3 0.9998 0.9970 0.9866 0.9624 0.9192 0.8535 0.7650 0.6562
4 1.0000 0.9998 0.9988 0.9954 0.9868 0.9694 0.9389 0.8906
5 1.0000 1.0000 1.0000 0.9998 0.9991 0.9972 0.9930 0.9844
6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
7 0 0.6365 0.3927 0.2338 0.1335 0.0726 0.0373 0.0178 0.0078
1 0.9335 0.7854 0.6114 0.4449 0.3036 0.1937 0.1148 0.0625
2 0.9929 0.9537 0.8728 0.7564 0.6186 0.4753 0.3412 0.2266
3 0.9995 0.9938 0.9733 0.9294 0.8572 0.7570 0.6346 0.5000
4 1.0000 0.9995 0.9965 0.9871 0.9656 0.9260 0.8628 0.7734
5 1.0000 1.0000 0.9997 0.9987 0.9952 0.9868 0.9693 0.9375
6 1.0000 1.0000 1.0000 0.9999 0.9997 0.9990 0.9969 0.9922
7 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
8 0 0.5967 0.3436 0.1899 0.1001 0.0499 0.0233 0.0100 0.0039
1 0.9150 0.7363 0.5406 0.3671 0.2314 0.1350 0.0724 0.0352
2 0.9892 0.9327 0.8238 0.6785 0.5201 0.3697 0.2422 0.1445
3 0.9991 0.9888 0.9545 0.8862 0.7826 0.6514 0.5062 0.3633
4 1.0000 0.9988 0.9922 0.9727 0.9318 0.8626 0.7630 0.6367

5 1.0000 0.9999 0.9991 0.9958 0.9860 0.9640 0.9227 0.8555
6 1.0000 1.0000 0.9999 0.9996 0.9983 0.9944 0.9849 0.9648
7 1.0000 1.0000 1.0000 1.0000 0.9999 0.9996 0.9987 0.9961
8 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
9 0 0.5594 0.3007 0.1543 0.0751 0.0343 0.0146 0.0056 0.0020
1 0.8951 0.6872 0.4748 0.3003 0.1747 0.0931 0.0451 0.0195
2 0.9846 0.9081 0.7707 0.6007 0.4299 0.2817 0.1679 0.0898
3 0.9985 0.9817 0.9300 0.8343 0.7006 0.5458 0.3907 0.2539
4 0.9999 0.9975 0.9851 0.9511 0.8851 0.7834 0.6506 0.5000
5 1.0000 0.9998 0.9978 0.9900 0.9690 0.9260 0.8528 0.7461
6 1.0000 1.0000 0.9998 0.9987 0.9945 0.9830 0.9577 0.9102
7 1.0000 1.0000 1.0000 0.9999 0.9994 0.9977 0.9926 0.9805
8 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9994 0.9980
9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
10 0 0.5245 0.2631 0.1254 0.0563 0.0236 0.0091 0.0032 0.0010
1 0.8741 0.6389 0.4147 0.2440 0.1308 0.0637 0.0278 0.0107
2 0.9790 0.8805 0.7152 0.5256 0.3501 0.2110 0.1142 0.0547
3 0.9976 0.9725 0.9001 0.7759 0.6160 0.4467 0.2932 0.1719
4 0.9998 0.9955 0.9748 0.9219 0.8275 0.6943 0.5369 0.3770
5 1.0000 0.9995 0.9955 0.9803 0.9428 0.8725 0.7644 0.6230
6 1.0000 1.0000 0.9994 0.9965 0.9865 0.9616 0.9118 0.8281
7 1.0000 1.0000 1.0000 0.9996 0.9979 0.9922 0.9773 0.9453
8 1.0000 1.0000 1.0000 1.0000 0.9998 0.9990 0.9964 0.9893
Tables 513
Table 1
(continued)
p
nk1/16 2/16 3/16 4/16 5/16 6/16 7/16 8/16
10 9 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9990
10 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

11 0 0.4917 0.2302 0.1019 0.0422 0.0162 0.0057 0.0018 0.0005
1 0.8522 0.5919 0.3605 0.1971 0.0973 0.0432 0.0170 0.0059
2 0.9724 0.8503 0.6589 0.4552 0.2816 0.1558 0.0764 0.0327
3 0.9965 0.9610 0.8654 0.7133 0.5329 0.3583 0.2149 0.1133
4 0.9997 0.9927 0.9608 0.8854 0.7614 0.6014 0.4303 0.2744
5 1.0000 0.9990 0.9916 0.9657 0.9068 0.8057 0.6649 0.5000
6 1.0000 0.9999 0.9987 0.9924 0.9729 0.9282 0.8473 0.7256
7 1.0000 1.0000 0.9999 0.9988 0.9943 0.9807 0.9487 0.8867
8 1.0000 1.0000 1.0000 0.9999 0.9992 0.9965 0.9881 0.9673
9 1.0000 1.0000 1.0000 1.0000 0.9999 0.9996 0.9983 0.9941
10 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9995
11 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
12 0 0.4610 0.2014 0.0828 0.0317 0.0111 0.0036 0.0010 0.0002
1 0.8297 0.5467 0.3120 0.1584 0.0720 0.0291 0.0104 0.0032
2 0.9649 0.8180 0.6029 0.3907 0.2240 0.1135 0.0504 0.0193
3 0.9950 0.9472 0.8267 0.6488 0.4544 0.2824 0.1543 0.0730
4 0.9995 0.9887 0.9429 0.8424 0.6900 0.5103 0.3361 0.1938
5 1.0000 0.9982 0.9858 0.9456 0.8613 0.7291 0.5622 0.3872
6 1.0000 0.9998 0.9973 0.9857 0.9522 0.8822 0.7675 0.6128
7 1.0000 1.0000 0.9996 0.9972 0.9876 0.9610 0.9043 0.8062
8 1.0000 1.0000 1.0000 0.9996 0.9977 0.9905 0.9708 0.9270
9 1.0000 1.0000 1.0000 1.0000 0.9997 0.9984 0.9938 0.9807
10 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9992 0.9968
11 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998
12 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
13 0 0.4321 0.1762 0.0673 0.0238 0.0077 0.0022 0.0006 0.0001
1 0.8067 0.5035 0.2690 0.1267 0.0530 0.0195 0.0063 0.0017
2 0.9565 0.7841 0.5484 0.3326 0.1765 0.0819 0.0329 0.0112
3 0.9931 0.9310 0.7847 0.5843 0.3824 0.2191 0.1089 0.0461
4 0.9992 0.9835 0.9211 0.7940 0.6164 0.4248 0.2565 0.1334

