γλώσσες
Σελίδες
Νομικός
ADVANCED SUBSIDIARY GENERAL CERTIFICATE OF EDUCATION
ADVANCED GENERAL CERTIFICATE OF EDUCATION
MATHEMATICS
LIST OF FORMULAE
AND
STATISTICAL TABLES
(List MF1)
MF1CST252
January 2007
Pure Mathematics
Mensuration
Surface area of sphere = 4πr2
Area of curved surface of cone = πr × slant height
Trigonometry
a2 = b2 + c2 − 2bc cos A
Arithmetic Series
un = a + (n − 1)dSn = 1
2n(a + l) = 1
2n{2a + (n − 1)d}
Geometric Series
un = arn−1
Sn = a(1 − rn)1 − r
S∞ = a1 − r
for |r | < 1
Summations
n
∑r=1
r2 = 16n(n + 1)(2n + 1)
n
∑r=1
r3 = 14n2(n + 1)2
Binomial Series
(nr) + ( n
r + 1) = (n + 1
r + 1)
(a + b)n = an + (n1)an−1b + (n
2)an−2b2 + . . . + (n
r)an−rbr + . . . + bn (n ∈ ),
where (nr) = nCr = n!
r!(n − r)!(1 + x)n = 1 + nx + n(n − 1)
1.2x2 + . . . + n(n − 1) . . . (n − r + 1)
1.2.3 . . . rxr + . . . (|x| < 1, n ∈ �)
Logarithms and exponentials
ex ln a = ax
Complex Numbers
{r(cos θ + i sin θ)}n = rn(cos nθ + i sin nθ)eiθ = cos θ + i sin θ
The roots of �n = 1 are given by � = e2πki
n , for k = 0, 1, 2, . . . , n − 1
2
Maclaurin’s Series
f(x) = f(0) + xf ′(0) + x2
2!f ′′(0) + . . . + xr
r!f (r)(0) + . . .
ex = exp(x) = 1 + x + x2
2!+ . . . + xr
r!+ . . . for all x
ln(1 + x) = x − x2
2+ x3
3− . . . + (−1)r+1 xr
r+ . . . (−1 < x ≤ 1)
sin x = x − x3
3!+ x5
5!− . . . + (−1)r x2r+1
(2r + 1)! + . . . for all x
cos x = 1 − x2
2!+ x4
4!− . . . + (−1)r x2r
(2r)! + . . . for all x
tan−1 x = x − x3
3+ x5
5− . . . + (−1)r x2r+1
2r + 1+ . . . (−1 ≤ x ≤ 1)
sinh x = x + x3
3!+ x5
5!+ . . . + x2r+1
(2r + 1)! + . . . for all x
cosh x = 1 + x2
2!+ x4
4!+ . . . + x2r
(2r)! + . . . for all x
tanh−1 x = x + x3
3+ x5
5+ . . . + x2r+1
2r + 1+ . . . (−1 < x < 1)
Hyperbolic Functions
cosh2 x − sinh2 x = 1
sinh 2x = 2 sinh x cosh x
cosh 2x = cosh2 x + sinh2 x
cosh−1 x = ln{x + √(x2 − 1)} (x ≥ 1)sinh−1 x = ln{x + √(x2 + 1)}tanh−1 x = 1
2ln(1 + x
1 − x) (|x| < 1)
Coordinate Geometry
The perpendicular distance from (h, k) to ax + by + c = 0 is|ah + bk + c|√(a2 + b2)
The acute angle between lines with gradients m1 and m2 is tan−1 ∣∣∣∣ m1 − m2
1 +m1m2
∣∣∣∣Trigonometric Identities
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
tan(A ± B) = tan A ± tan B1 ∓ tan A tan B
(A ± B ≠ (k + 12)π)
For t = tan 12A: sin A = 2t
1 + t2, cos A = 1 − t2
1 + t2
sin A + sin B = 2 sinA + B
2cos
A − B2
sin A − sin B = 2 cosA + B
2sin
A − B2
cos A + cos B = 2 cosA + B
2cos
A − B2
cos A − cos B = −2 sinA + B
2sin
A − B2
3
Vectors
The resolved part of a in the direction of b isa.b|b|
The point dividing AB in the ratio λ : µ isµa + λb
λ + µ
Vector product: a × b = |a| |b| sin θ n̂ = ∣∣∣∣∣i a1 b1j a2 b2k a
3b
3
∣∣∣∣∣ = (a2b3 − a3b2a3b1 − a1b3a
1b
2− a
2b
1
)If A is the point with position vector a = a1i + a2j + a3k and the direction vector b is given by
b = b1i + b
2j + b
3k, then the straight line through A with direction vector b has cartesian equation
x − a1
b1
= y − a2
b2
= � − a3
b3
(= λ )The plane through A with normal vector n = n1i + n2j + n3k has cartesian equation
n1x + n2y + n3� + d = 0, where d = −a.n
The plane through non-collinear points A, B and C has vector equation
r = a + λ (b − a) + µ(c − a) = (1 − λ − µ)a + λb + µc
The plane through the point with position vector a and parallel to b and c has equation r = a + sb + tc
The perpendicular distance of (α, β , γ ) from n1x + n2y + n3� + d = 0 is∣∣n1α + n2β + n3γ + d ∣∣√(n2
1 + n22 + n2
3)Matrix transformations
Anticlockwise rotation through θ about O: ( cos θ − sin θsin θ cos θ )
Reflection in the line y = (tan θ)x: ( cos 2θ sin 2θsin 2θ − cos 2θ )
Differentiation
f(x) f ′(x)tan kx k sec2 kx
sin−1 x1√(1 − x2)
cos−1 x − 1√(1 − x2)tan−1 x
1
1 + x2
sec x sec x tan xcot x − cosec2 x
cosec x − cosec x cot xsinh x cosh xcosh x sinh xtanh x sech2 x
sinh−1 x1√(1 + x2)
cosh−1 x1√(x2 − 1)
tanh−1 x1
1 − x2
If y = f(x)g(x) then
dydx
= f ′(x)g(x) − f(x)g′(x){g(x)}2
4
Integration ( + constant; a > 0 where relevant)
f(x) � f(x) dx
sec2 kx1k
tan kx
tan x ln |sec x|cot x ln |sin x|
cosec x − ln |cosec x + cot x| = ln ∣∣tan 12x∣∣
sec x ln |sec x + tan x| = ln ∣∣tan(12x + 1
4π)∣∣
sinh x cosh xcosh x sinh xtanh x ln cosh x
1√(a2 − x2) sin−1(xa) (|x| < a)
1
a2 + x2
1a
tan−1(xa)
1√(x2 − a2) cosh−1(xa) or ln{x + √(x2 − a2)} (x > a)
1√(a2 + x2) sinh−1(xa) or ln{x + √(x2 + a2)}
1
a2 − x2
12a
ln ∣∣∣ a + xa − x
∣∣∣ = 1a
tanh−1(xa) (|x| < a)
1
x2 − a2
12a
ln ∣∣∣ x − ax + a
∣∣∣� u
dvdx
dx = uv − � vdudx
dx
Area of a sector
A = 12� r2 dθ (polar coordinates)
A = 12� (x dy
dt− y
dxdt) dt (parametric form)
Numerical Mathematics
Numerical integration
The trapezium rule: � b
ay dx ≈ 1
2h{(y0 + yn) + 2(y1 + y2 + . . . + yn−1)}, where h = b − a
n
Simpson’s Rule: � b
ay dx ≈ 1
3h{(y0 + yn) + 4(y1 + y3 + . . . + yn−1) + 2(y2 + y4 + . . . + yn−2)},
where h = b − an
and n is even
Numerical Solution of Equations
The Newton-Raphson iteration for solving f(x) = 0: xn+1 = xn − f(xn)f′(xn)
5
Mechanics
Motion in a circle
Transverse velocity: v = rθ̇
Transverse acceleration: v̇ = rθ̈
Radial acceleration: − rθ̇2 = −v2
r
Centres of Mass (for uniform bodies)
Triangular lamina: 23
along median from vertex
Solid hemisphere, radius r: 38r from centre
Hemispherical shell, radius r: 12r from centre
Circular arc, radius r, angle at centre 2α:r sin α
αfrom centre
Sector of circle, radius r, angle at centre 2α:2r sin α
3αfrom centre
Solid cone or pyramid of height h: 14h above the base on the line from centre of base to vertex
Conical shell of height h: 13h above the base on the line from centre of base to vertex
Moments of Inertia (for uniform bodies of mass m)
Thin rod, length 2l, about perpendicular axis through centre: 13ml2
Rectangular lamina about axis in plane bisecting edges of length 2l: 13ml2
Thin rod, length 2l, about perpendicular axis through end: 43ml2
Rectangular lamina about edge perpendicular to edges of length 2l: 43ml2
Rectangular lamina, sides 2a and 2b, about perpendicular axis through centre: 13m(a2 + b2)
Hoop or cylindrical shell of radius r about axis: mr2
Hoop of radius r about a diameter: 12mr2
Disc or solid cylinder of radius r about axis: 12mr2
Disc of radius r about a diameter: 14mr2
Solid sphere, radius r, about diameter: 25mr2
Spherical shell of radius r about a diameter: 23mr2
Parallel axes theorem: IA = IG +m(AG)2
Perpendicular axes theorem: I� = Ix + Iy (for a lamina in the x-y plane)
6
Probability & Statistics
Probability
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)P(A ∩ B) = P(A)P(B | A)P(A | B) = P(B | A)P(A)
P(B | A)P(A) + P(B | A′)P(A′)Bayes’ Theorem: P(Aj | B) = P(Aj)P(B | Aj)
ΣP(Ai)P(B | Ai)Discrete distributions
For a discrete random variable X taking values xi with probabilities pi
Expectation (mean): E(X) = µ = Σ xip
i
Variance: Var(X) = σ2 = Σ(xi − µ)2pi = Σ x2i pi − µ2
For a function g(X): E(g(X)) = Σ g(xi)pi
The probability generating function of X is GX(t) = E(tX), and
E(X) = G′X(1)
Var(X) = G′′X(1) + G′
X(1) − {G′X(1)}2
For Z = X + Y , where X and Y are independent: GZ(t) = GX(t)GY(t)Standard discrete distributions
Distribution of X P(X = x) Mean Variance P.G.F.
