Sixth Term Examination Papers MATHEMATICS LIST OF FORMULAE AND STATISTICAL TABLES

Sixth Term Examiatio Papers MATHEMATICS LIST OF FORMULAE AND STATISTICAL TABLES

Pure Mathematics Mesuratio Surface area of sphere = 4πr Area of curved surface of coe = πr slat height Trigoometry a = b + c bc cos A Arithmetic Series u = a +( )d S = (a + l) = {a +( )d} Geometric Series u = ar S = a( r ) r S = a for r < r Summatios r= r= r = ( + )( + ) 6 r 3 = 4 ( + ) Biomial Series ( r ) + ( r + ) = ( + r + ) (a + b) = a + ( ) a b + ( ) a b +... + ( r ) a r b r +... + b ( ), where ( r ) = C r = ( + x) = + x +! r!( r)! ( ) x +... +. ( )... ( r + ) x r +.....3... r ( x <, ) Logarithms ad expoetials e x l a = a x Complex Numbers {r(cos θ + isiθ)} = r (cos θ + isiθ) e iθ = cos θ + isiθ πki The roots of = aregiveby =e,fork = 0,,,...,

Maclauri s Series f(x) =f(0)+xf (0)+ x! f (0)+... + xr r! f (r) (0)+... e x = exp(x) = + x + x xr +... + +...! r! forallx l( + x) =x x + x3 3... x r +( )r+ +... r ( < x ) si x = x x3 3! + x5 5!... x r+ +( )r +... (r + )! forallx cos x = x! + x4 4!... x r +( )r (r)! +... forallx ta x = x x3 3 + x5 5... x r+ +( )r +... ( x ) r + sih x = x + x3 3! + x5 xr+ +... + +... forallx 5! (r + )! cosh x = + x! + x4 xr +... + 4! (r)! +... forallx tah x = x + x3 3 + x5 xr+ +... + +... ( < x < ) 5 r + Hyperbolic Fuctios cosh x sih x = sih x = sihx cosh x cosh x = cosh x + sih x cosh x = l{x + (x )} (x ) sih x = l{x + (x + )} tah x = Coordiate Geometry + x l ( ) ( x <) x The perpedicular distace from (h, k) to ax + by + c = 0is ah + bk + c (a + b ) The acute agle betwee lies with gradiets m ad m is ta m m + m m Trigoometric Idetities si(a ± B) =si A cos B ± cos A si B cos(a ± B) =cos A cos B si A si B ta A ± ta B ta(a ± B) = ta A ta B (A ± B (k + )π) For t = ta A: sia = t si A + si B = si A + B si A si B = cos A + B cos A + cos B = cos A + B cos A cos B = si A + B t,cosa = + t + t cos A B si A B cos A B si A B 3

Vectors The resolved part of a i the directio of b is a.b b The poit dividig AB i the ratio λ : μ is μa + λb λ + μ a b 3 a 3 b = ( a 3 b a b 3 ) a b a b i a b Vector product: a b = a b si θ ˆ = j a b k a 3 b 3 If A is the poit with positio vector a = a i + a j + a 3 k ad the directio vector b is give by b = b i + b j + b 3 k, the the straight lie through A with directio vector b has cartesia equatio x a b = y a b = a 3 b 3 (= λ ) The plae through A with ormal vector = i + j + 3 k has cartesia equatio x + y + 3 +d = 0, where d = a. The plae through o-colliear poits A, B ad C has vector equatio r = a + λ (b a)+μ(c a) =( λ μ)a + λb + μc The plae through the poit with positio vector a ad parallel to b ad c has equatio r = a + sb + tc The perpedicular distace of (α, β, γ ) from x + y + 3 +d = 0is α + β + 3 γ + d ( + + 3 ) Matrix trasformatios Aticlockwise rotatio through θ about O: ( cos θ si θ si θ cos θ ) cos θ si θ Reflectioitheliey =(ta θ)x: ( si θ cos θ ) Differetiatio f(x) ta kx si x cos x ta x sec x cot x cosec x sihx cosh x tah x sih x cosh x tah x f (x) k sec kx ( x ) ( x ) + x sec x ta x cosec x cosec x cot x cosh x sih x sech x ( + x ) (x ) x 4

Itegratio ( + costat; a > 0 where relevat) f(x) f(x) dx sec kx ta kx k ta x l sec x cot x l si x cosec x l cosec x + cot x =l ta x sec x l sec x + ta x =l ta(x + π) 4 sih x cosh x cosh x sih x tah x l cosh x (a x ) si ( x ) a ( x < a) a + x a ta ( x a ) (x a ) cosh ( x a ) or l{x + (x a )} (x > a) (a + x ) sih ( x a ) or l{x + (x + a )} a x a l a + x a x = a tah ( x ) ( x < a) a x a a l x a x + a u dv dx dx = uv v du dx dx Area of a sector A = r dθ (polar coordiates) A = (x dy dt y dx dt ) dt (parametric form) Numerical Mathematics Numerical itegratio b The trapezium rule: y dx h{(y 0 + y )+(y + y +... + y b a )}, whereh = a b Simpso s Rule: y dx h{(y 3 0 + y )+4(y + y 3 +... + y )+(y + y 4 +... + y )}, a where h = b a ad is eve Numerical Solutio of Equatios The Newto-Raphso iteratio for solvig f(x) =0: x + = x f(x ) f (x ) 5