5 0.9999 0.9970 0.9778 0.9198 0.8078 0.6470 0.4633 0.2905
6 1.0000 0.9996 0.9952 0.9757 0.9238 0.8248 0.6777 0.5000
7 1.0000 1.0000 0.9992 0.9944 0.9765 0.9315 0.8445 0.7095
8 1.0000 1.0000 0.9999 0.9990 0.9945 0.9795 0.9417 0.8666
9 1.0000 1.0000 1.0000 0.9999 0.9991 0.9955 0.9838 0.9539
10 1.0000 1.0000 1.0000 1.0000 0.9999 0.9993 0.9968 0.9888
11 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9996 0.9983
12 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999
13 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
14 0 0.4051 0.1542 0.0546 0.0178 0.0053 0.0014 0.0003 0.0001
1 0.7833 0.4626 0.2312 0.1010 0.0388 0.0130 0.0038 0.0009
514 Appendix III Tables
Table 1
(continued)
p
nk1/16 2/16 3/16 4/16 5/16 6/16 7/16 8/16
14 2 0.9471 0.7490 0.4960 0.2811 0.1379 0.0585 0.0213 0.0065
3 0.9908 0.9127 0.7404 0.5213 0.3181 0.1676 0.0756 0.0287
4 0.9988 0.9970 0.8955 0.7415 0.5432 0.3477 0.1919 0.0898
5 0.9999 0.9953 0.9671 0.8883 0.7480 0.5637 0.3728 0.2120
6 1.0000 0.9993 0.9919 0.9167 0.8876 0.7581 0.5839 0.3953
7 1.0000 0.9999 0.9985 0.9897 0.9601 0.8915 0.7715 0.6047
8 1.0000 1.0000 0.9998 0.9978 0.9889 0.9615 0.8992 0.7880
9 1.0000 1.0000 1.0000 0.9997 0.9976 0.9895 0.9654 0.9102
10 1.0000 1.0000 1.0000 1.0000 0.9996 0.9979 0.9911 0.9713
11 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9984 0.9935
12 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9991
13 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999
14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
15 0 0.3798 0.1349 0.0444 0.0134 0.0036 0.0009 0.0002 0.0000

1 0.7596 0.4241 0.1981 0.0802 0.0283 0.0087 0.0023 0.0005
2 0.9369 0.7132 0.4463 0.2361 0.1069 0.0415 0.0136 0.0037
3 0.9881 0.8922 0.6946 0.4613 0.2618 0.1267 0.0518 0.0176
4 0.9983 0.9689 0.8665 0.6865 0.4729 0.2801 0.1410 0.0592
5 0.9998 0.9930 0.9537 0.8516 0.6840 0.4827 0.2937 0.1509
6 1.0000 0.9988 0.9873 0.9434 0.8435 0.6852 0.4916 0.3036
7 1.0000 0.9998 0.9972 0.9827 0.9374 0.8415 0.6894 0.5000
8 1.0000 1.0000 0.9995 0.9958 0.9799 0.9352 0.8433 0.6964
9 1.0000 1.0000 0.9999 0.9992 0.9949 0.9790 0.9364 0.8491
10 1.0000 1.0000 1.0000 0.9999 0.9990 0.9947 0.9799 0.9408
11 1.0000 1.0000 1.0000 1.0000 0.9999 0.9990 0.9952 0.9824
12 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9992 0.9963
13 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9995
14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
16 0 0.3561 0.1181 0.0361 0.0100 0.0025 0.0005 0.0001 0.0000
1 0.7359 0.3879 0.1693 0.0635 0.0206 0.0057 0.0014 0.0003
2 0.9258 0.6771 0.3998 0.1971 0.0824 0.0292 0.0086 0.0021
3 0.9849 0.8698 0.6480 0.4050 0.2134 0.0947 0.0351 0.0106
4 0.9977 0.9593 0.8342 0.6302 0.4069 0.2226 0.1020 0.0384
5 0.9997 0.9900 0.9373 0.8103 0.6180 0.4067 0.2269 0.1051
6 1.0000 0.9981 0.9810 0.9204 0.7940 0.6093 0.4050 0.2272
7 1.0000 0.9997 0.9954 0.9729 0.9082 0.7829 0.6029 0.4018
8 1.0000 1.0000 0.9991 0.9925 0.9666 0.9001 0.7760 0.5982
9 1.0000 1.0000 0.9999 0.9984 0.9902 0.9626 0.8957 0.7728
10 1.0000 1.0000 1.0000 0.9997 0.9977 0.9888 0.9609 0.8949
11 1.0000 1.0000 1.0000 1.0000 0.9996 0.9974 0.9885 0.9616
12 1.0000 1.0000 1.0000 1.0000 0.9999 0.9995 0.9975 0.9894
13 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9996 0.9979
Tables 515

Table 1
(continued)
p
nk1/16 2/16 3/16 4/16 5/16 6/16 7/16 8/16
16 14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997
15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
16 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
17 0 0.3338 0.1033 0.0293 0.0075 0.0017 0.0003 0.0001 0.0000
1 0.7121 0.3542 0.1443 0.0501 0.0149 0.0038 0.0008 0.0001
2 0.9139 0.6409 0.3566 0.1637 0.0631 0.0204 0.0055 0.0012
3 0.9812 0.8457 0.6015 0.3530 0.1724 0.0701 0.0235 0.0064
4 0.9969 0.9482 0.7993 0.5739 0.3464 0.1747 0.0727 0.0245
5 0.9996 0.9862 0.9180 0.7653 0.5520 0.3377 0.1723 0.0717
6 1.0000 0.9971 0.9728 0.8929 0.7390 0.5333 0.3271 0.1662
7 1.0000 0.9995 0.9927 0.9598 0.8725 0.7178 0.5163 0.3145
8 1.0000 0.9999 0.9984 0.9876 0.9484 0.8561 0.7002 0.5000
9 1.0000 1.0000 0.9997 0.9969 0.9828 0.9391 0.8433 0.6855
10 1.0000 1.0000 1.0000 0.9994 0.9954 0.9790 0.9323 0.8338
11 1.0000 1.0000 1.0000 0.9999 0.9990 0.9942 0.9764 0.9283
12 1.0000 1.0000 1.0000 1.0000 0.9998 0.9987 0.9935 0.9755
13 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9987 0.9936
14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9988
15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999
16 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
18 0 0.3130 0.0904 0.0238 0.0056 0.0012 0.0002 0.0000 0.0000
1 0.6885 0.3228 0.1227 0.0395 0.0108 0.0025 0.0005 0.0001
2 0.9013 0.6051 0.3168 0.1353 0.0480 0.0142 0.0034 0.0007
3 0.9770 0.8201 0.5556 0.3057 0.1383 0.0515 0.0156 0.0038
4 0.9959 0.9354 0.7622 0.5187 0.2920 0.1355 0.0512 0.0154
5 0.9994 0.9814 0.8958 0.7175 0.4878 0.2765 0.1287 0.0481