Binomial B(n, p) (nx)px(1 − p)n−x np np(1 − p) (1 − p + pt)n
Poisson Po(λ ) e−λ λ x
x!λ λ eλ (t−1)
Geometric Geo(p) on 1, 2, … p(1 − p)x−1 1p
1 − pp2
pt1 − (1 − p)t
Continuous distributions
For a continuous random variable X having probability density function f
Expectation (mean): E(X) = µ = � xf(x) dx
Variance: Var(X) = σ2 = � (x − µ)2f(x) dx = � x2f(x) dx − µ2
For a function g(X): E(g(X)) = � g(x)f(x) dx
Cumulative distribution function: F(x) = P(X ≤ x) = � x
−∞f(t) dt
The moment generating function of X is MX(t) = E(etX) and
E(X) = M′X(0)
E(Xn) = M(n)X (0)
Var(X) = M′′X(0) − {M′
X(0)}2
For Z = X + Y , where X and Y are independent: MZ(t) = MX(t)MY(t)
7
Standard continuous distributions
Distribution of X P.D.F. Mean Variance M.G.F.
Uniform (Rectangular) on [a, b] 1b − a
12(a + b) 1
12(b − a)2 ebt − eat
(b − a)tExponential λe−λx 1
λ1
λ 2
λλ − t
Normal N(µ, σ2) 1σ√(2π)e−1
2( x−µ
σ )2
µ σ2 eµt+12σ2t2
Expectation algebra
Covariance: Cov(X, Y) = E((X − µX)(Y − µY)) = E(XY) − µXµY
Var(aX ± bY) = a2 Var(X) + b2 Var(Y) ± 2ab Cov(X, Y)Product moment correlation coefficient: ρ = Cov(X, Y)
σX
σY
If X = aX′ + b and Y = cY ′ + d, then Cov(X, Y) = ac Cov(X′, Y ′)For independent random variables X and Y
E(XY) = E(X)E(Y)Var(aX ± bY) = a2 Var(X) + b2 Var(Y)
Sampling distributions
For a random sample X1, X2, . . . , Xn of n independent observations from a distribution having mean µand variance σ2
X is an unbiased estimator of µ, with Var(X) = σ2
n
S2 is an unbiased estimator of σ2, where S2 = Σ(Xi − X)2
n − 1
For a random sample of n observations from N(µ, σ2)X − µσ/√n
∼ N(0, 1)X − µS/√n
∼ tn−1 (also valid in matched-pairs situations)
If X is the observed number of successes in n independent Bernoulli trials in each of which the
probability of success is p, and Y = Xn
, then
E(Y) = p and Var(Y) = p(1 − p)n
For a random sample of nx observations from N(µx, σ2x ) and, independently, a random sample of
ny observations from N(µy, σ2y )(X − Y) − (µx − µy)√(σ2
x
nx
+ σ2y
ny
) ∼ N(0, 1)
If σ2x = σ2
y = σ2 (unknown) then(X − Y) − (µx − µy)√{S2
p( 1
nx
+ 1ny
)} ∼ tnx+n
y−2,
where S2p = (nx − 1)S2
x + (ny − 1)S2y
nx + ny − 2
8
Correlation and regression
For a set of n pairs of values (xi, yi)Sxx = Σ(xi − x)2 = Σ x2
i − (Σ xi)2
n
Syy = Σ(yi − y)2 = Σ y2i − (Σ yi)2
n
Sxy= Σ(x
i− x)(y
i− y) = Σ x
iy
i− (Σ xi)(Σ yi)
nThe product moment correlation coefficient is
r = Sxy√(SxxSyy) =Σ(xi − x)(yi − y)√{(Σ(x
i− x)2)(Σ(y
i− y)2)} = Σ xiyi − (Σ xi)(Σ yi)
n√{(Σ x2i − (Σ xi)2
n)(Σ y2
i − (Σ yi)2
n)}
Spearman’s rank correlation coefficient is rs = 1 − 6Σ d2
n(n2 − 1)The regression coefficient of y on x is b = Sxy
Sxx
= Σ(xi − x)(yi − y)Σ(xi − x)2
Least squares regression line of y on x is y = a + bx where a = y − bx
Distribution-free (non-parametric) tests
Goodness-of-fit test and contingency tables: ∑ (Oi − Ei)2
Ei
∼ χ2ν
Approximate distributions for large samples
Wilcoxon Signed Rank test: T ∼ N(14n(n + 1), 1
24n(n + 1)(2n + 1))
Wilcoxon Rank Sum test (samples of sizes m and n, with m ≤ n):
W ∼ N(12m(m + n + 1), 1
12mn(m + n + 1))
9
CUMULATIVEBINOMIALPROBABILITIES
n=5 p
0.05
0.1
0.15
1/60.
20.
250.
31/3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
2/30.
70.
750.
85/6
0.85
0.9
0.95
x=0
0.77
380.
5905
0.44
370.
4019
0.32
770.
2373
0.16
810.
1317
0.11
600.
0778
0.05
030.
0313
0.01
850.
0102
0.00
530.
0041
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240.
0010
0.00
030.
0001
0.00
010.
0000
0.00
001
0.97
740.
9185
0.83
520.
8038
0.73
730.
6328
0.52
820.
4609
0.42
840.
3370
0.25
620.
1875
0.13
120.
0870
0.05
400.
0453
0.03
080.
0156
0.00
670.
0033
0.00
220.
0005
0.00
002
0.99
880.
9914
0.97
340.
9645
0.94
210.
8965
0.83
690.
7901
0.76
480.
6826
0.59
310.
5000
0.40
690.
3174
0.23
520.
2099
0.16
310.
1035
0.05
790.
0355
0.02
660.
0086
0.00
123
1.00
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0.99
780.
9967
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330.
9844
0.96
920.
9547
0.94
600.
9130
0.86
880.
8125
0.74
380.
6630
0.57
160.
5391
0.47
180.
3672
0.26
270.
1962
0.16
480.
0815
0.02
264
1.00
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0000
0.99
990.
9999
0.99
970.
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760.
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470.
9898
0.98
150.
9688
0.94
970.
9222
0.88
400.
8683
0.83
190.
7627
0.67
230.
5981
0.55
630.
4095
0.22
625
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001.
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00
n=6 p
0.05
0.1
0.15
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20.
250.
31/3
0.35
0.4
0.45
0.5
0.55
0.6
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70.
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85/6
0.85
0.9
0.95
x=0
0.73
510.
5314
0.37
710.
3349
0.26
210.
1780
0.11
760.
0878
0.07
540.
0467
0.02
770.
0156
0.00
830.
0041
0.00
180.
0014
0.00
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0002
0.00
010.
0000
0.00
000.
0000
0.00
001
0.96
720.
8857
0.77
650.
7368
0.65
540.
5339
0.42
020.
3512
0.31
910.
2333
0.16
360.
1094
0.06
920.
0410
0.02
230.
0178
0.01
090.
0046
0.00
160.
0007
0.00
040.
0001
0.00
002
0.99
780.
9842
0.95
270.
9377
0.90
110.
8306
0.74
430.
6804
0.64
710.
5443
0.44
150.
3438
0.25
530.
1792
0.11
740.
1001
0.07
050.
0376
0.01
700.
0087
0.00
590.
0013
0.00
013
0.99
990.
9987
0.99
410.
9913
0.98
300.
9624
0.92
950.
8999
0.88
260.
8208
0.74
470.
6563
0.55
850.
4557
0.35
290.
3196
0.25
570.
1694
0.09
890.
0623
0.04
730.
0159
0.00
224
1.00
000.
9999
0.99
960.
9993
0.99
840.
9954
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910.
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0.97
770.
9590
0.93
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0.83
640.
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0.68
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0.57
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0.34
460.
2632
0.22
350.
1143
0.03
285
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170.
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0.88
240.
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0.73
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0.62
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n=7 p
0.05
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20.
250.
31/3
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0.4
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0.5
0.55
0.6
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85/6
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x=0
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830.
4783
0.32
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560.
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0.71
660.
6698
0.57
670.
4449
0.32
940.
2634
0.23
380.
1586
0.10
240.
0625
0.03
570.
0188
0.00
900.
0069
0.00
380.
0013
0.00
040.
0001
0.00
010.
0000
0.00
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0.99
620.
9743
0.92
620.
9042
0.85
200.
7564
0.64
710.
5706
0.53
230.
4199
0.31
640.
2266
0.15
290.
0963
0.05
560.
0453
0.02
880.
0129
0.00
470.
0020
0.00
120.
0002
0.00
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0.99
980.
9973
0.98
790.
9824
0.96
670.
9294
0.87
400.
8267
0.80
020.
7102
0.60
830.
5000
0.39
170.
2898
0.19
980.
1733
0.12
600.
0706
0.03
330.
0176
0.01
210.
0027
0.00
024
1.00
000.
9998
0.99
880.
9980
0.99
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0.97
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0.94
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9037
0.84
710.
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0.68
360.
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0.46
770.
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0.35
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0.14
800.
0958
0.07
380.
0257
0.00
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1.00
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0000
0.99
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9999
0.99
960.
9987
0.99
620.
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9812
0.96
430.
9375
0.89
760.
8414
0.76
620.
7366
0.67
060.
5551
0.42
330.
3302
0.28
340.
1497
0.04
446
1.00
001.
0000
1.00
001.
0000
1.00
000.
9999
0.99
980.
9995
0.99
940.
9984
0.99
630.
9922
0.98
480.
9720
0.95
100.
9415
0.91
760.
8665
0.79
030.
7209
0.67
940.
5217
0.30
177
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
00
n=8 p
0.05
0.1
0.15
1/60.
20.
250.
31/3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
2/30.
70.
750.
85/6
0.85
0.9
0.95
x=0
0.66
340.
4305
0.27
250.
2326
0.16
780.
1001
0.05
760.
0390
0.03
190.
0168
0.00
840.
0039
0.00
170.
0007
0.00
020.