Mechaics Motio i a circle Trasverse velocity: v = r θ Trasverse acceleratio: v = r θ Radial acceleratio: r θ = v r Cetres of Mass (for uiform bodies) Triagular lamia: 3 alogmediafromvertex Solid hemisphere, radius r: 3 r from cetre 8 Hemispherical shell, radius r: r from cetre Circular arc, radius r, agleatcetreα: r si α α Sector of circle, radius r, agleatcetreα: r si α 3α from cetre from cetre Solid coe or pyramid of height h: h above the base o the lie from cetre of base to vertex 4 Coical shell of height h: h above the base o the lie from cetre of base to vertex 3 Momets of Iertia (for uiform bodies of mass m) Thi rod, legth l, about perpedicular axis through cetre: 3 ml Rectagular lamia about axis i plae bisectig edges of legth l: 3 ml Thi rod, legth l, about perpedicular axis through ed: 4 3 ml Rectagular lamia about edge perpedicular to edges of legth l: 4 3 ml Rectagular lamia, sides a ad b, about perpedicular axis through cetre: 3 m(a + b ) Hoop or cylidrical shell of radius r about axis: mr Hoop of radius r about a diameter: mr Disc or solid cylider of radius r about axis: mr Disc of radius r about a diameter: 4 mr Solid sphere, radius r, about diameter: 5 mr Spherical shell of radius r about a diameter: 3 mr Parallel axes theorem: I A = I G + m(ag) Perpedicular axes theorem: I = I x + I y (foralamiaithex-y plae) 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(A j B) = P(A j )P(B A j ) ΣP(A i )P(B A i ) Discrete distributios For a discrete radom variable X takig values x i with probabilities p i Expectatio (mea): E(X) =μ = Σ x i p i Variace: Var(X) =σ = Σ(x i μ) p i = Σ x i p i μ For a fuctio g(x): E(g(X)) = Σ g(x i )p i The probability geeratig fuctio of X is G X (t) =E(t X ),ad E(X) =G X () Var(X) =G X ()+G X () {G X ()} For Z = X + Y,whereX ad Y are idepedet: G Z (t) =G X (t)g Y (t) Stadard discrete distributios Distributio of X P(X = x) Mea Variace P.G.F. Biomial B(, p) ( x ) px ( p) x p p( p) ( p + pt) Poisso Po(λ ) e λ λ x x! Geometric Geo(p) o,, p( p) x p λ λ e λ(t ) p p pt ( p)t Cotiuous distributios For a cotiuous radom variable X havig probability desity fuctio f Expectatio (mea): E(X) =μ = xf(x) dx Variace: Var(X) =σ = (x μ) f(x) dx = x f(x) dx μ For a fuctio g(x): E(g(X)) = g(x)f(x) dx x Cumulative distributio fuctio: F(x) =P(X x) = f(t) dt The momet geeratig fuctio of X is M X (t) =E(e tx ) ad E(X) =M X (0) E(X )=M () X (0) Var(X) =M X (0) {M X (0)} For Z = X + Y,whereX ad Y are idepedet: M Z (t) =M X (t)m Y (t) 7

Stadard cotiuous distributios Distributio of X P.D.F. Mea Variace M.G.F. Uiform (Rectagular) o [a, b] b a Expoetial λe λx λ Normal N(μ, σ ) (a + b) (b e bt e at a) (b a)t λ λ λ t σ ( x μ (π) e σ ) μ σ e μt+ σ t Expectatio algebra Covariace: Cov(X, Y) =E((X μ X )(Y μ Y )) = E(XY) μ X μ Y Var(aX ± by) =a Var(X)+b Var(Y)±ab Cov(X, Y) Product momet correlatio coefficiet: ρ = Cov(X, Y) σ X σ Y If X = ax + b ad Y = cy + d, the Cov(X, Y) =ac Cov(X, Y ) For idepedet radom variables X ad Y E(XY) =E(X)E(Y) Var(aX ± by) =a Var(X)+b Var(Y) Samplig distributios For a radom sample X, X,..., X of idepedet observatios from a distributio havig mea μ ad variace σ X is a ubiased estimator of μ, with Var(X) = σ S is a ubiased estimator of σ,wheres = Σ(X i X) For a radom sample of observatios from N(μ, σ ) X μ σ/ N(0, ) X μ S/ t (also valid i matched-pairs situatios) If X is the observed umber of successes i idepedet Beroulli trials i each of which the probability of success is p, ady = X,the p( p) E(Y) =p ad Var(Y) = For a radom sample of x observatios from N(μ x, σ ) ad, idepedetly, a radom sample of x y observatios from N(μ y, σ y ) (X Y) (μ x μ y ) ( σ x + σ N(0, ) y ) x y If σ x = σ y = σ (ukow) the (X Y) (μ x μ y ) {Sp ( + )} x y t x + y, where S p = ( x )S x +( y )S y x + y 8