6 0.9999 0.9957 0.9625 0.8610 0.6806 0.4600 0.2593 0.1189
7 1.0000 0.9992 0.9889 0.9431 0.8308 0.6486 0.4335 0.2403
8 1.0000 0.9999 0.9973 0.9807 0.9247 0.8042 0.6198 0.4073
9 1.0000 1.0000 0.9995 0.9946 0.9721 0.9080 0.7807 0.5927
10 1.0000 1.0000 0.9999 0.9988 0.9915 0.9640 0.8934 0.7597
11 1.0000 1.0000 1.0000 0.9998 0.9979 0.9885 0.9571 0.8811
12 1.0000 1.0000 1.0000 1.0000 0.9996 0.9970 0.9860 0.9519
13 1.0000 1.0000 1.0000 1.0000 0.9999 0.9994 0.9964 0.9846
14 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9993 0.9962
15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9993
16 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999
17 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
19 0 0.2934 0.0791 0.0193 0.0042 0.0008 0.0001 0.0000 0.0000
1 0.6650 0.2938 0.1042 0.0310 0.0078 0.0016 0.0003 0.0000
2 0.8880 0.5698 0.2804 0.1113 0.0364 0.0098 0.0021 0.0004
3 0.9722 0.7933 0.5108 0.2631 0.1101 0.0375 0.0103 0.0022
4 0.9947 0.9209 0.7235 0.4654 0.2440 0.1040 0.0356 0.0096
516 Appendix III Tables
Table 1
(continued)
p
nk1/16 2/16 3/16 4/16 5/16 6/16 7/16 8/16
19 5 0.9992 0.9757 0.8707 0.6678 0.4266 0.2236 0.0948 0.0318
6 0.9999 0.9939 0.9500 0.8251 0.6203 0.3912 0.2022 0.0835
7 1.0000 0.9988 0.9840 0.9225 0.7838 0.5779 0.3573 0.1796
8 1.0000 0.9998 0.9957 0.9713 0.8953 0.7459 0.5383 0.3238
9 1.0000 1.0000 0.9991 0.9911 0.9573 0.8691 0.7103 0.5000
10 1.0000 1.0000 0.9998 0.9977 0.9854 0.9430 0.8441 0.0672
11 1.0000 1.0000 1.0000 0.9995 0.9959 0.9793 0.9292 0.8204
12 1.0000 1.0000 1.0000 0.9999 0.9990 0.9938 0.9734 0.9165

13 1.0000 1.0000 1.0000 1.0000 0.9998 0.9985 0.9919 0.9682
14 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9980 0.9904
15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9996 0.9978
16 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9996
17 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
18 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
20 0 0.2751 0.0692 0.0157 0.0032 0.0006 0.0001 0.0000 0.0000
1 0.6148 0.2669 0.0883 0.0243 0.0056 0.0011 0.0002 0.0000
2 0.8741 0.5353 0.2473 0.0913 0.0275 0.0067 0.0013 0.0002
3 0.9670 0.7653 0.4676 0.2252 0.0870 0.0271 0.0067 0.0013
4 0.9933 0.9050 0.6836 0.4148 0.2021 0.0790 0.0245 0.0059
5 0.9989 0.9688 0.8431 0.6172 0.3695 0.1788 0.0689 0.0207
6 0.9999 0.9916 0.9351 0.7858 0.5598 0.3284 0.1552 0.0577
7 1.0000 0.9981 0.9776 0.8982 0.7327 0.5079 0.2894 0.1316
8 1.0000 0.9997 0.9935 0.9591 0.8605 0.6829 0.4591 0.2517
9 1.0000 0.9999 0.9984 0.9861 0.9379 0.8229 0.6350 0.4119
10 1.0000 1.0000 0.9997 0.9961 0.9766 0.9153 0.7856 0.5881
11 1.0000 1.0000 0.9999 0.9991 0.9926 0.9657 0.8920 0.7483
12 1.0000 1.0000 1.0000 0.9998 0.9981 0.9884 0.9541 0.8684
13 1.0000 1.0000 1.0000 1.0000 0.9996 0.9968 0.9838 0.9423
14 1.0000 1.0000 1.0000 1.0000 0.9999 0.9993 0.9953 0.9793
15 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9989 0.9941
16 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9987
17 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998
18 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
19 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
21 0 0.2579 0.0606 0.0128 0.0024 0.0004 0.0001 0.0000 0.0000
1 0.6189 0.2422 0.0747 0.0190 0.0040 0.0007 0.0001 0.0000
2 0.8596 0.5018 0.2175 0.0745 0.0206 0.0046 0.0008 0.0001
3 0.9612 0.7366 0.4263 0.1917 0.0684 0.0195 0.0044 0.0007

4 0.9917 0.8875 0.6431 0.3674 0.1662 0.0596 0.0167 0.0036
5 0.9986 0.9609 0.8132 0.5666 0.3172 0.1414 0.0495 0.0133
6 0.9998 0.9888 0.9179 0.7436 0.5003 0.2723 0.1175 0.0392
7 1.0000 0.9973 0.9696 0.8701 0.6787 0.4405 0.2307 0.0946
8 1.0000 0.9995 0.9906 0.9439 0.8206 0.6172 0.3849 0.1917
Tables 517
Table 1
(continued)
p
nk1/16 2/16 3/16 4/16 5/16 6/16 7/16 8/16
21 9 1.0000 0.9999 0.9975 0.9794 0.9137 0.7704 0.5581 0.3318
10 1.0000 1.0000 0.9995 0.9936 0.9645 0.8806 0.7197 0.5000
11 1.0000 1.0000 0.9999 0.9983 0.9876 0.9468 0.8454 0.6682
12 1.0000 1.0000 1.0000 0.9996 0.9964 0.9799 0.9269 0.8083
13 1.0000 1.0000 1.0000 0.9999 0.9991 0.9936 0.9708 0.9054
14 1.0000 1.0000 1.0000 1.0000 0.9998 0.9983 0.9903 0.9605
15 1.0000 1.0000 1.0000 1.0000 1.0000 0.9996 0.9974 0.9867
16 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9994 0.9964
17 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9993
18 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999
19 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
20 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
22 0 0.2418 0.0530 0.0104 0.0018 0.0003 0.0000 0.0000 0.0000
1 0.5963 0.2195 0.0631 0.0149 0.0029 0.0005 0.0001 0.0000
2 0.8445 0.4693 0.1907 0.0606 0.0154 0.0031 0.0005 0.0001
3 0.9548 0.7072 0.3871 0.1624 0.0535 0.0139 0.0028 0.0004
4 0.9898 0.8687 0.6024 0.3235 0.1356 0.0445 0.0133 0.0022
5 0.9981 0.9517 0.7813 0.5168 0.2700 0.1107 0.0352 0.0085
6 0.9997 0.9853 0.8983 0.6994 0.4431 0.2232 0.0877 0.0267
7 1.0000 0.9963 0.9599 0.8385 0.6230 0.3774 0.1812 0.0669

8 1.0000 0.9992 0.9866 0.9254 0.7762 0.5510 0.3174 0.1431
9 1.0000 0.9999 0.9962 0.9705 0.8846 0.7130 0.4823 0.2617
10 1.0000 1.0000 0.9991 0.9900 0.9486 0.8393 0.6490 0.4159
11 1.0000 1.0000 0.9998 0.9971 0.9804 0.9220 0.7904 0.5841
12 1.0000 1.0000 1.0000 0.9993 0.9936 0.9675 0.8913 0.7383
13 1.0000 1.0000 1.0000 0.9999 0.9982 0.9885 0.9516 0.8569
14 1.0000 1.0000 1.0000 1.0000 0.9996 0.9966 0.9818 0.9331
15 1.0000 1.0000 1.0000 1.0000 0.9999 0.9991 0.9943 0.9739
16 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9985 0.9915
17 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9978
18 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9995
19 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999
20 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
23 0 0.2266 0.0464 0.0084 0.0013 0.0002 0.0000 0.0000 0.0000
1 0.5742 0.1987 0.0532 0.0116 0.0021 0.0003 0.0000 0.0000
2 0.8290 0.4381 0.1668 0.0492 0.0115 0.0021 0.0003 0.0000
3 0.9479 0.6775 0.3503 0.1370 0.0416 0.0099 0.0018 0.0002
4 0.9876 0.8485 0.5621 0.2832 0.1100 0.0330 0.0076 0.0013
5 0.9976 0.9413 0.7478 0.4685 0.2280 0.0859 0.0247 0.0053
6 0.9996 0.9811 0.8763 0.6537 0.3890 0.1810 0.0647 0.0173
7 1.0000 0.9949 0.9484 0.8037 0.5668 0.3196 0.1403 0.0466
8 1.0000 0.9988 0.9816 0.9037 0.7283 0.4859 0.2578 0.1050
9 1.0000 0.9998 0.9944 0.9592 0.8507 0.6522 0.4102 0.2024
518 Appendix III Tables
Table 1
(continued)
p
nk1/16 2/16 3/16 4/16 5/16 6/16 7/16 8/16
23 10 1.0000 1.0000 0.9986 0.9851 0.9286 0.7919 0.5761 0.3388
11 1.0000 1.0000 0.9997 0.9954 0.9705 0.8910 0.7285 0.5000