0002
0.00
010.
0000
0.00
000.
0000
0.00
000.
0000
0.00
001
0.94
280.
8131
0.65
720.
6047
0.50
330.
3671
0.25
530.
1951
0.16
910.
1064
0.06
320.
0352
0.01
810.
0085
0.00
360.
0026
0.00
130.
0004
0.00
010.
0000
0.00
000.
0000
0.00
002
0.99
420.
9619
0.89
480.
8652
0.79
690.
6785
0.55
180.
4682
0.42
780.
3154
0.22
010.
1445
0.08
850.
0498
0.02
530.
0197
0.01
130.
0042
0.00
120.
0004
0.00
020.
0000
0.00
003
0.99
960.
9950
0.97
860.
9693
0.94
370.
8862
0.80
590.
7414
0.70
640.
5941
0.47
700.
3633
0.26
040.
1737
0.10
610.
0879
0.05
800.
0273
0.01
040.
0046
0.00
290.
0004
0.00
004
1.00
000.
9996
0.99
710.
9954
0.98
960.
9727
0.94
200.
9121
0.89
390.
8263
0.73
960.
6367
0.52
300.
4059
0.29
360.
2586
0.19
410.
1138
0.05
630.
0307
0.02
140.
0050
0.00
045
1.00
001.
0000
0.99
980.
9996
0.99
880.
9958
0.98
870.
9803
0.97
470.
9502
0.91
150.
8555
0.77
990.
6846
0.57
220.
5318
0.44
820.
3215
0.20
310.
1348
0.10
520.
0381
0.00
586
1.00
001.
0000
1.00
001.
0000
0.99
990.
9996
0.99
870.
9974
0.99
640.
9915
0.98
190.
9648
0.93
680.
8936
0.83
090.
8049
0.74
470.
6329
0.49
670.
3953
0.34
280.
1869
0.05
727
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
0.99
990.
9998
0.99
980.
9993
0.99
830.
9961
0.99
160.
9832
0.96
810.
9610
0.94
240.
8999
0.83
220.
7674
0.72
750.
5695
0.33
668
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
00
10
CUMULATIVEBINOMIALPROBABILITIES
n=9 p
0.05
0.1
0.15
1/60.
20.
250.
31/3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
2/30.
70.
750.
85/6
0.85
0.9
0.95
x=0
0.63
020.
3874
0.23
160.
1938
0.13
420.
0751
0.04
040.
0260
0.02
070.
0101
0.00
460.
0020
0.00
080.
0003
0.00
010.
0001
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
001
0.92
880.
7748
0.59
950.
5427
0.43
620.
3003
0.19
600.
1431
0.12
110.
0705
0.03
850.
0195
0.00
910.
0038
0.00
140.
0010
0.00
040.
0001
0.00
000.
0000
0.00
000.
0000
0.00
002
0.99
160.
9470
0.85
910.
8217
0.73
820.
6007
0.46
280.
3772
0.33
730.
2318
0.14
950.
0898
0.04
980.
0250
0.01
120.
0083
0.00
430.
0013
0.00
030.
0001
0.00
000.
0000
0.00
003
0.99
940.
9917
0.96
610.
9520
0.91
440.
8343
0.72
970.
6503
0.60
890.
4826
0.36
140.
2539
0.16
580.
0994
0.05
360.
0424
0.02
530.
0100
0.00
310.
0011
0.00
060.
0001
0.00
004
1.00
000.
9991
0.99
440.
9910
0.98
040.
9511
0.90
120.
8552
0.82
830.
7334
0.62
140.
5000
0.37
860.
2666
0.17
170.
1448
0.09
880.
0489
0.01
960.
0090
0.00
560.
0009
0.00
005
1.00
000.
9999
0.99
940.
9989
0.99
690.
9900
0.97
470.
9576
0.94
640.
9006
0.83
420.
7461
0.63
860.
5174
0.39
110.
3497
0.27
030.
1657
0.08
560.
0480
0.03
390.
0083
0.00
066
1.00
001.
0000
1.00
000.
9999
0.99
970.
9987
0.99
570.
9917
0.98
880.
9750
0.95
020.
9102
0.85
050.
7682
0.66
270.
6228
0.53
720.
3993
0.26
180.
1783
0.14
090.
0530
0.00
847
1.00
001.
0000
1.00
001.
0000
1.00
000.
9999
0.99
960.
9990
0.99
860.
9962
0.99
090.
9805
0.96
150.
9295
0.87
890.
8569
0.80
400.
6997
0.56
380.
4573
0.40
050.
2252
0.07
128
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
000.
9999
0.99
990.
9997
0.99
920.
9980
0.99
540.
9899
0.97
930.
9740
0.95
960.
9249
0.86
580.
8062
0.76
840.
6126
0.36
989
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
00
n=1
0p
0.05
0.1
0.15
1/60.
20.
250.
31/3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
2/30.
70.
750.
85/6
0.85
0.9
0.95
x=0
0.59
870.
3487
0.19
690.
1615
0.10
740.
0563
0.02
820.
0173
0.01
350.
0060
0.00
250.
0010
0.00
030.
0001
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
001
0.91
390.
7361
0.54
430.
4845
0.37
580.
2440
0.14
930.
1040
0.08
600.
0464
0.02
330.
0107
0.00
450.
0017
0.00
050.
0004
0.00
010.
0000
0.00
000.
0000
0.00
000.
0000
0.00
002
0.98
850.
9298
0.82
020.
7752
0.67
780.
5256
0.38
280.
2991
0.26
160.
1673
0.09
960.
0547
0.02
740.
0123
0.00
480.
0034
0.00
160.
0004
0.00
010.
0000
0.00
000.
0000
0.00
003
0.99
900.
9872
0.95
000.
9303
0.87
910.
7759
0.64
960.
5593
0.51
380.
3823
0.26
600.
1719
0.10
200.
0548
0.02
600.
0197
0.01
060.
0035
0.00
090.
0003
0.00
010.
0000
0.00
004
0.99
990.
9984
0.99
010.
9845
0.96
720.
9219
0.84
970.
7869
0.75
150.
6331
0.50
440.
3770
0.26
160.
1662
0.09
490.
0766
0.04
730.
0197
0.00
640.
0024
0.00
140.
0001
0.00
005
1.00
000.
9999
0.99
860.
9976
0.99
360.
9803
0.95
270.
9234
0.90
510.
8338
0.73
840.
6230
0.49
560.
3669
0.24
850.
2131
0.15
030.
0781
0.03
280.
0155
0.00
990.
0016
0.00
016
1.00
001.
0000
0.99
990.
9997
0.99
910.
9965
0.98
940.
9803
0.97
400.
9452
0.89
800.
8281
0.73
400.
6177
0.48
620.
4407
0.35
040.
2241
0.12
090.
0697
0.05
000.
0128
0.00
107
1.00
001.
0000
1.00
001.
0000
0.99
990.
9996
0.99
840.
9966
0.99
520.
9877
0.97
260.
9453
0.90
040.
8327
0.73
840.
7009
0.61
720.
4744
0.32
220.
2248
0.17
980.
0702
0.01
158
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
0.99
990.
9996
0.99
950.
9983
0.99
550.
9893
0.97
670.
9536
0.91
400.
8960
0.85
070.
7560
0.62
420.
5155
0.45
570.
2639
0.08
619
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
000.
9999
0.99
970.
9990
0.99
750.
9940
0.98
650.
9827
0.97
180.
9437
0.89
260.
8385
0.80
310.
6513
0.40
1310
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
00
n=1
2p
0.05
0.1
0.15
1/60.
20.
250.
31/3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
2/30.
70.
750.
85/6
0.85
0.9
0.95
x=0
0.54
040.
2824
0.14
220.
1122
0.06
870.
0317
0.01
380.
0077
0.00
570.
0022
0.00
080.
0002
0.00
010.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
001
0.88
160.
6590
0.44
350.
3813
0.27
490.
1584
0.08
500.
0540
0.04
240.
0196
0.00
830.
0032
0.00
110.
0003
0.00
010.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
002
0.98
040.
8891
0.73
580.
6774
0.55
830.
3907
0.25
280.
1811
0.15
130.
0834
0.04
210.
0193
0.00
790.
0028
0.00
080.
0005
0.00
020.
0000
0.00
000.
0000
0.00
000.
0000
0.00
003
0.99
780.
9744
0.90
780.
8748
0.79
460.
6488
0.49
250.
3931
0.34
670.
2253
0.13
450.
0730
0.03
560.
0153
0.00
560.
0039
0.00
170.
0004
0.00
010.
0000
0.00
000.
0000
0.00
004
0.99
980.
9957
0.97
610.
9636
0.92
740.
8424
0.72
370.
6315
0.58
330.
4382
0.30
440.
1938
0.11
170.
0573
0.02
550.
0188
0.00
950.
0028
0.00
060.
0002
0.00
010.
0000
0.00
005
1.00
000.
9995
0.99
540.
9921
0.98
060.
9456
0.88
220.
8223
0.78
730.
6652
0.52
690.
3872
0.26
070.
1582
0.08
460.
0664
0.03
860.
0143
0.00
390.
0013
0.00
070.
0001
0.00
006
1.00
000.
9999
0.99
930.
9987
0.99
610.
9857
0.96
140.
9336
0.91
540.
8418
0.73
930.
6128
0.47
310.
3348
0.21
270.
1777
0.11
780.
0544
0.01
940.
0079
0.00
460.
0005
0.00
007
1.00
001.
0000
0.99
990.
9998
0.99
940.
9972
0.99
050.
9812
0.97
450.
9427
0.88
830.
8062
0.69
560.
5618
0.41
670.
3685
0.27
630.
1576
0.07
260.
0364
0.02
390.
0043
0.00
028
1.00
001.
0000
1.00
001.
0000
0.99
990.
9996
0.99
830.
9961
0.99
440.
9847
0.96
440.
9270
0.86
550.
7747
0.65
330.
6069
0.50
750.
3512
0.20
540.
1252
0.09
220.
0256
0.00
229
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
0.99
980.
9995
0.99
920.
9972
0.99
210.
9807
0.95
790.
9166
0.84
870.