Correlatio ad regressio For a set of pairs of values (x i, y i ) S xx = Σ(x i x) = Σ x i (Σ x i ) S yy = Σ(y i y) = Σ y i (Σ y i ) S xy = Σ(x i x)(y i y) =Σ x i y i (Σ x i )(Σ y i ) The product momet correlatio coefficiet is r = S xy (Sxx S yy ) = Σ(x i x)(y i y) {(Σ(xi x) )(Σ(y i y) )} = Spearma s rak correlatio coefficiet is r s = (Σ x Σ x i y i i )(Σ y i ) {(Σ x i (Σ x i ) 6Σ d ( ) The regressio coefficiet of y o x is b = S xy S xx = Σ(x i x)(y i y) Σ(x i x) Least squares regressio lie of y o x is y = a + bx where a = y bx )(Σ y i (Σ y i ) )} Distributio-free (o-parametric) tests (O Goodess-of-fit test ad cotigecy tables: i E i ) χ ν Approximate distributios for large samples Wilcoxo Siged Rak test: T N( ( + ), ( + )( + )) 4 4 Wilcoxo Rak Sum test (samples of sizes m ad, withm ): W N( m(m + + ), m(m + + )) E i 9

CUMULATIVE BINOMIAL PROBABILITIES = 5 p 0.05 0. 0.5 /6 0. 0.5 0.3 /3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 /3 0.7 0.75 0.8 5/6 0.85 0.9 0.95 x = 0 0.7738 0.5905 0.4437 0.409 0.377 0.373 0.68 0.37 0.60 0.0778 0.0503 0.033 0.085 0.00 0.0053 0.004 0.004 0.000 0.0003 0.000 0.000 0.0000 0.0000 0.9774 0.985 0.835 0.8038 0.7373 0.638 0.58 0.4609 0.484 0.3370 0.56 0.875 0.3 0.0870 0.0540 0.0453 0.0308 0.056 0.0067 0.0033 0.00 0.0005 0.0000 0.9988 0.994 0.9734 0.9645 0.94 0.8965 0.8369 0.790 0.7648 0.686 0.593 0.5000 0.4069 0.374 0.35 0.099 0.63 0.035 0.0579 0.0355 0.066 0.0086 0.00 3.0000 0.9995 0.9978 0.9967 0.9933 0.9844 0.969 0.9547 0.9460 0.930 0.8688 0.85 0.7438 0.6630 0.576 0.539 0.478 0.367 0.67 0.96 0.648 0.085 0.06 4.0000.0000 0.9999 0.9999 0.9997 0.9990 0.9976 0.9959 0.9947 0.9898 0.985 0.9688 0.9497 0.9 0.8840 0.8683 0.839 0.767 0.673 0.598 0.5563 0.4095 0.6 5.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 = 6 p 0.05 0. 0.5 /6 0. 0.5 0.3 /3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 /3 0.7 0.75 0.8 5/6 0.85 0.9 0.95 x = 0 0.735 0.534 0.377 0.3349 0.6 0.780 0.76 0.0878 0.0754 0.0467 0.077 0.056 0.0083 0.004 0.008 0.004 0.0007 0.000 0.000 0.0000 0.0000 0.0000 0.0000 0.967 0.8857 0.7765 0.7368 0.6554 0.5339 0.40 0.35 0.39 0.333 0.636 0.094 0.069 0.040 0.03 0.078 0.009 0.0046 0.006 0.0007 0.0004 0.000 0.0000 0.9978 0.984 0.957 0.9377 0.90 0.8306 0.7443 0.6804 0.647 0.5443 0.445 0.3438 0.553 0.79 0.74 0.00 0.0705 0.0376 0.070 0.0087 0.0059 0.003 0.000 3 0.9999 0.9987 0.994 0.993 0.9830 0.964 0.995 0.8999 0.886 0.808 0.7447 0.6563 0.5585 0.4557 0.359 0.396 0.557 0.694 0.0989 0.063 0.0473 0.059 0.00 4.0000 0.9999 0.9996 0.9993 0.9984 0.9954 0.989 0.98 0.9777 0.9590 0.9308 0.8906 0.8364 0.7667 0.6809 0.6488 0.5798 0.466 0.3446 0.63 0.35 0.43 0.038 5.0000.0000.0000.0000 0.9999 0.9998 0.9993 0.9986 0.998 0.9959 0.997 0.9844 0.973 0.9533 0.946 0.9 0.884 0.80 0.7379 0.665 0.69 0.4686 0.649 6.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 = 7 p 0.05 0. 0.5 /6 0. 0.5 0.3 /3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 /3 0.7 0.75 0.8 5/6 0.85 0.9 0.95 x = 0 0.6983 0.4783 0.306 0.79 0.097 0.335 0.084 0.0585 0.0490 0.080 0.05 0.0078 0.0037 0.006 0.0006 0.0005 0.000 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9556 0.8503 0.766 0.6698 0.5767 0.4449 0.394 0.634 0.338 0.586 0.04 0.065 0.0357 0.088 0.0090 0.0069 0.0038 0.003 0.0004 0.000 0.000 0.0000 0.0000 0.996 0.9743 0.96 0.904 0.850 0.7564 0.647 0.5706 0.533 0.499 0.364 0.66 0.59 0.0963 0.0556 0.0453 0.088 0.09 0.0047 0.000 0.00 0.000 0.0000 3 0.9998 0.9973 0.9879 0.984 0.9667 0.994 0.8740 0.867 0.800 0.70 0.6083 0.5000 0.397 0.898 0.998 0.733 0.60 0.0706 0.0333 0.076 0.0 0.007 0.000 4.0000 0.9998 0.9988 0.9980 0.9953 0.987 0.97 0.9547 0.9444 0.9037 0.847 0.7734 0.6836 0.580 0.4677 0.494 0.359 0.436 0.480 0.0958 0.0738 0.057 0.0038 5.0000.0000 0.9999 0.9999 0.9996 0.9987 0.996 0.993 0.990 0.98 0.9643 0.9375 0.8976 0.844 0.766 0.7366 0.6706 0.555 0.433 0.330 0.834 0.497 0.0444 6.0000.0000.0000.0000.0000 0.9999 0.9998 0.9995 0.9994 0.9984 0.9963 0.99 0.9848 0.970 0.950 0.945 0.976 0.8665 0.7903 0.709 0.6794 0.57 0.307 7.