12 1.0000 1.0000 0.9999 0.9988 0.9895 0.9504 0.8471 0.6612
13 1.0000 1.0000 1.0000 0.9997 0.9968 0.9806 0.9252 0.7976
14 1.0000 1.0000 1.0000 0.9999 0.9992 0.9935 0.9686 0.8950
15 1.0000 1.0000 1.0000 1.0000 0.9998 0.9982 0.9888 0.9534
16 1.0000 1.0000 1.0000 1.0000 1.0000 0.9996 0.9967 0.9827
17 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9992 0.9947
18 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9987
19 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998
20 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
21 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
24 0 0.2125 0.0406 0.0069 0.0010 0.0001 0.0000 0.0000 0.0000
1 0.5524 0.1797 0.0448 0.0090 0.0015 0.0002 0.0000 0.0000
2 0.8131 0.4082 0.1455 0.0398 0.0086 0.0014 0.0002 0.0000
3 0.9405 0.6476 0.3159 0.1150 0.0322 0.0070 0.0011 0.0001
4 0.9851 0.8271 0.5224 0.2466 0.0886 0.0243 0.0051 0.0008
5 0.9970 0.9297 0.7130 0.4222 0.1911 0.0661 0.0172 0.0033
6 0.9995 0.9761 0.8522 0.6074 0.3387 0.1453 0.0472 0.0113
7 0.9999 0.9932 0.9349 0.7662 0.5112 0.2676 0.1072 0.0320
8 1.0000 0.9983 0.9754 0.8787 0.6778 0.4235 0.2064 0.0758
9 1.0000 0.9997 0.9920 0.9453 0.8125 0.5898 0.3435 0.1537
10 1.0000 0.9999 0.9978 0.9787 0.9043 0.7395 0.5035 0.2706
11 1.0000 1.0000 0.9995 0.9928 0.9574 0.8538 0.6618 0.4194
12 1.0000 1.0000 0.9999 0.9979 0.9835 0.9281 0.7953 0.5806
13 1.0000 1.0000 1.0000 0.9995 0.9945 0.9693 0.8911 0.7294
14 1.0000 1.0000 1.0000 0.9999 0.9984 0.9887 0.9496 0.8463
15 1.0000 1.0000 1.0000 1.0000 0.9996 0.9964 0.9799 0.9242
16 1.0000 1.0000 1.0000 1.0000 0.9999 0.9990 0.9932 0.9680
17 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9981 0.9887
18 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9996 0.9967
19 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9992

20 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999
21 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
22 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
25 0 0.1992 0.0355 0.0056 0.0008 0.0001 0.0000 0.0000 0.0000
1 0.5132 0.1623 0.0377 0.0070 0.0011 0.0001 0.0000 0.0000
2 0.7968 0.3796 0.1266 0.0321 0.0064 0.0010 0.0001 0.0000
3 0.9325 0.6176 0.2840 0.0962 0.0248 0.0049 0.0007 0.0001
4 0.9823 0.8047 0.4837 0.2137 0.0710 0.0178 0.0033 0.0005
5 0.9962 0.9169 0.6772 0.3783 0.1591 0.0504 0.0119 0.0028
6 0.9993 0.9703 0.8261 0.5611 0.2926 0.1156 0.0341 0.0073
7 0.9999 0.9910 0.9194 0.7265 0.4573 0.2218 0.0810 0.0216
Tables 519
Table 1
(continued)
p
nk1/16 2/16 3/16 4/16 5/16 6/16 7/16 8/16
25 8 1.0000 0.9977 0.9678 0.8506 0.6258 0.3651 0.1630 0.0539
9 1.0000 0.9995 0.9889 0.9287 0.7704 0.5275 0.2835 0.1148
10 1.0000 0.9999 0.9967 0.9703 0.8756 0.6834 0.4335 0.2122
11 1.0000 1.0000 0.9992 0.9893 0.9408 0.8110 0.5926 0.3450
12 1.0000 1.0000 0.9998 0.9966 0.9754 0.9003 0.7369 0.5000
13 1.0000 1.0000 1.0000 0.9991 0.9911 0.9538 0.8491 0.6550
14 1.0000 1.0000 1.0000 0.9998 0.9972 0.9814 0.9240 0.7878
15 1.0000 1.0000 1.0000 1.0000 0.9992 0.9935 0.9667 0.8852
16 1.0000 1.0000 1.0000 1.0000 0.9998 0.9981 0.9874 0.9462
17 1.0000 1.0000 1.0000 1.0000 1.0000 0.9995 0.9960 0.9784
18 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9989 0.9927
19 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9980
20 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9995
21 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999

22 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
520 Appendix III Tables
Table 2 The Cumulative Poisson Distribution
The tabulated quantity is
j
k
j
e
j
=
−
∑
0
λ
λ
!
.
λλ
λλ
λ
k 0.001 0.005 0.010 0.015 0.020 0.025
0 0.9990 0050 0.9950 1248 0.9900 4983 0.9851 1194 0.9801 9867 0.9753 099
1 0.9999 9950 0.9999 8754 0.9999 5033 0.9998 8862 0.9998 0264 0.9996 927
2 1.0000 0000 0.9999 9998 0.9999 9983 0.9999 9945 0.9999 9868 0.9999 974
3 1.0000 0000 1.0000 0000 1.0000 0000 0.9999 9999 1.0000 000
4 1.0000 0000 1.0000 000
λλ
λλ
λ
k 0.030 0.035 0.040 0.045 0.050 0.055

0 0.970 446 0.965 605 0.960 789 0.955 997 0.951 229 0.946 485
1 0.999 559 0.999 402 0.999 221 0.999 017 0.998 791 0.998 542
2 0.999 996 0.999 993 0.999 990 0.999 985 0.999 980 0.999 973
3 1.000 000 1.000 000 1.000 000 1.000 000 1.000 000 1.000 000
λλ
λλ
λ
k 0.060 0.065 0.070 0.075 0.080 0.085
0 0.941 765 0.937 067 0.932 394 0.927 743 0.923 116 0.918 512
1 0.998 270 0.997 977 0.997 661 0.997 324 0.996 966 0.996 586
2 0.999 966 0.999 956 0.999 946 0.999 934 0.999 920 0.999 904
3 0.999 999 0.999 999 0.999 999 0.999 999 0.999 998 0.999 998
4 1.000 000 1.000 000 1.000 000 1.000 000 1.000 000 1.000 000
λλ
λλ
λ
k 0.090 0.095 0.100 0.200 0.300 0.400
0 0.913 931 0.909 373 0.904 837 0.818 731 0.740 818 0.670 320
1 0.996 185 0.995 763 0.995 321 0.982 477 0.963 064 0.938 448
2 0.999 886 0.999 867 0.999 845 0.998 852 0.996 401 0.992 074
3 0.999 997 0.999 997 0.999 996 0.999 943 0.999 734 0.999 224
4 1.000 000 1.000 000 1.000 000 0.999 998 0.999 984 0.999 939
5 1.000 000 0.999 999 0.999 996
6 1.000 000 1.000 000
λλ
λλ
λ
k 0.500 0.600 0.700 0.800 0.900 1.000
0 0.606 531 0.548 812 0.496 585 0.449 329 0.406 329 0.367 879
1 0.909 796 0.878 099 0.844 195 0.808 792 0.772 482 0.735 759