8189
0.74
720.
6093
0.44
170.
3226
0.26
420.
1109
0.01
9610
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
0.99
990.
9997
0.99
890.
9968
0.99
170.
9804
0.95
760.
9460
0.91
500.
8416
0.72
510.
6187
0.55
650.
3410
0.11
8411
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
0.99
990.
9998
0.99
920.
9978
0.99
430.
9923
0.98
620.
9683
0.93
130.
8878
0.85
780.
7176
0.45
9612
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
00
11
CUMULATIVEBINOMIALPROBABILITIES
n=1
4p
0.05
0.1
0.15
1/60.
20.
250.
31/3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
2/30.
70.
750.
85/6
0.85
0.9
0.95
x=0
0.48
770.
2288
0.10
280.
0779
0.04
400.
0178
0.00
680.
0034
0.00
240.
0008
0.00
020.
0001
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
001
0.84
700.
5846
0.35
670.
2960
0.19
790.
1010
0.04
750.
0274
0.02
050.
0081
0.00
290.
0009
0.00
030.
0001
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
002
0.96
990.
8416
0.64
790.
5795
0.44
810.
2811
0.16
080.
1053
0.08
390.
0398
0.01
700.
0065
0.00
220.
0006
0.00
010.
0001
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
003
0.99
580.
9559
0.85
350.
8063
0.69
820.
5213
0.35
520.
2612
0.22
050.
1243
0.06
320.
0287
0.01
140.
0039
0.00
110.
0007
0.00
020.
0000
0.00
000.
0000
0.00
000.
0000
0.00
004
0.99
960.
9908
0.95
330.
9310
0.87
020.
7415
0.58
420.
4755
0.42
270.
2793
0.16
720.
0898
0.04
260.
0175
0.00
600.
0040
0.00
170.
0003
0.00
000.
0000
0.00
000.
0000
0.00
005
1.00
000.
9985
0.98
850.
9809
0.95
610.
8883
0.78
050.
6898
0.64
050.
4859
0.33
730.
2120
0.11
890.
0583
0.02
430.
0174
0.00
830.
0022
0.00
040.
0001
0.00
000.
0000
0.00
006
1.00
000.
9998
0.99
780.
9959
0.98
840.
9617
0.90
670.
8505
0.81
640.
6925
0.54
610.
3953
0.25
860.
1501
0.07
530.
0576
0.03
150.
0103
0.00
240.
0007
0.00
030.
0000
0.00
007
1.00
001.
0000
0.99
970.
9993
0.99
760.
9897
0.96
850.
9424
0.92
470.
8499
0.74
140.
6047
0.45
390.
3075
0.18
360.
1495
0.09
330.
0383
0.01
160.
0041
0.00
220.
0002
0.00
008
1.00
001.
0000
1.00
000.
9999
0.99
960.
9978
0.99
170.
9826
0.97
570.
9417
0.88
110.
7880
0.66
270.
5141
0.35
950.
3102
0.21
950.
1117
0.04
390.
0191
0.01
150.
0015
0.00
009
1.00
001.
0000
1.00
001.
0000
1.00
000.
9997
0.99
830.
9960
0.99
400.
9825
0.95
740.
9102
0.83
280.
7207
0.57
730.
5245
0.41
580.
2585
0.12
980.
0690
0.04
670.
0092
0.00
0410
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
0.99
980.
9993
0.99
890.
9961
0.98
860.
9713
0.93
680.
8757
0.77
950.
7388
0.64
480.
4787
0.30
180.
1937
0.14
650.
0441
0.00
4211
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
000.
9999
0.99
990.
9994
0.99
780.
9935
0.98
300.
9602
0.91
610.
8947
0.83
920.
7189
0.55
190.
4205
0.35
210.
1584
0.03
0112
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
000.
9999
0.99
970.
9991
0.99
710.
9919
0.97
950.
9726
0.95
250.
8990
0.80
210.
7040
0.64
330.
4154
0.15
3013
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
000.
9999
0.99
980.
9992
0.99
760.
9966
0.99
320.
9822
0.95
600.
9221
0.89
720.
7712
0.51
2314
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
00
n=1
6p
0.05
0.1
0.15
1/60.
20.
250.
31/3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
2/30.
70.
750.
85/6
0.85
0.9
0.95
x=0
0.44
010.
1853
0.07
430.
0541
0.02
810.
0100
0.00
330.
0015
0.00
100.
0003
0.00
010.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
001
0.81
080.
5147
0.28
390.
2272
0.14
070.
0635
0.02
610.
0137
0.00
980.
0033
0.00
100.
0003
0.00
010.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
002
0.95
710.
7892
0.56
140.
4868
0.35
180.
1971
0.09
940.
0594
0.04
510.
0183
0.00
660.
0021
0.00
060.
0001
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
003
0.99
300.
9316
0.78
990.
7291
0.59
810.
4050
0.24
590.
1659
0.13
390.
0651
0.02
810.
0106
0.00
350.
0009
0.00
020.
0001
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
004
0.99
910.
9830
0.92
090.
8866
0.79
820.
6302
0.44
990.
3391
0.28
920.
1666
0.08
530.
0384
0.01
490.
0049
0.00
130.
0008
0.00
030.
0000
0.00
000.
0000
0.00
000.
0000
0.00
005
0.99
990.
9967
0.97
650.
9622
0.91
830.
8103
0.65
980.
5469
0.49
000.
3288
0.19
760.
1051
0.04
860.
0191
0.00
620.
0040
0.00
160.
0003
0.00
000.
0000
0.00
000.
0000
0.00
006
1.00
000.
9995
0.99
440.
9899
0.97
330.
9204
0.82
470.
7374
0.68
810.
5272
0.36
600.
2272
0.12
410.
0583
0.02
290.
0159
0.00
710.
0016
0.00
020.
0000
0.00
000.
0000
0.00
007
1.00
000.
9999
0.99
890.
9979
0.99
300.
9729
0.92
560.
8735
0.84
060.
7161
0.56
290.
4018
0.25
590.
1423
0.06
710.
0500
0.02
570.
0075
0.00
150.
0004
0.00
020.
0000
0.00
008
1.00
001.
0000
0.99
980.
9996
0.99
850.
9925
0.97
430.
9500
0.93
290.
8577
0.74
410.
5982
0.43
710.
2839
0.15
940.
1265
0.07
440.
0271
0.00
700.
0021
0.00
110.
0001
0.00
009
1.00
001.
0000
1.00
001.
0000
0.99
980.
9984
0.99
290.
9841
0.97
710.
9417
0.87
590.
7728
0.63
400.
4728
0.31
190.
2626
0.17
530.
0796
0.02
670.
0101
0.00
560.
0005
0.00
0010
1.00
001.
0000
1.00
001.
0000
1.00
000.
9997
0.99
840.
9960
0.99
380.
9809
0.95
140.
8949
0.80
240.
6712
0.51
000.
4531
0.34
020.
1897
0.08
170.
0378
0.02
350.
0033
0.00
0111
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
0.99
970.
9992
0.99
870.
9951
0.98
510.
9616
0.91
470.
8334
0.71
080.
6609
0.55
010.
3698
0.20
180.
1134
0.07
910.
0170
0.00
0912
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
000.
9999
0.99
980.
9991
0.99
650.
9894
0.97
190.
9349
0.86
610.
8341
0.75
410.
5950
0.40
190.
2709
0.21
010.
0684
0.00
7013
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
000.
9999
0.99
940.
9979
0.99
340.
9817
0.95
490.
9406
0.90
060.
8029
0.64
820.
5132
0.43
860.
2108
0.04
2914
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
0.99
990.
9997
0.99
900.
9967
0.99
020.
9863
0.97
390.
9365
0.85
930.
7728
0.71
610.
4853
0.18
9215
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
0.99
990.
9997
0.99
900.
9985
0.99
670.
9900
0.97
190.
9459
0.92
570.
8147
0.55
9916
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
00
12
CUMULATIVEBINOMIALPROBABILITIES
n=1
8p
0.05
0.1
0.15
1/60.
20.
250.
31/3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
2/30.
70.
750.
85/6
0.85
0.9
0.95
x=0
0.39
720.
1501
0.05
360.
0376
0.01
800.
0056
0.00
160.
0007
0.00
040.
0001
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
001
0.77
350.
4503
0.22
410.
1728
0.09
910.
0395
0.01
420.
0068
0.00
460.
0013
0.00
030.
0001
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
002
0.94
190.
7338
0.47
970.
4027
0.27
130.
1353
0.06
000.
0326
0.02
360.
0082
0.00
250.
0007
0.00
010.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
003
0.98
910.
9018
0.72
020.
6479
0.50
100.
3057
0.16
460.
1017
0.07
830.
0328
0.01
200.
0038
0.00
100.
0002
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
004
0.99
850.
9718
0.87
940.
8318
0.71
640.
5187
0.33
270.
2311
0.18
860.
0942
0.04
110.
0154
0.00
490.
0013
0.00
030.
0001
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
005
0.99
980.
9936
0.95
810.
9347
0.86
710.
7175
0.53
440.
4122
0.35
500.
2088
0.10
770.
0481
0.01
830.
0058
0.00
140.
0009
0.00
030.
0000
0.00
000.
0000
0.00
000.
0000
0.00
006
1.00
000.
9988
0.98
820.
9794
0.94
870.
8610
0.72
170.
6085
0.54
910.
3743
0.22
580.
1189
0.05
370.
0203
0.00
620.
0039
0.00
140.
0002
0.00
000.
0000
0.00
000.
0000
0.00
007
1.00
000.
9998
0.99
730.
9947
0.98
370.
9431
0.85
930.
7767
0.72
830.
5634
0.39
150.
2403
0.12
800.
0576
0.02
120.
0144
0.00
610.
0012
0.00
020.
0000
0.00
000.
0000
0.00
008
1.00
001.
0000
0.99
950.
9989
0.99
570.
9807
0.94
040.
8924
0.86
090.
7368
0.57
780.
4073
0.25
270.
1347
0.05
970.
0433
0.02
100.
0054
0.00
090.
0002
0.00
010.
0000
0.00
009
1.00
001.