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 = 8 p 0.05 0. 0.5 /6 0. 0.5 0.3 /3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 /3 0.7 0.75 0.8 5/6 0.85 0.9 0.95 x = 0 0.6634 0.4305 0.75 0.36 0.678 0.00 0.0576 0.0390 0.039 0.068 0.0084 0.0039 0.007 0.0007 0.000 0.000 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.948 0.83 0.657 0.6047 0.5033 0.367 0.553 0.95 0.69 0.064 0.063 0.035 0.08 0.0085 0.0036 0.006 0.003 0.0004 0.000 0.0000 0.0000 0.0000 0.0000 0.994 0.969 0.8948 0.865 0.7969 0.6785 0.558 0.468 0.478 0.354 0.0 0.445 0.0885 0.0498 0.053 0.097 0.03 0.004 0.00 0.0004 0.000 0.0000 0.0000 3 0.9996 0.9950 0.9786 0.9693 0.9437 0.886 0.8059 0.744 0.7064 0.594 0.4770 0.3633 0.604 0.737 0.06 0.0879 0.0580 0.073 0.004 0.0046 0.009 0.0004 0.0000 4.0000 0.9996 0.997 0.9954 0.9896 0.977 0.940 0.9 0.8939 0.863 0.7396 0.6367 0.530 0.4059 0.936 0.586 0.94 0.38 0.0563 0.0307 0.04 0.0050 0.0004 5.0000.0000 0.9998 0.9996 0.9988 0.9958 0.9887 0.9803 0.9747 0.950 0.95 0.8555 0.7799 0.6846 0.57 0.538 0.448 0.35 0.03 0.348 0.05 0.038 0.0058 6.0000.0000.0000.0000 0.9999 0.9996 0.9987 0.9974 0.9964 0.995 0.989 0.9648 0.9368 0.8936 0.8309 0.8049 0.7447 0.639 0.4967 0.3953 0.348 0.869 0.057 7.0000.0000.0000.0000.0000.0000 0.9999 0.9998 0.9998 0.9993 0.9983 0.996 0.996 0.983 0.968 0.960 0.944 0.8999 0.83 0.7674 0.775 0.5695 0.3366 8.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0

CUMULATIVE BINOMIAL PROBABILITIES = 9 p 0.05 0. 0.5 /6 0. 0.5 0.3 /3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 /3 0.7 0.75 0.8 5/6 0.85 0.9 0.95 x = 0 0.630 0.3874 0.36 0.938 0.34 0.075 0.0404 0.060 0.007 0.00 0.0046 0.000 0.0008 0.0003 0.000 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.988 0.7748 0.5995 0.547 0.436 0.3003 0.960 0.43 0. 0.0705 0.0385 0.095 0.009 0.0038 0.004 0.000 0.0004 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.996 0.9470 0.859 0.87 0.738 0.6007 0.468 0.377 0.3373 0.38 0.495 0.0898 0.0498 0.050 0.0 0.0083 0.0043 0.003 0.0003 0.000 0.0000 0.0000 0.0000 3 0.9994 0.997 0.966 0.950 0.944 0.8343 0.797 0.6503 0.6089 0.486 0.364 0.539 0.658 0.0994 0.0536 0.044 0.053 0.000 0.003 0.00 0.0006 0.000 0.0000 4.0000 0.999 0.9944 0.990 0.9804 0.95 0.90 0.855 0.883 0.7334 0.64 0.5000 0.3786 0.666 0.77 0.448 0.0988 0.0489 0.096 0.0090 0.0056 0.0009 0.0000 5.0000 0.9999 0.9994 0.9989 0.9969 0.9900 0.9747 0.9576 0.9464 0.9006 0.834 0.746 0.6386 0.574 0.39 0.3497 0.703 0.657 0.0856 0.0480 0.0339 0.0083 0.0006 6.0000.0000.0000 0.9999 0.9997 0.9987 0.9957 0.997 0.9888 0.9750 0.950 0.90 0.8505 0.768 0.667 0.68 0.537 0.3993 0.68 0.783 0.409 0.0530 0.0084 7.0000.0000.0000.0000.0000 0.9999 0.9996 0.9990 0.9986 0.996 0.9909 0.9805 0.965 0.995 0.8789 0.8569 0.8040 0.6997 0.5638 0.4573 0.4005 0.5 0.07 8.0000.0000.0000.0000.0000.0000.0000 0.9999 0.9999 0.9997 0.999 0.9980 0.9954 0.9899 0.9793 0.9740 0.9596 0.949 0.8658 0.806 0.7684 0.66 0.3698 9.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 = 0 p 0.05 0. 0.5 /6 0. 