2 0.985 612 0.976 885 0.965 858 0.952 577 0.937 143 0.919 699
Tables 521
Table 2
(continued)
λλ
λλ
λ
k 0.500 0.600 0.700 0.800 0.900 1.000
3 0.998 248 0.996 642 0.994 247 0.990 920 0.986 541 0.981 012
4 0.999 828 0.999 606 0.999 214 0.998 589 0.997 656 0.996 340
5 0.999 986 0.999 961 0.999 910 0.999 816 0.999 657 0.999 406
6 0.999 999 0.999 997 0.999 991 0.999 979 0.999 957 0.999 917
7 1.000 000 1.000 000 0.999 999 0.999 998 0.999 995 0.999 990
8 1.000 000 1.000 000 1.000 000 0.999 999
9 1.000 000
λλ
λλ
λ
k 1.20 1.40 1.60 1.80 2.00 2.50 3.00 3.50
0 0.3012 0.2466 0.2019 0.1653 0.1353 0.0821 0.0498 0.0302
1 0.6626 0.5918 0.5249 0.4628 0.4060 0.2873 0.1991 0.1359
2 0.8795 0.8335 0.7834 0.7306 0.6767 0.5438 0.4232 0.3208
3 0.9662 0.9463 0.9212 0.8913 0.8571 0.7576 0.6472 0.5366
4 0.9923 0.9857 0.9763 0.9636 0.9473 0.8912 0.8153 0.7254
5 0.9985 0.9968 0.9940 0.9896 0.9834 0.9580 0.9161 0.8576
6 0.9997 0.9994 0.9987 0.9974 0.9955 0.9858 0.9665 0.9347
7 1.0000 0.9999 0.9997 0.9994 0.9989 0.9958 0.9881 0.9733
8 1.0000 1.0000 0.9999 0.9998 0.9989 0.9962 0.9901
9 1.0000 1.0000 0.9997 0.9989 0.9967
10 0.9999 0.9997 0.9990

11 1.0000 0.9999 0.9997
12 1.0000 0.9999
13 1.0000
λλ
λλ
λ
k 4.00 4.50 5.00 6.00 7.00 8.00 9.00 10.00
0 0.0183 0.0111 0.0067 0.0025 0.0009 0.0003 0.0001 0.0000
1 0.0916 0.0611 0.0404 0.0174 0.0073 0.0030 0.0012 0.0005
2 0.2381 0.1736 0.1247 0.0620 0.0296 0.0138 0.0062 0.0028
3 0.4335 0.3423 0.2650 0.1512 0.0818 0.0424 0.0212 0.0103
4 0.6288 0.5321 0.4405 0.2851 0.1730 0.0996 0.0550 0.0293
5 0.7851 0.7029 0.6160 0.4457 0.3007 0.1912 0.1157 0.0671
6 0.8893 0.8311 0.7622 0.6063 0.4497 0.3134 0.2068 0.1301
7 0.9489 0.9134 0.8666 0.7440 0.5987 0.4530 0.3239 0.2202
8 0.9786 0.9597 0.9319 0.8472 0.7291 0.5925 0.4577 0.3328
9 0.9919 0.9829 0.9682 0.9161 0.8305 0.7166 0.5874 0.4579
10 0.9972 0.9933 0.9863 0.9574 0.9015 0.8159 0.7060 0.5830
11 0.9991 0.9976 0.9945 0.9799 0.9467 0.8881 0.8030 0.6968
12 0.9997 0.9992 0.9980 0.9912 0.9730 0.9362 0.8758 0.7916
13 0.9999 0.9997 0.9993 0.9964 0.9872 0.9658 0.9261 0.8645
14 1.0000 0.9999 0.9998 0.9986 0.9943 0.9827 0.9585 0.9165
522 Appendix III Tables
Table 2
(continued)
λλ
λλ
λ
k 4.00 4.50 5.00 6.00 7.00 8.00 9.00 10.00
15 1.0000 0.9999 0.9995 0.9976 0.9918 0.9780 0.9513

16 1.0000 0.9998 0.9990 0.9963 0.9889 0.9730
17 0.9999 0.9996 0.9984 0.9947 0.9857
18 1.0000 0.9999 0.9993 0.9976 0.9928
19 0.9997 0.9989 0.9965
20 1.0000 0.9999 0.9996 0.9984
21 1.0000 0.9998 0.9993
22 0.9999 0.9997
23 1.0000 0.9999
24 1.0000
Tables 523
Table 3 The Normal Distribution
The tabulated quantity is
Φ
ΦΦ
xedt
xx
x
t
(
)
=
−
(
)
=−
(
)
[]
−∞
−

∫
1
2
1
2
2
π
.
.
x
ΦΦ
ΦΦ
Φ(x) x
ΦΦ
ΦΦ
Φ(x) x
ΦΦ
ΦΦ
Φ(x) x
ΦΦ
ΦΦ
Φ(x)
0.00 0.500000 0.35 0.636831 0.70 0.758036 1.05 0.853141
0.01 0.503989 0.36 0.640576 0.71 0.761148 1.06 0.855428
0.02 0.507978 0.37 0.644309 0.72 0.764238 1.07 0.857690
0.03 0.511966 0.38 0.648027 0.73 0.767305 1.08 0.859929
0.04 0.515953 0.39 0.651732 0.74 0.770350 1.09 0.862143
0.05 0.519939 0.40 0.655422 0.75 0.773373 1.10 0.864334
0.06 0.523922 0.41 0.659097 0.76 0.776373 1.11 0.866500
0.07 0.527903 0.42 0.662757 0.77 0.779350 1.12 0.868643