0000
0.99
990.
9998
0.99
910.
9946
0.97
900.
9567
0.94
030.
8653
0.74
730.
5927
0.42
220.
2632
0.13
910.
1076
0.05
960.
0193
0.00
430.
0011
0.00
050.
0000
0.00
0010
1.00
001.
0000
1.00
001.
0000
0.99
980.
9988
0.99
390.
9856
0.97
880.
9424
0.87
200.
7597
0.60
850.
4366
0.27
170.
2233
0.14
070.
0569
0.01
630.
0053
0.00
270.
0002
0.00
0011
1.00
001.
0000
1.00
001.
0000
1.00
000.
9998
0.99
860.
9961
0.99
380.
9797
0.94
630.
8811
0.77
420.
6257
0.45
090.
3915
0.27
830.
1390
0.05
130.
0206
0.01
180.
0012
0.00
0012
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
0.99
970.
9991
0.99
860.
9942
0.98
170.
9519
0.89
230.
7912
0.64
500.
5878
0.46
560.
2825
0.13
290.
0653
0.04
190.
0064
0.00
0213
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
000.
9999
0.99
970.
9987
0.99
510.
9846
0.95
890.
9058
0.81
140.
7689
0.66
730.
4813
0.28
360.
1682
0.12
060.
0282
0.00
1514
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
000.
9998
0.99
900.
9962
0.98
800.
9672
0.92
170.
8983
0.83
540.
6943
0.49
900.
3521
0.27
980.
0982
0.01
0915
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
0.99
990.
9993
0.99
750.
9918
0.97
640.
9674
0.94
000.
8647
0.72
870.
5973
0.52
030.
2662
0.05
8116
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
000.
9999
0.99
970.
9987
0.99
540.
9932
0.98
580.
9605
0.90
090.
8272
0.77
590.
5497
0.22
6517
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
000.
9999
0.99
960.
9993
0.99
840.
9944
0.98
200.
9624
0.94
640.
8499
0.60
2818
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
00
n=2
0p
0.05
0.1
0.15
1/60.
20.
250.
31/3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
2/30.
70.
750.
85/6
0.85
0.9
0.95
x=0
0.35
850.
1216
0.03
880.
0261
0.01
150.
0032
0.00
080.
0003
0.00
020.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
001
0.73
580.
3917
0.17
560.
1304
0.06
920.
0243
0.00
760.
0033
0.00
210.
0005
0.00
010.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
002
0.92
450.
6769
0.40
490.
3287
0.20
610.
0913
0.03
550.
0176
0.01
210.
0036
0.00
090.
0002
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
003
0.98
410.
8670
0.64
770.
5665
0.41
140.
2252
0.10
710.
0604
0.04
440.
0160
0.00
490.
0013
0.00
030.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
004
0.99
740.
9568
0.82
980.
7687
0.62
960.
4148
0.23
750.
1515
0.11
820.
0510
0.01
890.
0059
0.00
150.
0003
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
005
0.99
970.
9887
0.93
270.
8982
0.80
420.
6172
0.41
640.
2972
0.24
540.
1256
0.05
530.
0207
0.00
640.
0016
0.00
030.
0002
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
006
1.00
000.
9976
0.97
810.
9629
0.91
330.
7858
0.60
800.
4793
0.41
660.
2500
0.12
990.
0577
0.02
140.
0065
0.00
150.
0009
0.00
030.
0000
0.00
000.
0000
0.00
000.
0000
0.00
007
1.00
000.
9996
0.99
410.
9887
0.96
790.
8982
0.77
230.
6615
0.60
100.
4159
0.25
200.
1316
0.05
800.
0210
0.00
600.
0037
0.00
130.
0002
0.00
000.
0000
0.00
000.
0000
0.00
008
1.00
000.
9999
0.99
870.
9972
0.99
000.
9591
0.88
670.
8095
0.76
240.
5956
0.41
430.
2517
0.13
080.
0565
0.01
960.
0130
0.00
510.
0009
0.00
010.
0000
0.00
000.
0000
0.00
009
1.00
001.
0000
0.99
980.
9994
0.99
740.
9861
0.95
200.
9081
0.87
820.
7553
0.59
140.
4119
0.24
930.
1275
0.05
320.
0376
0.01
710.
0039
0.00
060.
0001
0.00
000.
0000
0.00
0010
1.00
001.
0000
1.00
000.
9999
0.99
940.
9961
0.98
290.
9624
0.94
680.
8725
0.75
070.
5881
0.40
860.
2447
0.12
180.
0919
0.04
800.
0139
0.00
260.
0006
0.00
020.
0000
0.00
0011
1.00
001.
0000
1.00
001.
0000
0.99
990.
9991
0.99
490.
9870
0.98
040.
9435
0.86
920.
7483
0.58
570.
4044
0.23
760.
1905
0.11
330.
0409
0.01
000.
0028
0.00
130.
0001
0.00
0012
1.00
001.
0000
1.00
001.
0000
1.00
000.
9998
0.99
870.
9963
0.99
400.
9790
0.94
200.
8684
0.74
800.
5841
0.39
900.
3385
0.22
770.
1018
0.03
210.
0113
0.00
590.
0004
0.00
0013
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
0.99
970.
9991
0.99
850.
9935
0.97
860.
9423
0.87
010.
7500
0.58
340.
5207
0.39
200.
2142
0.08
670.
0371
0.02
190.
0024
0.00
0014
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
000.
9998
0.99
970.
9984
0.99
360.
9793
0.94
470.
8744
0.75
460.
7028
0.58
360.
3828
0.19
580.
1018
0.06
730.
0113
0.00
0315
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
000.
9997
0.99
850.
9941
0.98
110.
9490
0.88
180.
8485
0.76
250.
5852
0.37
040.
2313
0.17
020.
0432
0.00
2616
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
0.99
970.
9987
0.99
510.
9840
0.95
560.
9396
0.89
290.
7748
0.58
860.
4335
0.35
230.
1330
0.01
5917
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
000.
9998
0.99
910.
9964
0.98
790.
9824
0.96
450.
9087
0.79
390.
6713
0.59
510.
3231
0.07
5518
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
0.99
990.
9995
0.99
790.
9967
0.99
240.
9757
0.93
080.
8696
0.82
440.
6083
0.26
4219
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
0.99
980.
9997
0.99
920.
9968
0.98
850.
9739
0.96
120.
8784
0.64
1520
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
00
13
CUMULATIVEBINOMIALPROBABILITIES
n=2
5p
0.05
0.1
0.15
1/60.
20.
250.
31/3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
2/30.
70.
750.
85/6
0.85
0.9
0.95
x=0
0.27
740.
0718
0.01
720.
0105
0.00
380.
0008
0.00
010.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
001
0.64
240.
2712
0.09
310.
0629
0.02
740.
0070
0.00
160.
0005
0.00
030.
0001
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
002
0.87
290.
5371
0.25
370.
1887
0.09
820.
0321
0.00
900.
0035
0.00
210.
0004
0.00
010.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
003
0.96
590.
7636
0.47
110.
3816
0.23
400.
0962
0.03
320.
0149
0.00
970.
0024
0.00
050.
0001
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
004
0.99
280.
9020
0.68
210.
5937
0.42
070.
2137
0.09
050.
0462
0.03
200.
0095
0.00
230.
0005
0.00
010.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
005
0.99
880.
9666
0.83
850.
7720
0.61
670.
3783
0.19
350.
1120
0.08
260.
0294
0.00
860.
0020
0.00
040.
0001
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
006
0.99
980.
9905
0.93
050.
8908
0.78
000.
5611
0.34
070.
2215
0.17
340.
0736
0.02
580.
0073
0.00
160.
0003
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
007
1.00
000.
9977
0.97
450.
9553
0.89
090.
7265
0.51
180.
3703
0.30
610.
1536
0.06
390.
0216
0.00
580.
0012
0.00
020.
0001
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
008
1.00
000.
9995
0.99
200.
9843
0.95
320.
8506
0.67
690.
5376
0.46
680.
2735
0.13
400.
0539
0.01
740.
0043
0.00
080.
0004
0.00
010.
0000
0.00
000.
0000
0.00
000.
0000
0.00
009
1.00
000.
9999
0.99
790.
9953
0.98
270.
9287
0.81
060.
6956
0.63
030.
4246
0.24
240.
1148
0.04
400.
0132
0.00
290.
0016
0.00
050.
0000
0.00
000.
0000
0.00
000.
0000
0.00
0010
1.00
001.
0000
0.99
950.
9988
0.99
440.
9703
0.90
220.
8220
0.77
120.
5858
0.38
430.
2122
0.09
600.
0344
0.00
930.
0056
0.00
180.
0002
0.00
000.
0000
0.00
000.
0000
0.00
0011
1.00
001.
0000
0.99
990.
9997
0.99
850.
9893
0.95
580.
9082
0.87
460.
7323
0.54
260.
3450
0.18
270.
0778
0.02
550.
0164
0.00
600.
0009
0.00
010.
0000
0.00
000.
0000
0.00
0012
1.00
001.
0000
1.00
000.
9999
0.99
960.
9966
0.98
250.
9585
0.93
960.
8462
0.69
370.
5000
0.30
630.
1538
0.06
040.
0415
0.01
750.
0034
0.00
040.
0001
0.00
000.
0000
0.00
0013
1.00
001.
0000
1.00
001.
0000
0.99
990.
9991
0.99
400.
9836
0.97
450.
9222
0.81
730.
6550
0.45
740.
2677
0.12
540.
0918
0.04
420.
0107
0.00
150.
0003
0.00
010.
0000
0.00
0014
1.00
001.
0000
1.00
001.
0000
1.00
000.
9998
0.99
820.
9944
0.99
070.
9656
0.90
400.
7878
0.61
570.
4142
0.22
880.
1780
0.09
780.
0297
0.00
560.
0012
0.00
050.
0000
0.00
0015
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
0.99
950.
9984
0.99
710.
9868
0.95
600.
8852
0.75
760.
5754
0.36
970.
3044
0.18
940.
0713
0.01
730.
0047
0.00
210.
0001
0.00
0016
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
0.99
990.