0.5 0.3 /3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 /3 0.7 0.75 0.8 5/6 0.85 0.9 0.95 x = 0 0.5987 0.3487 0.969 0.65 0.074 0.0563 0.08 0.073 0.035 0.0060 0.005 0.000 0.0003 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.939 0.736 0.5443 0.4845 0.3758 0.440 0.493 0.040 0.0860 0.0464 0.033 0.007 0.0045 0.007 0.0005 0.0004 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9885 0.998 0.80 0.775 0.6778 0.556 0.388 0.99 0.66 0.673 0.0996 0.0547 0.074 0.03 0.0048 0.0034 0.006 0.0004 0.000 0.0000 0.0000 0.0000 0.0000 3 0.9990 0.987 0.9500 0.9303 0.879 0.7759 0.6496 0.5593 0.538 0.383 0.660 0.79 0.00 0.0548 0.060 0.097 0.006 0.0035 0.0009 0.0003 0.000 0.0000 0.0000 4 0.9999 0.9984 0.990 0.9845 0.967 0.99 0.8497 0.7869 0.755 0.633 0.5044 0.3770 0.66 0.66 0.0949 0.0766 0.0473 0.097 0.0064 0.004 0.004 0.000 0.0000 5.0000 0.9999 0.9986 0.9976 0.9936 0.9803 0.957 0.934 0.905 0.8338 0.7384 0.630 0.4956 0.3669 0.485 0.3 0.503 0.078 0.038 0.055 0.0099 0.006 0.000 6.0000.0000 0.9999 0.9997 0.999 0.9965 0.9894 0.9803 0.9740 0.945 0.8980 0.88 0.7340 0.677 0.486 0.4407 0.3504 0.4 0.09 0.0697 0.0500 0.08 0.000 7.0000.0000.0000.0000 0.9999 0.9996 0.9984 0.9966 0.995 0.9877 0.976 0.9453 0.9004 0.837 0.7384 0.7009 0.67 0.4744 0.3 0.48 0.798 0.070 0.05 8.0000.0000.0000.0000.0000.0000 0.9999 0.9996 0.9995 0.9983 0.9955 0.9893 0.9767 0.9536 0.940 0.8960 0.8507 0.7560 0.64 0.555 0.4557 0.639 0.086 9.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9999 0.9997 0.9990 0.9975 0.9940 0.9865 0.987 0.978 0.9437 0.896 0.8385 0.803 0.653 0.403 0.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 = p 0.05 0. 0.5 /6 0. 0.5 0.3 /3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 /3 0.7 0.75 0.8 5/6 0.85 0.9 0.95 x = 0 0.5404 0.84 0.4 0. 0.0687 0.037 0.038 0.0077 0.0057 0.00 0.0008 0.000 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.886 0.6590 0.4435 0.383 0.749 0.584 0.0850 0.0540 0.044 0.096 0.0083 0.003 0.00 0.0003 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9804 0.889 0.7358 0.6774 0.5583 0.3907 0.58 0.8 0.53 0.0834 0.04 0.093 0.0079 0.008 0.0008 0.0005 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3 0.9978 0.9744 0.9078 0.8748 0.7946 0.6488 0.495 0.393 0.3467 0.53 0.345 0.0730 0.0356 0.053 0.0056 0.0039 0.007 0.0004 0.000 0.0000 0.0000 0.0000 0.0000 4 0.9998 0.9957 0.976 0.9636 0.974 0.844 0.737 0.635 0.5833 0.438 0.3044 0.938 0.7 0.0573 0.055 0.088 0.0095 0.008 0.0006 0.000 0.000 0.0000 0.0000 5.0000 0.9995 0.9954 0.99 0.9806 0.9456 0.88 0.83 0.7873 0.665 0.569 0.387 0.607 0.58 0.0846 0.0664 0.0386 0.043 0.0039 0.003 0.0007 0.000 0.0000 6.0000 0.9999 0.9993 0.9987 0.996 0.9857 0.964 0.9336 0.954 0.848 0.7393 0.68 0.473 0.3348 0.7 0.777 0.78 0.0544 0.094 0.0079 0.0046 0.0005 0.0000 7.0000.0000 0.9999 0.9998 0.9994 0.997 0.9905 0.98 0.9745 0.947 0.8883 0.806 0.6956 0.568 0.467 0.3685 0.763 0.576 0.076 0.0364 0.039 0.0043 0.000 8.0000.0000.0000.0000 0.9999 0.9996 0.9983 0.996 0.9944 0.9847 0.9644 0.970 0.8655 0.7747 0.6533 0.6069 0.5075 0.35 0.054 0.5 0.09 0.056 0.00 9.0000.0000.0000.0000.0000.0000 0.9998 0.9995 0.999 0.997 0.99 0.9807 0.9579 0.966 0.8487 0.889 0.747 0.6093 0.447 0.36 0.64 0.09 0.096 0.0000.0000.0000.0000.0000.0000.0000.0000 0.9999 0.9997 0.9989 0.9968 0.997 0.9804 0.9576 0.9460 0.950 0.846 0.75 0.687 0.5565 0.340 0.84.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9999 0.9998 0.999 0.9978 0.9943 0.993 0.986 0.9683 0.933 0.8878 0.8578 0.776 0.4596.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000

CUMULATIVE BINOMIAL PROBABILITIES = 4 p 0.05 0. 0.5 /6 0. 0.5 0.3 /3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 /3 0.7 0.75 0.8 5/6 0.85 0.9 0.95 x = 0 0.4877 0.88 0.08 0.0779 0.0440 0.078 0.0068 0.0034 0.004 0.0008 0.000 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.8470 0.5846 0.3567 0.960 0.979 0.00 0.0475 0.074 0.005 0.008 0.009 0.0009 0.0003 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9699 0.846 0.6479 0.5795 0.448 0.8 0.608 0.053 0.0839 0.0398 0.070 0.0065 0.00 0.0006 0.000 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3 0.9958 0.9559 0.8535 0.8063 0.698 0.53 0.355 0.6 0.05 0.43 0.063 0.087 0.04 0.0039 0.00 0.0007 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 4 0.9996 0.9908 0.9533 0.930 0.870 0.745 0.584 