0.08 0.531881 0.43 0.666402 0.78 0.782305 1.13 0.870762
0.09 0.535856 0.44 0.670031 0.79 0.785236 1.14 0.872857
0.10 0.539828 0.45 0.673645 0.80 0.788145 1.15 0.874928
0.11 0.543795 0.46 0.677242 0.81 0.791030 1.16 0.876976
0.12 0.547758 0.47 0.680822 0.82 0.793892 1.17 0.879000
0.13 0.551717 0.48 0.684386 0.83 0.796731 1.18 0.881000
0.14 0.555670 0.49 0.687933 0.84 0.799546 1.19 0.882977
0.15 0.559618 0.50 0.691462 0.85 0.802337 1.20 0.884930
0.16 0.563559 0.51 0.694974 0.86 0.805105 1.21 0.886861
0.17 0.567495 0.52 0.698468 0.87 0.807850 1.22 0.888768
0.18 0.571424 0.53 0.701944 0.88 0.810570 1.23 0.890651
0.19 0.575345 0.54 0.705401 0.89 0.813267 1.24 0.892512
0.20 0.579260 0.55 0.708840 0.90 0.815940 1.25 0.894350
0.21 0.583166 0.56 0.712260 0.91 0.818589 1.26 0.896165
0.22 0.587064 0.57 0.715661 0.92 0.821214 1.27 0.897958
0.23 0.590954 0.58 0.719043 0.93 0.823814 1.28 0.899727
0.24 0.594835 0.59 0.722405 0.94 0.826391 1.29 0.901475
0.25 0.598706 0.60 0.725747 0.95 0.828944 1.30 0.903200
0.26 0.602568 0.61 0.279069 0.96 0.831472 1.31 0.904902
0.27 0.606420 0.62 0.732371 0.97 0.833977 1.32 0.906582
0.28 0.610261 0.63 0.735653 0.98 0.836457 1.33 0.908241
0.29 0.614092 0.64 0.738914 0.99 0.838913 1.34 0.909877
0.30 0.617911 0.65 0.742154 1.00 0.841345 1.35 0.911492
0.31 0.621720 0.66 0.745373 1.01 0.843752 1.36 0.913085
0.32 0.625516 0.67 0.748571 1.02 0.846136 1.37 0.914657
0.33 0.629300 0.68 0.751748 1.03 0.848495 1.38 0.916207
0.34 0.633072 0.69 0.754903 1.04 0.850830 1.39 0.917736
524 Appendix III Tables
Table 3
(continued)

x
ΦΦ
ΦΦ
Φ(x) x
ΦΦ
ΦΦ
Φ(x) x
ΦΦ
ΦΦ
Φ(x) x
ΦΦ
ΦΦ
Φ(x)
1.40 0.919243 1.85 0.967843 2.30 0.989276 2.75 0.997020
1.41 0.920730 1.86 0.968557 2.31 0.989556 2.76 0.997110
1.42 0.922196 1.87 0.969258 2.32 0.989830 2.77 0.997197
1.43 0.923641 1.88 0.969946 2.33 0.990097 2.78 0.997282
1.44 0.925066 1.89 0.970621 2.34 0.990358 2.79 0.997365
1.45 0.926471 1.90 0.971283 2.35 0.990613 2.80 0.997445
1.46 0.927855 1.91 0.971933 2.36 0.990863 2.81 0.997523
1.47 0.929219 1.92 0.972571 2.37 0.991106 2.82 0.997599
1.48 0.930563 1.93 0.973197 2.38 0.991344 2.83 0.997673
1.49 0.931888 1.94 0.973810 2.39 0.991576 2.84 0.997744
1.50 0.933193 1.95 0.974412 2.40 0.991802 2.85 0.997814
1.51 0.934478 1.96 0.975002 2.41 0.992024 2.86 0.997882
1.52 0.935745 1.97 0.975581 2.42 0.992240 2.87 0.997948
1.53 0.936992 1.98 0.976148 2.43 0.992451 2.88 0.998012
1.54 0.938220 1.99 0.976705 2.44 0.992656 2.89 0.998074
1.55 0.939429 2.00 0.977250 2.45 0.992857 2.90 0.998134
1.56 0.940620 2.01 0.977784 2.46 0.993053 2.91 0.998193

1.57 0.941792 2.02 0.978308 2.47 0.993244 2.92 0.998250
1.58 0.942947 2.03 0.978822 2.48 0.993431 2.93 0.998305
1.59 0.944083 2.04 0.979325 2.49 0.993613 2.54 0.998359
1.60 0.945201 2.05 0.979818 2.50 0.993790 2.95 0.998411
1.61 0.946301 2.06 0.980301 2.51 0.993963 2.96 0.998462
1.62 0.947384 2.07 0.980774 2.52 0.994132 2.97 0.998511
1.63 0.948449 2.08 0.981237 2.53 0.994297 2.98 0.998559
1.64 0.949497 2.09 0.981691 2.54 0.994457 2.99 0.998605
1.65 0.950529 2.10 0.982136 2.55 0.994614 3.00 0.998650
1.66 0.951543 2.11 0.982571 2.56 0.994766 3.01 0.998694
1.67 0.952540 2.12 0.982997 2.57 0.994915 3.02 0.998736
1.68 0.953521 2.13 0.983414 2.58 0.995060 3.03 0.998777
1.69 0.954486 2.14 0.983823 2.59 0.995201 3.04 0.998817
1.70 0.955435 2.15 0.984222 2.60 0.995339 3.05 0.998856
1.71 0.956367 2.16 0.984614 2.61 0.995473 3.06 0.998893
1.72 0.957284 2.17 0.984997 2.62 0.995604 3.07 0.998930
1.73 0.958185 2.18 0.985371 2.63 0.995731 3.08 0.998965
1.74 0.959070 2.19 0.985738 2.64 0.995855 3.09 0.998999
1.75 0.959941 2.20 0.986097 2.65 0.995975 3.10 0.999032
1.76 0.960796 2.21 0.986447 2.66 0.996093 3.11 0.999065
1.77 0.961636 2.22 0.986791 2.67 0.996207 3.12 0.999096
1.78 0.962462 2.23 0.987126 2.68 0.996319 3.13 0.999126
1.79 0.963273 2.24 0.987455 2.69 0.996427 3.14 0.999155
1.80 0.964070 2.25 0.987776 2.70 0.996533 3.15 0.999184
1.81 0.964852 2.26 0.988089 2.71 0.996636 3.16 0.999211
1.82 0.965620 2.27 0.988396 2.72 0.996736 3.17 0.999238
1.83 0.966375 2.28 0.988696 2.73 0.996833 3.18 0.999264
1.84 0.967116 2.29 0.988989 2.74 0.996928 3.19 0.999289
Tables 525
Table 3

(continued)
x
ΦΦ
ΦΦ
Φ(x) x
ΦΦ
ΦΦ
Φ(x) x
ΦΦ
ΦΦ
Φ(x) x
ΦΦ
ΦΦ
Φ(x)
3.20 0.999313 3.40 0.999663 3.60 0.999841 3.80 0.999928
3.21 0.999336 3.41 0.999675 3.61 0.999847 3.81 0.999931
3.22 0.999359 3.42 0.999687 3.62 0.999853 3.82 0.999933
3.23 0.999381 3.43 0.999698 3.63 0.999858 3.83 0.999936
3.24 0.999402 3.44 0.999709 3.64 0.999864 3.84 0.999938
3.25 0.999423 3.45 0.999720 3.65 0.999869 3.85 0.999941
3.26 0.999443 3.46 0.999730 3.66 0.999874 3.86 0.999943
3.27 0.999462 3.47 0.999740 3.67 0.999879 3.87 0.999946
3.28 0.999481 3.48 0.999749 3.68 0.999883 3.88 0.999948
3.29 0.999499 3.49 0.999758 3.69 0.999888 3.89 0.999950
3.30 0.999517 3.50 0.999767 3.70 0.999892 3.90 0.999952
3.31 0.999534 3.51 0.999776 3.71 0.999896 3.91 0.999954
3.32 0.999550 3.52 0.999784 3.72 0.999900 3.92 0.999956
3.33 0.999566 3.53 0.999792 3.73 0.999904 3.93 0.999958
3.34 0.999581 3.53 0.999800 3.74 0.999908 3.94 0.999959
3.35 0.999596 3.55 0.999807 3.75 0.999912 3.95 0.999961