9996
0.99
920.
9957
0.98
260.
9461
0.86
600.
7265
0.53
320.
4624
0.32
310.
1494
0.04
680.
0157
0.00
800.
0005
0.00
0017
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
000.
9999
0.99
980.
9988
0.99
420.
9784
0.93
610.
8464
0.69
390.
6297
0.48
820.
2735
0.10
910.
0447
0.02
550.
0023
0.00
0018
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
000.
9997
0.99
840.
9927
0.97
420.
9264
0.82
660.
7785
0.65
930.
4389
0.22
000.
1092
0.06
950.
0095
0.00
0219
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
000.
9999
0.99
960.
9980
0.99
140.
9706
0.91
740.
8880
0.80
650.
6217
0.38
330.
2280
0.16
150.
0334
0.00
1220
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
0.99
990.
9995
0.99
770.
9905
0.96
800.
9538
0.90
950.
7863
0.57
930.
4063
0.31
790.
0980
0.00
7221
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
000.
9999
0.99
950.
9976
0.99
030.
9851
0.96
680.
9038
0.76
600.
6184
0.52
890.
2364
0.03
4122
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
0.99
990.
9996
0.99
790.
9965
0.99
100.
9679
0.90
180.
8113
0.74
630.
4629
0.12
7123
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
000.
9999
0.99
970.
9995
0.99
840.
9930
0.97
260.
9371
0.90
690.
7288
0.35
7624
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
0.99
990.
9992
0.99
620.
9895
0.98
280.
9282
0.72
2625
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
001.
0000
1.00
00
14
CUMULATIVEBINOMIALPROBABILITIES
n=3
0p
0.05
0.1
0.15
1/60.
20.
250.
31/3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
2/30.
70.
750.
85/6
0.85
0.9
0.95
x=0
0.21
460.
0424
0.00
760.
0042
0.00
120.
0002
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
001
0.55
350.
1837
0.04
800.
0295
0.01
050.
0020
0.00
030.
0001
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
002
0.81
220.
4114
0.15
140.
1028
0.04
420.
0106
0.00
210.
0007
0.00
030.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
003
0.93
920.
6474
0.32
170.
2396
0.12
270.
0374
0.00
930.
0033
0.00
190.
0003
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
004
0.98
440.
8245
0.52
450.
4243
0.25
520.
0979
0.03
020.
0122
0.00
750.
0015
0.00
020.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
005
0.99
670.
9268
0.71
060.
6164
0.42
750.
2026
0.07
660.
0355
0.02
330.
0057
0.00
110.
0002
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
006
0.99
940.
9742
0.84
740.
7765
0.60
700.
3481
0.15
950.
0838
0.05
860.
0172
0.00
400.
0007
0.00
010.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
007
0.99
990.
9922
0.93
020.
8863
0.76
080.
5143
0.28
140.
1668
0.12
380.
0435
0.01
210.
0026
0.00
040.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
008
1.00
000.
9980
0.97
220.
9494
0.87
130.
6736
0.43
150.
2860
0.22
470.
0940
0.03
120.
0081
0.00
160.
0002
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
009
1.00
000.
9995
0.99
030.
9803
0.93
890.
8034
0.58
880.
4317
0.35
750.
1763
0.06
940.
0214
0.00
500.
0009
0.00
010.
0000
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
0010
1.00
000.
9999
0.99
710.
9933
0.97
440.
8943
0.73
040.
5848
0.50
780.
2915
0.13
500.
0494
0.01
380.
0029
0.00
040.
0002
0.00
000.
0000
0.00
000.
0000
0.00
000.
0000
0.00
0011
1.00
001.
0000
0.99
920.
9980
0.99
050.
9493
0.84
070.
7239
0.65
480.
4311
0.23
270.
1002
0.03
340.
0083
0.00
140.
0007
0.00
020.
0000
0.00
000.
0000
0.00
000.
0000
0.00
0012
1.00
001.
0000
0.99
980.
9995
0.99
690.
9784
0.91
550.
8340
0.78
020.
5785
0.35
920.
1808
0.07
140.
0212
0.00
450.
0025
0.00
060.
0001
0.00
000.
0000
0.00
000.
0000
0.00
0013
1.00
001.
0000
1.00
000.
9999
0.99
910.
9918
0.95
990.
9102
0.87
370.
7145
0.50
250.
2923
0.13
560.
0481
0.01
240.
0072
0.00
210.
0002
0.00
000.
0000
0.00
000.
0000
0.00
0014
1.00
001.
0000
1.00
001.
0000
0.99
980.
9973
0.98
310.
9565
0.93
480.
8246
0.64
480.
4278
0.23
090.
0971
0.03
010.
0188
0.00
640.
0008
0.00
010.
0000
0.00
000.
0000
0.00
0015
1.00
001.
0000
1.00
001.
0000
0.99
990.
9992
0.99
360.
9812
0.96
990.
9029
0.76
910.
5722
0.35
520.
1754
0.06
520.
0435
0.01
690.
0027
0.00
020.
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0.99
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0.98
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9519
0.86
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0.49
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0.12
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0898
0.04
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0.99
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0.92
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8192
0.64
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0.21
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0.08
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0216
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310.
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0.76
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0.34
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0020
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9506
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0.49
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4152
0.26
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0.02
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0.64
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5683
0.41
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0.06
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9060
0.77
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7140
0.56
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0.12
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0506
0.02
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0.00
0022
1.00
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0.98
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0.87
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0.71
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0.23
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0.94
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9162
0.84
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0.39
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0.15
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0.99
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0.97
670.
9645
0.92
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0.57
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0.28
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0.74
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0.47
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9958
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9576
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00
15
CUMULATIVE POISSON PROBABILITIES
λ 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09x = 0 0.9900 0.9802 0.9704 0.9608 0.9512 0.9418 0.9324 0.9231 0.9139
1 1.0000 0.9998 0.9996 0.9992 0.9988 0.9983 0.9977 0.9970 0.99622 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.99993 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
λ 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90x = 0 0.9048 0.8187 0.7408 0.6703 0.6065 0.5488 0.4966 0.4493 0.4066
1 0.9953 0.9825 0.9631 0.9384 0.9098 0.8781 0.8442 0.8088 0.77252 0.9998 0.9989 0.9964 0.9921 0.9856 0.9769 0.9659 0.9526 0.93713 1.0000 0.9999 0.9997 0.9992 0.9982 0.9966 0.9942 0.9909 0.98654 1.0000 1.0000 1.0000 0.9999 0.9998 0.9996 0.9992 0.9986 0.99775 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.99976 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
λ 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90x = 0 0.3679 0.3329 0.3012 0.2725 0.2466 0.2231 0.2019 0.1827 0.1653 0.1496
1 0.7358 0.6990 0.6626 0.6268 0.5918 0.5578 0.5249 0.4932 0.4628 0.43372 0.9197 0.9004 0.8795 0.8571 0.8335 0.8088 0.7834 0.7572 0.7306 0.70373 0.9810 0.9743 0.9662 0.9569 0.9463 0.9344 0.9212 0.9068 0.8913 0.87474 0.9963 0.9946 0.9923 0.9893 0.9857 0.9814 0.9763 0.9704 0.9636 0.95595 0.9994 0.9990 0.9985 0.9978 0.9968 0.9955 0.9940 0.9920 0.9896 0.98686 0.9999 0.9999 0.9997 0.9996 0.9994 0.9991 0.9987 0.9981 0.9974 0.99667 1.0000 1.0000 1.0000 0.9999 0.9999 0.9998 0.9997 0.9996 0.9994 0.99928 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.99989 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