0.4755 0.47 0.793 0.67 0.0898 0.046 0.075 0.0060 0.0040 0.007 0.0003 0.0000 0.0000 0.0000 0.0000 0.0000 5.0000 0.9985 0.9885 0.9809 0.956 0.8883 0.7805 0.6898 0.6405 0.4859 0.3373 0.0 0.89 0.0583 0.043 0.074 0.0083 0.00 0.0004 0.000 0.0000 0.0000 0.0000 6.0000 0.9998 0.9978 0.9959 0.9884 0.967 0.9067 0.8505 0.864 0.695 0.546 0.3953 0.586 0.50 0.0753 0.0576 0.035 0.003 0.004 0.0007 0.0003 0.0000 0.0000 7.0000.0000 0.9997 0.9993 0.9976 0.9897 0.9685 0.944 0.947 0.8499 0.744 0.6047 0.4539 0.3075 0.836 0.495 0.0933 0.0383 0.06 0.004 0.00 0.000 0.0000 8.0000.0000.0000 0.9999 0.9996 0.9978 0.997 0.986 0.9757 0.947 0.88 0.7880 0.667 0.54 0.3595 0.30 0.95 0.7 0.0439 0.09 0.05 0.005 0.0000 9.0000.0000.0000.0000.0000 0.9997 0.9983 0.9960 0.9940 0.985 0.9574 0.90 0.838 0.707 0.5773 0.545 0.458 0.585 0.98 0.0690 0.0467 0.009 0.0004 0.0000.0000.0000.0000.0000.0000 0.9998 0.9993 0.9989 0.996 0.9886 0.973 0.9368 0.8757 0.7795 0.7388 0.6448 0.4787 0.308 0.937 0.465 0.044 0.004.0000.0000.0000.0000.0000.0000.0000 0.9999 0.9999 0.9994 0.9978 0.9935 0.9830 0.960 0.96 0.8947 0.839 0.789 0.559 0.405 0.35 0.584 0.030.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9999 0.9997 0.999 0.997 0.999 0.9795 0.976 0.955 0.8990 0.80 0.7040 0.6433 0.454 0.530 3.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9999 0.9998 0.999 0.9976 0.9966 0.993 0.98 0.9560 0.9 0.897 0.77 0.53 4.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 = 6 p 0.05 0. 0.5 /6 0. 0.5 0.3 /3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 /3 0.7 0.75 0.8 5/6 0.85 0.9 0.95 x = 0 0.440 0.853 0.0743 0.054 0.08 0.000 0.0033 0.005 0.000 0.0003 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.808 0.547 0.839 0.7 0.407 0.0635 0.06 0.037 0.0098 0.0033 0.000 0.0003 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.957 0.789 0.564 0.4868 0.358 0.97 0.0994 0.0594 0.045 0.083 0.0066 0.00 0.0006 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3 0.9930 0.936 0.7899 0.79 0.598 0.4050 0.459 0.659 0.339 0.065 0.08 0.006 0.0035 0.0009 0.000 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 4 0.999 0.9830 0.909 0.8866 0.798 0.630 0.4499 0.339 0.89 0.666 0.0853 0.0384 0.049 0.0049 0.003 0.0008 0.0003 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 5 0.9999 0.9967 0.9765 0.96 0.983 0.803 0.6598 0.5469 0.4900 0.388 0.976 0.05 0.0486 0.09 0.006 0.0040 0.006 0.0003 0.0000 0.0000 0.0000 0.0000 0.0000 6.0000 0.9995 0.9944 0.9899 0.9733 0.904 0.847 0.7374 0.688 0.57 0.3660 0.7 0.4 0.0583 0.09 0.059 0.007 0.006 0.000 0.0000 0.0000 0.0000 0.0000 7.0000 0.9999 0.9989 0.9979 0.9930 0.979 0.956 0.8735 0.8406 0.76 0.569 0.408 0.559 0.43 0.067 0.0500 0.057 0.0075 0.005 0.0004 0.000 0.0000 0.0000 8.0000.0000 0.9998 0.9996 0.9985 0.995 0.9743 0.9500 0.939 0.8577 0.744 0.598 0.437 0.839 0.594 0.65 0.0744 0.07 0.0070 0.00 0.00 0.000 0.0000 9.0000.0000.0000.0000 0.9998 0.9984 0.999 0.984 0.977 0.947 0.8759 0.778 0.6340 0.478 0.39 0.66 0.753 0.0796 0.067 0.00 0.0056 0.0005 0.0000 0.0000.0000.0000.0000.0000 0.9997 0.9984 0.9960 0.9938 0.9809 0.954 0.8949 0.804 0.67 0.500 0.453 0.340 0.897 0.087 0.0378 0.035 0.0033 0.000.0000.0000.0000.0000.0000.0000 0.9997 0.999 0.9987 0.995 0.985 0.966 0.947 0.8334 0.708 0.6609 0.550 0.3698 0.08 0.34 0.079 0.070 0.0009.0000.0000.0000.0000.0000.0000.0000 0.9999 0.9998 0.999 0.9965 0.9894 0.979 0.9349 0.866 0.834 0.754 0.5950 0.409 0.709 0.0 0.0684 0.0070 3.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9999 0.9994 0.9979 0.9934 0.987 0.9549 0.9406 0.9006 0.809 0.648 0.53 0.4386 0.08 0.049 4.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9999 0.9997 0.9990 0.9967 0.990 0.9863 0.9739 0.9365 0.8593 0.778 0.76 0.4853 0.89 5.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9999 0.9997 0.9990 0.9985 0.9967 0.9900 0.979 0.9459 0.957 0.847 0.5599 6.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000