3.36 0.999610 3.56 0.999815 3.76 0.999915 3.96 0.999963
3.37 0.999624 3.57 0.999822 3.77 0.999918 3.97 0.999964
3.38 0.999638 3.58 0.999828 3.78 0.999922 3.98 0.999966
3.39 0.999651 3.59 0.999835 3.79 0.999925 3.99 0.999967
526 Appendix III Tables
Table 4 Critical Values for Student’s
t
-Distribution
Let t
r
be a random variable having the Student’s t-distribution with r degrees of
freedom. Then the tabulated quantities are the numbers x for which
Pt x
r
≤
()
=γ.
γγ
γγ
γ
r 0.75 0.90 0.95 0.975 0.99 0.995
1 1.0000 3.0777 6.3138 12.7062 31.8207 63.6574
2 0.8165 1.8856 2.9200 4.3027 6.9646 9.9248
3 0.7649 1.6377 2.3534 3.1824 4.5407 5.8409
4 0.7407 1.5332 2.1318 2.7764 3.7649 4.6041
5 0.7267 1.4759 2.0150 2.5706 3.3649 4.0322
6 0.7176 1.4398 1.9432 2.4469 3.1427 3.7074
7 0.7111 1.4149 1.8946 2.3646 2.9980 3.4995
8 0.7064 1.3968 1.8595 3.3060 2.8965 3.3554
9 0.7027 1.3830 1.8331 2.2622 2.8214 3.2498

10 0.6998 1.3722 1.8125 2.2281 2.7638 3.1693
11 0.6974 1.3634 1.7959 2.2010 2.7181 3.1058
12 0.6955 1.3562 1.7823 2.1788 2.6810 3.0545
13 0.6938 1.3502 1.7709 1.1604 2.6503 3.0123
14 0.6924 1.3450 1.7613 2.1448 2.6245 2.9768
15 0.6912 1.3406 1.7531 2.1315 2.6025 2.9467
16 0.6901 1.3368 1.7459 2.1199 2.5835 2.9208
17 0.6892 1.3334 1.7396 2.1098 2.5669 2.8982
18 0.6884 1.3304 1.7341 2.1009 2.5524 2.8784
19 0.6876 1.3277 1.7291 2.0930 2.5395 2.8609
20 0.6870 1.3253 1.7247 2.0860 2.5280 2.8453
21 0.6864 1.3232 1.7207 2.0796 2.5177 2.8314
22 0.6858 1.3212 1.7171 2.0739 2.5083 2.8188
23 0.6853 1.3195 1.7139 2.0687 2.4999 2.8073
24 0.6848 1.3178 1.7109 2.0639 2.4922 2.7969
25 0.6844 1.3163 1.7081 2.0595 2.4851 2.7874
26 0.6840 1.3150 1.7056 2.0555 2.4786 2.7787
27 0.6837 1.3137 1.7033 2.0518 2.4727 2.7707
28 0.6834 1.3125 1.7011 2.0484 2.4671 2.7633
29 0.6830 1.3114 1.6991 2.0452 2.4620 2.7564
30 0.6828 1.3104 1.6973 2.0423 2.4573 2.7500
31 0.6825 1.3095 1.6955 2.0395 2.4528 2.7440
32 0.6822 1.3086 1.6939 2.0369 2.4487 2.7385
33 0.6820 1.3077 1.6924 2.0345 2.4448 2.7333
34 0.6818 1.3070 1.6909 2.0322 2.4411 2.7284
35 0.6816 1.3062 1.6896 2.0301 2.4377 2.7238
36 0.6814 1.3055 1.6883 2.0281 2.4345 2.7195
37 0.6812 1.3049 1.6871 2.0262 2.4314 1.7154
38 0.6810 1.3042 1.6860 2.0244 2.4286 2.7116
Tables 527

Table 4
(continued)
γγ
γγ
γ
r 0.75 0.90 0.95 0.975 0.99 0.995
39 0.6808 1.3036 1.6849 2.0227 2.4258 2.7079
40 0.6807 1.3031 1.6839 2.0211 2.4233 2.7045
41 0.6805 1.3025 1.6829 2.0195 2.4208 2.7012
42 0.6804 1.3020 1.6820 2.0181 2.4185 2.6981
43 0.6802 1.3016 1.6811 2.0167 2.4163 2.6951
44 0.6801 1.3011 1.6802 2.0154 2.4141 2.6923
45 0.6800 1.3006 1.6794 2.0141 2.4121 2.6896
46 0.6799 1.3002 1.6787 2.0129 2.4102 2.6870
47 0.6797 1.2998 1.6779 2.0117 2.4083 2.6846
48 0.6796 1.2994 1.6772 2.0106 2.4066 2.6822
49 0.6795 1.2991 1.6766 2.0096 2.4069 2.6800
50 0.6794 1.2987 1.6759 2.0086 2.4033 2.6778
51 0.6793 1.2984 1.6753 2.0076 2.4017 2.6757
52 0.6792 1.2980 1.6747 2.0066 2.4002 2.6737
53 0.6791 1.2977 1.6741 2.0057 2.3988 2.6718
54 0.6791 1.2974 1.6736 2.0049 2.3974 2.6700
55 0.6790 1.2971 1.6730 2.0040 2.3961 2.6682
56 0.6789 1.2969 1.6725 2.0032 2.3948 2.6665
57 0.6788 1.2966 1.6720 2.0025 2.3936 2.6649
58 0.6787 1.2963 1.6716 2.0017 2.3924 2.6633
59 0.6787 1.2961 1.6711 2.0010 2.3912 2.6618
60 0.6786 1.2958 1.6706 2.0003 2.3901 2.6603
61 0.6785 1.2956 1.6702 1.9996 2.3890 2.6589
62 0.6785 1.2954 1.6698 1.9990 2.3880 2.6575

63 0.6784 1.2951 1.6694 1.9983 2.3870 2.6561
64 0.6783 1.2949 1.6690 1.9977 2.3860 2.6549
65 0.6783 1.2947 1.6686 1.9971 2.3851 2.6536
66 0.6782 1.2945 1.6683 1.9966 2.3842 2.6524
67 0.6782 1.2943 1.6679 1.9960 2.3833 2.6512
68 0.6781 1.2941 1.6676 1.9955 2.3824 2.6501
69 0.6781 1.2939 1.6672 1.9949 2.3816 2.6490
70 0.6780 1.2938 1.6669 1.9944 2.3808 2.6479
71 0.6780 1.2936 1.6666 1.9939 2.3800 2.6469
72 0.6779 1.2934 1.6663 1.9935 2.3793 2.6459
73 0.6779 1.2933 1.6660 1.9930 2.3785 2.6449
74 0.6778 1.2931 1.6657 1.9925 2.3778 2.6439
75 0.6778 1.2929 1.6654 1.9921 2.3771 2.6430
76 0.6777 1.2928 1.6652 1.9917 2.3764 2.6421
77 0.6777 1.2926 1.6649 1.9913 2.3758 2.6412
78 0.6776 1.2925 1.6646 1.9908 2.3751 2.6403
79 0.6776 1.2924 1.6644 1.9905 2.3745 2.6395
80 0.6776 1.2922 1.6641 1.9901 2.3739 2.6387
81 0.6775 1.2921 1.6639 1.9897 2.3733 2.6379
528 Appendix III Tables
Table 4
(continued)
γγ
γγ
γ
r 0.75 0.90 0.95 0.975 0.99 0.995
82 0.6775 1.2920 1.6636 1.9893 2.3727 2.6371
83 0.6775 1.2918 1.6634 1.9890 2.3721 2.6364
84 0.6774 1.2917 1.6632 1.9886 2.3716 2.6356
85 0.6774 1.2916 1.6630 1.9883 2.3710 2.6349