λ 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90x = 0 0.1353 0.1225 0.1108 0.1003 0.0907 0.0821 0.0743 0.0672 0.0608 0.0550
1 0.4060 0.3796 0.3546 0.3309 0.3084 0.2873 0.2674 0.2487 0.2311 0.21462 0.6767 0.6496 0.6227 0.5960 0.5697 0.5438 0.5184 0.4936 0.4695 0.44603 0.8571 0.8386 0.8194 0.7993 0.7787 0.7576 0.7360 0.7141 0.6919 0.66964 0.9473 0.9379 0.9275 0.9162 0.9041 0.8912 0.8774 0.8629 0.8477 0.83185 0.9834 0.9796 0.9751 0.9700 0.9643 0.9580 0.9510 0.9433 0.9349 0.92586 0.9955 0.9941 0.9925 0.9906 0.9884 0.9858 0.9828 0.9794 0.9756 0.97137 0.9989 0.9985 0.9980 0.9974 0.9967 0.9958 0.9947 0.9934 0.9919 0.99018 0.9998 0.9997 0.9995 0.9994 0.9991 0.9989 0.9985 0.9981 0.9976 0.99699 1.0000 0.9999 0.9999 0.9999 0.9998 0.9997 0.9996 0.9995 0.9993 0.9991
10 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9998 0.999811 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.999912 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
λ 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90x = 0 0.0498 0.0450 0.0408 0.0369 0.0334 0.0302 0.0273 0.0247 0.0224 0.0202
1 0.1991 0.1847 0.1712 0.1586 0.1468 0.1359 0.1257 0.1162 0.1074 0.09922 0.4232 0.4012 0.3799 0.3594 0.3397 0.3208 0.3027 0.2854 0.2689 0.25313 0.6472 0.6248 0.6025 0.5803 0.5584 0.5366 0.5152 0.4942 0.4735 0.45324 0.8153 0.7982 0.7806 0.7626 0.7442 0.7254 0.7064 0.6872 0.6678 0.64845 0.9161 0.9057 0.8946 0.8829 0.8705 0.8576 0.8441 0.8301 0.8156 0.80066 0.9665 0.9612 0.9554 0.9490 0.9421 0.9347 0.9267 0.9182 0.9091 0.89957 0.9881 0.9858 0.9832 0.9802 0.9769 0.9733 0.9692 0.9648 0.9599 0.95468 0.9962 0.9953 0.9943 0.9931 0.9917 0.9901 0.9883 0.9863 0.9840 0.98159 0.9989 0.9986 0.9982 0.9978 0.9973 0.9967 0.9960 0.9952 0.9942 0.9931
10 0.9997 0.9996 0.9995 0.9994 0.9992 0.9990 0.9987 0.9984 0.9981 0.997711 0.9999 0.9999 0.9999 0.9998 0.9998 0.9997 0.9996 0.9995 0.9994 0.999312 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9999 0.9998 0.999813 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.999914 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
16
CUMULATIVE POISSON PROBABILITIES
λ 4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90
x = 0 0.0183 0.0166 0.0150 0.0136 0.0123 0.0111 0.0101 0.0091 0.0082 0.00741 0.0916 0.0845 0.0780 0.0719 0.0663 0.0611 0.0563 0.0518 0.0477 0.04392 0.2381 0.2238 0.2102 0.1974 0.1851 0.1736 0.1626 0.1523 0.1425 0.13333 0.4335 0.4142 0.3954 0.3772 0.3594 0.3423 0.3257 0.3097 0.2942 0.27934 0.6288 0.6093 0.5898 0.5704 0.5512 0.5321 0.5132 0.4946 0.4763 0.45825 0.7851 0.7693 0.7531 0.7367 0.7199 0.7029 0.6858 0.6684 0.6510 0.63356 0.8893 0.8786 0.8675 0.8558 0.8436 0.8311 0.8180 0.8046 0.7908 0.77677 0.9489 0.9427 0.9361 0.9290 0.9214 0.9134 0.9049 0.8960 0.8867 0.87698 0.9786 0.9755 0.9721 0.9683 0.9642 0.9597 0.9549 0.9497 0.9442 0.93829 0.9919 0.9905 0.9889 0.9871 0.9851 0.9829 0.9805 0.9778 0.9749 0.9717
10 0.9972 0.9966 0.9959 0.9952 0.9943 0.9933 0.9922 0.9910 0.9896 0.988011 0.9991 0.9989 0.9986 0.9983 0.9980 0.9976 0.9971 0.9966 0.9960 0.995312 0.9997 0.9997 0.9996 0.9995 0.9993 0.9992 0.9990 0.9988 0.9986 0.998313 0.9999 0.9999 0.9999 0.9998 0.9998 0.9997 0.9997 0.9996 0.9995 0.999414 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9999 0.9999 0.999815 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.999916 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
λ 5.00 5.50 6.00 6.50 7.00 7.50 8.00 8.50 9.00 9.50
x = 0 0.0067 0.0041 0.0025 0.0015 0.0009 0.0006 0.0003 0.0002 0.0001 0.00011 0.0404 0.0266 0.0174 0.0113 0.0073 0.0047 0.0030 0.0019 0.0012 0.00082 0.1247 0.0884 0.0620 0.0430 0.0296 0.0203 0.0138 0.0093 0.0062 0.00423 0.2650 0.2017 0.1512 0.1118 0.0818 0.0591 0.0424 0.0301 0.0212 0.01494 0.4405 0.3575 0.2851 0.2237 0.1730 0.1321 0.0996 0.0744 0.0550 0.04035 0.6160 0.5289 0.4457 0.3690 0.3007 0.2414 0.1912 0.1496 0.1157 0.08856 0.7622 0.6860 0.6063 0.5265 0.4497 0.3782 0.3134 0.2562 0.2068 0.16497 0.8666 0.8095 0.7440 0.6728 0.5987 0.5246 0.4530 0.3856 0.3239 0.26878 0.9319 0.8944 0.8472 0.7916 0.7291 0.6620 0.5925 0.5231 0.4557 0.39189 0.9682 0.9462 0.9161 0.8774 0.8305 0.7764 0.7166 0.6530 0.5874 0.5218
10 0.9863 0.9747 0.9574 0.9332 0.9015 0.8622 0.8159 0.7634 0.7060 0.645311 0.9945 0.9890 0.9799 0.9661 0.9467 0.9208 0.8881 0.8487 0.8030 0.752012 0.9980 0.9955 0.9912 0.9840 0.9730 0.9573 0.9362 0.9091 0.8758 0.836413 0.9993 0.9983 0.9964 0.9929 0.9872 0.9784 0.9658 0.9486 0.9261 0.898114 0.9998 0.9994 0.9986 0.9970 0.9943 0.9897 0.9827 0.9726 0.9585 0.940015 0.9999 0.9998 0.9995 0.9988 0.9976 0.9954 0.9918 0.9862 0.9780 0.966516 1.0000 0.9999 0.9998 0.9996 0.9990 0.9980 0.9963 0.9934 0.9889 0.982317 1.0000 1.0000 0.9999 0.9998 0.9996 0.9992 0.9984 0.9970 0.9947 0.991118 1.0000 1.0000 1.0000 0.9999 0.9999 0.9997 0.9993 0.9987 0.9976 0.995719 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9995 0.9989 0.998020 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9996 0.999121 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.999622 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.999923 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.999924 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
17
CUMULATIVE POISSON PROBABILITIES
λ 10.00 11.00 12.00 13.00 14.00 15.00 16.00 17.00 18.00 19.00
x = 0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.00001 0.0005 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.00002 0.0028 0.0012 0.0005 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 0.00003 0.0103 0.0049 0.0023 0.0011 0.0005 0.0002 0.0001 0.0000 0.0000 0.00004 0.0293 0.0151 0.0076 0.0037 0.0018 0.0009 0.0004 0.0002 0.0001 0.00005 0.0671 0.0375 0.0203 0.0107 0.0055 0.0028 0.0014 0.0007 0.0003 0.00026 0.1301 0.0786 0.0458 0.0259 0.0142 0.0076 0.0040 0.0021 0.0010 0.00057 0.2202 0.1432 0.0895 0.0540 0.0316 0.0180 0.0100 0.0054 0.0029 0.00158 0.3328 0.2320 0.1550 0.0998 0.0621 0.0374 0.0220 0.0126 0.0071 0.00399 0.4579 0.3405 0.2424 0.1658 0.1094 0.0699 0.0433 0.0261 0.0154 0.0089
10 0.5830 0.4599 0.3472 0.2517 0.1757 0.1185 0.0774 0.0491 0.0304 0.018311 0.6968 0.5793 0.4616 0.3532 0.2600 0.1848 0.1270 0.0847 0.0549 0.034712 0.7916 0.6887 0.5760 0.4631 0.3585 0.2676 0.1931 0.1350 0.0917 0.060613 0.8645 0.7813 0.6815 0.5730 0.4644 0.3632 0.2745 0.2009 0.1426 0.098414 0.9165 0.8540 0.7720 0.6751 0.5704 0.4657 0.3675 0.2808 0.2081 0.149715 0.9513 0.9074 0.8444 0.7636 0.6694 0.5681 0.4667 0.3715 0.2867 0.214816 0.9730 0.9441 0.8987 0.8355 0.7559 0.6641 0.5660 0.4677 0.3751 0.292017 0.9857 0.9678 0.9370 0.8905 0.8272 0.7489 0.6593 0.5640 0.4686 0.378418 0.9928 0.9823 0.9626 0.9302 0.8826 0.8195 0.7423 0.6550 0.5622 0.469519 0.9965 0.9907 0.9787 0.9573 0.9235 0.8752 0.8122 0.7363 0.6509 0.560620 0.9984 0.9953 0.9884 0.9750 0.9521 0.9170 0.8682 0.8055 0.7307 0.647221 0.9993 0.9977 0.9939 0.9859 0.9712 0.9469 0.9108 0.8615 0.7991 0.725522 0.9997 0.9990 0.9970 0.9924 0.9833 0.9673 0.9418 0.9047 0.8551 0.793123 0.9999 0.9995 0.9985 0.9960 0.9907 0.9805 0.9633 0.9367 0.8989 0.849024 1.0000 0.9998 0.9993 0.9980 0.9950 0.9888 0.9777 0.9594 0.9317 0.893325 1.0000 0.9999 0.9997 0.9990 0.9974 0.9938 0.9869 0.9748 0.9554 0.926926 1.0000 1.0000 0.9999 0.9995 0.9987 0.9967 0.9925 0.9848 0.9718 0.951427 1.0000 1.0000 0.9999 0.9998 0.9994 0.9983 0.9959 0.9912 0.9827 0.968728 1.0000 1.0000 1.0000 0.9999 0.9997 0.9991 0.9978 0.9950 0.9897 0.980529 1.0000 1.0000 1.0000 1.0000 0.9999 0.9996 0.9989 0.9973 0.9941 0.988230 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9994 0.9986 0.9967 0.993031 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9993 0.9982 0.996032 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9996 0.9990 0.997833 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9995 0.998834 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.999435 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.999736 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.999837 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.999938 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
18
THE NORMAL DISTRIBUTION FUNCTION
If Z has a normal distribution with mean 0 andvariance 1 then, for each value of �, the table givesthe value of Φ(�), where
Φ(�) = P(Z ≤ �).For negative values of � use Φ(−�) = 1 − Φ(�).