CUMULATIVE BINOMIAL PROBABILITIES = 8 p 0.05 0. 0.5 /6 0. 0.5 0.3 /3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 /3 0.7 0.75 0.8 5/6 0.85 0.9 0.95 x = 0 0.397 0.50 0.0536 0.0376 0.080 0.0056 0.006 0.0007 0.0004 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.7735 0.4503 0.4 0.78 0.099 0.0395 0.04 0.0068 0.0046 0.003 0.0003 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.949 0.7338 0.4797 0.407 0.73 0.353 0.0600 0.036 0.036 0.008 0.005 0.0007 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3 0.989 0.908 0.70 0.6479 0.500 0.3057 0.646 0.07 0.0783 0.038 0.00 0.0038 0.000 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 4 0.9985 0.978 0.8794 0.838 0.764 0.587 0.337 0.3 0.886 0.094 0.04 0.054 0.0049 0.003 0.0003 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 5 0.9998 0.9936 0.958 0.9347 0.867 0.775 0.5344 0.4 0.3550 0.088 0.077 0.048 0.083 0.0058 0.004 0.0009 0.0003 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 6.0000 0.9988 0.988 0.9794 0.9487 0.860 0.77 0.6085 0.549 0.3743 0.58 0.89 0.0537 0.003 0.006 0.0039 0.004 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 7.0000 0.9998 0.9973 0.9947 0.9837 0.943 0.8593 0.7767 0.783 0.5634 0.395 0.403 0.80 0.0576 0.0 0.044 0.006 0.00 0.000 0.0000 0.0000 0.0000 0.0000 8.0000.0000 0.9995 0.9989 0.9957 0.9807 0.9404 0.894 0.8609 0.7368 0.5778 0.4073 0.57 0.347 0.0597 0.0433 0.00 0.0054 0.0009 0.000 0.000 0.0000 0.0000 9.0000.0000 0.9999 0.9998 0.999 0.9946 0.9790 0.9567 0.9403 0.8653 0.7473 0.597 0.4 0.63 0.39 0.076 0.0596 0.093 0.0043 0.00 0.0005 0.0000 0.0000 0.0000.0000.0000.0000 0.9998 0.9988 0.9939 0.9856 0.9788 0.944 0.870 0.7597 0.6085 0.4366 0.77 0.33 0.407 0.0569 0.063 0.0053 0.007 0.000 0.0000.0000.0000.0000.0000.0000 0.9998 0.9986 0.996 0.9938 0.9797 0.9463 0.88 0.774 0.657 0.4509 0.395 0.783 0.390 0.053 0.006 0.08 0.00 0.0000.0000.0000.0000.0000.0000.0000 0.9997 0.999 0.9986 0.994 0.987 0.959 0.893 0.79 0.6450 0.5878 0.4656 0.85 0.39 0.0653 0.049 0.0064 0.000 3.0000.0000.0000.0000.0000.0000.0000 0.9999 0.9997 0.9987 0.995 0.9846 0.9589 0.9058 0.84 0.7689 0.6673 0.483 0.836 0.68 0.06 0.08 0.005 4.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9998 0.9990 0.996 0.9880 0.967 0.97 0.8983 0.8354 0.6943 0.4990 0.35 0.798 0.098 0.009 5.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9999 0.9993 0.9975 0.998 0.9764 0.9674 0.9400 0.8647 0.787 0.5973 0.503 0.66 0.058 6.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9999 0.9997 0.9987 0.9954 0.993 0.9858 0.9605 0.9009 0.87 0.7759 0.5497 0.65 7.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9999 0.9996 0.9993 0.9984 0.9944 0.980 0.964 0.9464 0.8499 0.608 8.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 = 0 p 0.05 0. 0.5 /6 0. 0.5 0.3 /3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 /3 0.7 0.75 0.8 5/6 0.85 0.9 0.95 x = 0 0.3585 0.6 0.0388 0.06 0.05 0.003 0.0008 0.0003 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.7358 0.397 0.756 0.304 0.069 0.043 0.0076 0.0033 0.00 0.0005 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.945 0.6769 0.4049 0.387 0.06 0.093 0.0355 0.076 0.0 0.0036 0.0009 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3 0.984 0.8670 0.6477 0.5665 0.44 0.5 0.07 0.0604 0.0444 0.060 0.0049 0.003 0.0003 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 4 0.9974 0.9568 0.898 0.7687 0.696 0.448 0.375 0.55 0.8 0.050 0.089 0.0059 0.005 0.0003 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 5 0.9997 0.9887 0.937 0.898 0.804 0.67 0.464 0.97 0.454 0.56 0.0553 0.007 0.0064 0.006 0.0003 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 6.0000 0.9976 0.978 0.969 0.933 0.7858 0.6080 0.4793 0.466 0.500 0.99 0.0577 0.04 0.0065 0.005 0.0009 0.0003 