86 0.6774 1.2915 1.6628 1.9879 2.3705 2.6342
87 0.6773 1.2914 1.6626 1.9876 2.3700 2.6335
88 0.6773 1.2912 1.6624 1.9873 2.3695 2.6329
89 0.6773 1.2911 1.6622 1.9870 2.3690 2.6322
90 0.6772 1.2910 1.6620 1.9867 2.3685 2.6316
Tables 529
Table 5 Critical Values for the Chi-Square Distribution
Let
χ
2
r
be a random variable having the chi-square distribution with r degrees of
freedom. Then the tabulated quantities are the numbers x for which
Px
r
χγ
2
≤
(
)
= .
γγ
γγ
γ
r 0.005 0.01 0.025 0.05 0.10 0.25
1 — — 0.001 0.004 0.016 0.102
2 0.010 0.020 0.051 0.103 0.211 0.575
3 0.072 0.115 0.216 0.352 0.584 1.213
4 0.207 0.297 0.484 0.711 1.064 1.923
5 0.412 0.554 0.831 1.145 1.610 2.675

6 0.676 0.872 1.237 1.635 2.204 3.455
7 0.989 1.239 1.690 2.167 2.833 4.255
8 1.344 1.646 2.180 2.733 3.490 5.071
9 1.735 2.088 2.700 2.325 4.168 5.899
10 2.156 2.558 3.247 3.940 4.865 6.737
11 2.603 3.053 3.816 4.575 5.578 7.584
12 3.074 3.571 4.404 5.226 6.304 9.438
13 3.565 4.107 5.009 5.892 7.042 9.299
14 4.075 4.660 5.629 6.571 7.790 10.165
15 4.601 5.229 6.262 7.261 8.547 11.037
16 5.142 5.812 6.908 7.962 9.312 11.912
17 5.697 6.408 7.564 8.672 10.085 12.792
18 6.265 7.015 8.231 8.390 10.865 13.675
19 6.844 7.633 8.907 10.117 11.651 14.562
20 7.434 8.260 9.591 10.851 12.443 15.452
21 8.034 8.897 10.283 11.591 13.240 16.344
22 8.643 9.542 10.982 12.338 14.042 17.240
23 9.260 10.196 11.689 13.091 14.848 18.137
24 9.886 10.856 12.401 13.848 15.659 19.037
25 10.520 11.524 13.120 14.611 16.473 19.939
26 11.160 12.198 13.844 13.379 17.292 20.843
27 11.808 12.879 14.573 16.151 18.114 21.749
28 12.461 13.565 15.308 16.928 18.939 22.657
29 13.121 14.257 16.047 17.708 19.768 23.567
30 13.787 14.954 16.791 18.493 20.599 24.478
31 14.458 15.655 17.539 19.281 21.434 25.390
32 15.134 16.362 18.291 20.072 22.271 26.304
33 15.815 17.074 19.047 20.867 23.110 27.219
34 16.501 17.789 19.806 21.664 23.952 28.136
35 17.192 18.509 20.569 22.465 24.797 29.054

36 17.887 19.233 21.336 23.269 25.643 29.973
37 18.586 19.960 22.106 24.075 26.492 30.893
38 19.289 20.691 22.878 24.884 27.343 31.815
530 Appendix III Tables
Table 5
(continued)
γγ
γγ
γ
r 0.005 0.01 0.025 0.05 0.10 0.25
39 19.996 21.426 23.654 25.695 28.196 32.737
40 20.707 22.164 24.433 26.509 29.051 33.660
41 21.421 22.906 25.215 27.326 29.907 34.585
42 22.138 23.650 25.999 28.144 30.765 35.510
43 22.859 24.398 26.785 28.965 31.625 36.436
44 23.584 25.148 27.575 29.787 32.487 37.363
45 24.311 25.901 28.366 30.612 33.350 38.291
γγ
γγ
γ
r 0.75 0.90 0.95 0.975 0.99 0.995
1 1.323 2.706 3.841 5.024 6.635 7.879
2 2.773 4.605 5.991 7.378 9.210 10.597
3 4.108 6.251 7.815 9.348 11.345 12.838
4 5.385 7.779 9.488 11.143 13.277 14.860
5 6.626 9.236 11.071 12.833 15.086 16.750
6 7.841 10.645 12.592 14.449 16.812 18.548
7 9.037 12.017 14.067 16.013 18.475 20.278
8 10.219 13.362 15.507 17.535 20.090 21.955
9 11.389 14.684 16.919 19.023 21.666 23.589

10 12.549 15.987 18.307 20.483 23.209 25.188
11 13.701 17.275 19.675 21.920 24.725 26.757
12 14.845 18.549 21.026 23.337 26.217 28.299
13 15.984 19.812 23.362 24.736 27.688 29.819
14 17.117 21.064 23.685 26.119 29.141 31.319
15 18.245 22.307 24.996 27.488 30.578 32.801
16 19.369 23.542 26.296 28.845 32.000 34.267
17 20.489 24.769 27.587 30.191 33.409 35.718
18 21.605 25.989 28.869 31.526 34.805 37.156
19 22.718 27.204 30.144 32.852 36.191 38.582
20 23.828 28.412 31.410 34.170 37.566 39.997
21 24.935 29.615 32.671 35.479 38.932 41.401
22 26.039 30.813 33.924 36.781 40.289 42.796
23 27.141 32.007 35.172 38.076 41.638 44.181
24 28.241 33.196 36.415 39.364 42.980 45.559
25 29.339 34.382 37.652 40.646 44.314 46.928
26 30.435 35.563 38.885 41.923 45.642 48.290
27 31.528 36.741 40.113 43.194 46.963 49.645
28 32.620 37.916 41.337 44.641 48.278 50.993
29 33.711 39.087 42.557 45.722 49.588 52.336
30 34.800 40.256 43.773 46.979 50.892 53.672
31 35.887 41.422 44.985 48.232 51.191 55.003
32 36.973 42.585 46.194 49.480 53.486 56.328
Tables 531
Table 5
(continued)
γγ
γγ
γ
r 0.75 0.90 0.95 0.975 0.99 0.995

33 38.058 43.745 47.400 50.725 54.776 57.648
34 39.141 44.903 48.602 51.966 56.061 58.964
35 40.223 46.059 49.802 53.203 57.342 60.275
36 41.304 47.212 50.998 54.437 58.619 61.581
37 42.383 48.363 52.192 55.668 59.892 62.883
38 43.462 49.513 53.384 56.896 61.162 64.181
39 44.539 50.660 54.572 58.120 62.428 65.476
40 45.616 51.805 55.758 59.342 63.691 66.766
41 46.692 52.949 56.942 60.561 64.950 68.053
42 47.766 54.090 58.124 61.777 66.206 69.336
43 48.840 55.230 59.304 62.990 67.459 70.616
44 49.913 56.369 60.481 64.201 68.710 71.893
45 50.985 57.505 61.656 65.410 69.957 73.166