� 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9ADD
0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 4 8 12 16 20 24 28 32 360.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 4 8 12 16 20 24 28 32 360.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 4 8 12 15 19 23 27 31 350.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 4 7 11 15 19 22 26 30 340.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 4 7 11 14 18 22 25 29 32
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 3 7 10 14 17 20 24 27 310.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 3 7 10 13 16 19 23 26 290.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 3 6 9 12 15 18 21 24 270.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 3 5 8 11 14 16 19 22 250.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 3 5 8 10 13 15 18 20 23
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 2 5 7 9 12 14 16 19 211.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 2 4 6 8 10 12 14 16 181.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 2 4 6 7 9 11 13 15 171.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 2 3 5 6 8 10 11 13 141.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 1 3 4 6 7 8 10 11 13
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 1 2 4 5 6 7 8 10 111.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 1 2 3 4 5 6 7 8 91.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 1 2 3 4 4 5 6 7 81.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706 1 1 2 3 4 4 5 6 61.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767 1 1 2 2 3 4 4 5 5
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817 0 1 1 2 2 3 3 4 42.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 0 1 1 2 2 2 3 3 42.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890 0 1 1 1 2 2 2 3 32.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 0 1 1 1 1 2 2 2 22.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936 0 0 1 1 1 1 1 2 2
2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952 0 0 0 1 1 1 1 1 12.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 0 0 0 0 1 1 1 1 12.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 0 0 0 0 0 1 1 1 12.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981 0 0 0 0 0 0 0 1 12.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986 0 0 0 0 0 0 0 0 0
Critical values for the normal distribution
If Z has a normal distribution with mean 0 andvariance 1 then, for each value of p, the table givesthe value of � such that
P(Z ≤ �) = p.
p 0.75 0.90 0.95 0.975 0.99 0.995 0.9975 0.999 0.9995
� 0.674 1.282 1.645 1.960 2.326 2.576 2.807 3.090 3.291
19
CRITICAL VALUES FOR THE t DISTRIBUTION
If T has a t distribution with v degrees of freedomthen, for each pair of values of p and v, the tablegives the value of t such that
P(T ≤ t) = p.
p 0.75 0.90 0.95 0.975 0.99 0.995 0.9975 0.999 0.9995
v = 1 1.000 3.078 6.314 12.71 31.82 63.66 127.3 318.3 636.62 0.816 1.886 2.920 4.303 6.965 9.925 14.09 22.33 31.603 0.765 1.638 2.353 3.182 4.541 5.841 7.453 10.21 12.924 0.741 1.533 2.132 2.776 3.747 4.604 5.598 7.173 8.610
5 0.727 1.476 2.015 2.571 3.365 4.032 4.773 5.894 6.8696 0.718 1.440 1.943 2.447 3.143 3.707 4.317 5.208 5.9597 0.711 1.415 1.895 2.365 2.998 3.499 4.029 4.785 5.4088 0.706 1.397 1.860 2.306 2.896 3.355 3.833 4.501 5.0419 0.703 1.383 1.833 2.262 2.821 3.250 3.690 4.297 4.781
10 0.700 1.372 1.812 2.228 2.764 3.169 3.581 4.144 4.58711 0.697 1.363 1.796 2.201 2.718 3.106 3.497 4.025 4.43712 0.695 1.356 1.782 2.179 2.681 3.055 3.428 3.930 4.31813 0.694 1.350 1.771 2.160 2.650 3.012 3.372 3.852 4.22114 0.692 1.345 1.761 2.145 2.624 2.977 3.326 3.787 4.140
15 0.691 1.341 1.753 2.131 2.602 2.947 3.286 3.733 4.07316 0.690 1.337 1.746 2.120 2.583 2.921 3.252 3.686 4.01517 0.689 1.333 1.740 2.110 2.567 2.898 3.222 3.646 3.96518 0.688 1.330 1.734 2.101 2.552 2.878 3.197 3.610 3.92219 0.688 1.328 1.729 2.093 2.539 2.861 3.174 3.579 3.883
20 0.687 1.325 1.725 2.086 2.528 2.845 3.153 3.552 3.85021 0.686 1.323 1.721 2.080 2.518 2.831 3.135 3.527 3.81922 0.686 1.321 1.717 2.074 2.508 2.819 3.119 3.505 3.79223 0.685 1.319 1.714 2.069 2.500 2.807 3.104 3.485 3.76824 0.685 1.318 1.711 2.064 2.492 2.797 3.091 3.467 3.745
25 0.684 1.316 1.708 2.060 2.485 2.787 3.078 3.450 3.72526 0.684 1.315 1.706 2.056 2.479 2.779 3.067 3.435 3.70727 0.684 1.314 1.703 2.052 2.473 2.771 3.057 3.421 3.68928 0.683 1.313 1.701 2.048 2.467 2.763 3.047 3.408 3.67429 0.683 1.311 1.699 2.045 2.462 2.756 3.038 3.396 3.660
30 0.683 1.310 1.697 2.042 2.457 2.750 3.030 3.385 3.64640 0.681 1.303 1.684 2.021 2.423 2.704 2.971 3.307 3.55160 0.679 1.296 1.671 2.000 2.390 2.660 2.915 3.232 3.460
120 0.677 1.289 1.658 1.980 2.358 2.617 2.860 3.160 3.373∞ 0.674 1.282 1.645 1.960 2.326 2.576 2.807 3.090 3.291
20
CRITICAL VALUES FOR THE χχχχχ2 DISTRIBUTION
If X has a χ2 distribution with v degrees of freedomthen, for each pair of values of p and v, the tablegives the value of x such that
P(X ≤ x) = p.
p 0.01 0.025 0.05 0.90 0.95 0.975 0.99 0.995 0.999
v = 1 0.031571 0.039821 0.023932 2.706 3.841 5.024 6.635 7.879 10.832 0.02010 0.05064 0.1026 4.605 5.991 7.378 9.210 10.60 13.823 0.1148 0.2158 0.3518 6.251 7.815 9.348 11.34 12.84 16.274 0.2971 0.4844 0.7107 7.779 9.488 11.14 13.28 14.86 18.47
5 0.5543 0.8312 1.145 9.236 11.07 12.83 15.09 16.75 20.516 0.8721 1.237 1.635 10.64 12.59 14.45 16.81 18.55 22.467 1.239 1.690 2.167 12.02 14.07 16.01 18.48 20.28 24.328 1.647 2.180 2.733 13.36 15.51 17.53 20.09 21.95 26.129 2.088 2.700 3.325 14.68 16.92 19.02 21.67 23.59 27.88
10 2.558 3.247 3.940 15.99 18.31 20.48 23.21 25.19 29.5911 3.053 3.816 4.575 17.28 19.68 21.92 24.73 26.76 31.2612 3.571 4.404 5.226 18.55 21.03 23.34 26.22 28.30 32.9113 4.107 5.009 5.892 19.81 22.36 24.74 27.69 29.82 34.5314 4.660 5.629 6.571 21.06 23.68 26.12 29.14 31.32 36.12
15 5.229 6.262 7.261 22.31 25.00 27.49 30.58 32.80 37.7016 5.812 6.908 7.962 23.54 26.30 28.85 32.00 34.27 39.2517 6.408 7.564 8.672 24.77 27.59 30.19 33.41 35.72 40.7918 7.015 8.231 9.390 25.99 28.87 31.53 34.81 37.16 42.3119 7.633 8.907 10.12 27.20 30.14 32.85 36.19 38.58 43.82
20 8.260 9.591 10.85 28.41 31.41 34.17 37.57 40.00 45.3121 8.897 10.28 11.59 29.62 32.67 35.48 38.93 41.40 46.8022 9.542 10.98 12.34 30.81 33.92 36.78 40.29 42.80 48.2723 10.20 11.69 13.09 32.01 35.17 38.08 41.64 44.18 49.7324 10.86 12.40 13.85 33.20 36.42 39.36 42.98 45.56 51.18
25 11.52 13.12 14.61 34.38 37.65 40.65 44.31 46.93 52.6230 14.95 16.79 18.49 40.26 43.77 46.98 50.89 53.67 59.7040 22.16 24.43 26.51 51.81 55.76 59.34 63.69 66.77 73.4050 29.71 32.36 34.76 63.17 67.50 71.42 76.15 79.49 86.6660 37.48 40.48 43.19 74.40 79.08 83.30 88.38 91.95 99.61
70 45.44 48.76 51.74 85.53 90.53 95.02 100.4 104.2 112.380 53.54 57.15 60.39 96.58 101.9 106.6 112.3 116.3 124.890 61.75 65.65 69.13 107.6 113.1 118.1 124.1 128.3 137.2
100 70.06 74.22 77.93 118.5 124.3 129.6 135.8 140.2 149.4
21
WILCOXON SIGNED RANK TEST
P is the sum of the ranks corresponding to the positive differences,
Q is the sum of the ranks corresponding to the negative differences,
T is the smaller of P and Q.
For each value of n the table gives the largest value of T which will lead to rejection of the null hypothesis at the levelof significance indicated.
Critical values of T
Level of significance
One Tail 0.05 0.025 0.01 0.005Two Tail 0.10 0.05 0.02 0.01
n = 6 2 07 3 2 08 5 3 1 09 8 5 3 1
10 10 8 5 311 13 10 7 512 17 13 9 713 21 17 12 914 25 21 15 1215 30 25 19 1516 35 29 23 1917 41 34 27 2318 47 40 32 2719 53 46 37 3220 60 52 43 37
For larger values of n, each of P and Q can be approximated by the normal distribution with mean 14n(n + 1) and
variance 124
n(n + 1)(2n + 1).
22
WILCOXON RANK SUM TEST
The two samples have sizes m and n, where m ≤ n.
Rm is the sum of the ranks of the items in the sample of size m.
W is the smaller of Rm and m(m + n + 1) − Rm.
For each pair of values of m and n, the table gives the largest value of W which will lead to rejection of the nullhypothesis at the level of significance indicated.
Critical values ofW
Level of significance
One Tail 0.05 0.025 0.01 0.05 0.025 0.01 0.05 0.025 0.01 0.05 0.025 0.01Two Tail 0.1 0.05 0.02 0.1 0.05 0.02 0.1 0.05 0.02 0.1 0.05 0.02
n m = 3 m = 4 m = 5 m = 6
3 6 − −4 6 − − 11 10 −5 7 6 − 12 11 10 19 17 166 8 7 − 13 12 11 20 18 17 28 26 24
7 8 7 6 14 13 11 21 20 18 29 27 258 9 8 6 15 14 12 23 21 19 31 29 279 10 8 7 16 14 13 24 22 20 33 31 28
10 10 9 7 17 15 13 26 23 21 35 32 29
Level of significance
One Tail 0.05 0.025 0.01 0.05 0.025 0.01 0.05 0.025 0.01 0.05 0.025 0.01Two Tail 0.1 0.05 0.02 0.1 0.05 0.02 0.1 0.05 0.02 0.1 0.05 0.02
n m = 7 m = 8 m = 9 m = 10
7 39 36 348 41 38 35 51 49 459 43 40 37 54 51 47 66 62 59
10 45 42 39 56 53 49 69 65 61 82 78 74
For larger values of m and n, the normal distribution with mean 12m(m + n + 1) and variance 1
12mn(m + n + 1) should
be used as an approximation to the distribution of Rm.
23
BLANK PAGE
24
SP (NF) T41021/4
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