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 7.0000 0.9996 0.994 0.9887 0.9679 0.898 0.773 0.665 0.600 0.459 0.50 0.36 0.0580 0.00 0.0060 0.0037 0.003 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 8.0000 0.9999 0.9987 0.997 0.9900 0.959 0.8867 0.8095 0.764 0.5956 0.443 0.57 0.308 0.0565 0.096 0.030 0.005 0.0009 0.000 0.0000 0.0000 0.0000 0.0000 9.0000.0000 0.9998 0.9994 0.9974 0.986 0.950 0.908 0.878 0.7553 0.594 0.49 0.493 0.75 0.053 0.0376 0.07 0.0039 0.0006 0.000 0.0000 0.0000 0.0000 0.0000.0000.0000 0.9999 0.9994 0.996 0.989 0.964 0.9468 0.875 0.7507 0.588 0.4086 0.447 0.8 0.099 0.0480 0.039 0.006 0.0006 0.000 0.0000 0.0000.0000.0000.0000.0000 0.9999 0.999 0.9949 0.9870 0.9804 0.9435 0.869 0.7483 0.5857 0.4044 0.376 0.905 0.33 0.0409 0.000 0.008 0.003 0.000 0.0000.0000.0000.0000.0000.0000 0.9998 0.9987 0.9963 0.9940 0.9790 0.940 0.8684 0.7480 0.584 0.3990 0.3385 0.77 0.08 0.03 0.03 0.0059 0.0004 0.0000 3.0000.0000.0000.0000.0000.0000 0.9997 0.999 0.9985 0.9935 0.9786 0.943 0.870 0.7500 0.5834 0.507 0.390 0.4 0.0867 0.037 0.09 0.004 0.0000 4.0000.0000.0000.0000.0000.0000.0000 0.9998 0.9997 0.9984 0.9936 0.9793 0.9447 0.8744 0.7546 0.708 0.5836 0.388 0.958 0.08 0.0673 0.03 0.0003 5.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9997 0.9985 0.994 0.98 0.9490 0.888 0.8485 0.765 0.585 0.3704 0.33 0.70 0.043 0.006 6.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9997 0.9987 0.995 0.9840 0.9556 0.9396 0.899 0.7748 0.5886 0.4335 0.353 0.330 0.059 7.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9998 0.999 0.9964 0.9879 0.984 0.9645 0.9087 0.7939 0.673 0.595 0.33 0.0755 8.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9999 0.9995 0.9979 0.9967 0.994 0.9757 0.9308 0.8696 0.844 0.6083 0.64 9.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9998 0.9997 0.999 0.9968 0.9885 0.9739 0.96 0.8784 0.645 0.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 3

CUMULATIVE BINOMIAL PROBABILITIES = 5 p 0.05 0. 0.5 /6 0. 0.5 0.3 /3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 /3 0.7 0.75 0.8 5/6 0.85 0.9 0.95 x = 0 0.774 0.078 0.07 0.005 0.0038 0.0008 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.644 0.7 0.093 0.069 0.074 0.0070 0.006 0.0005 0.0003 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.879 0.537 0.537 0.887 0.098 0.03 0.0090 0.0035 0.00 0.0004 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3 0.9659 0.7636 0.47 0.386 0.340 0.096 0.033 0.049 0.0097 0.004 0.0005 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 4 0.998 0.900 0.68 0.5937 0.407 0.37 0.0905 0.046 0.030 0.0095 0.003 0.0005 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 5 0.9988 0.9666 0.8385 0.770 0.667 0.3783 0.935 0.0 0.086 0.094 0.0086 0.000 0.0004 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 6 0.9998 0.9905 0.9305 0.8908 0.7800 0.56 0.3407 0.5 0.734 0.0736 0.058 0.0073 0.006 0.0003 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 7.0000 0.9977 0.9745 0.9553 0.8909 0.765 0.58 0.3703 0.306 0.536 0.0639 0.06 0.0058 0.00 0.000 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 8.0000 0.9995 0.990 0.9843 0.953 0.8506 0.6769 0.5376 0.4668 0.735 0.340 0.0539 0.074 0.0043 0.0008 0.0004 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 9.0000 0.9999 0.9979 0.9953 0.987 0.987 0.806 0.6956 0.6303 0.446 0.44 0.48 0.0440 0.03 0.009 0.006 0.0005 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000.0000 0.9995 0.9988 0.9944 0.9703 0.90 0.80 0.77 0.5858 0.3843 0. 0.0960 0.0344 0.0093 0.0056 0.008 0.000 0.0000 0.0000 0.0000 0.0000 0.0000.0000.0000 0.9999 0.9997 0.9985 0.9893 0.9558 0.908 0.8746 0.733 0.546 0.3450 0.87 0.0778 0.055 0.064 0.0060 0.0009 0.000 0.0000 0.0000 0.0000 0.0000.0000.0000.0000 0.9999 0.9996 0.9966 0.985 0.9585 0.9396 0.846 0.6937 0.5000 0.3063 0.538 0.0604 0.045 0.075 0.0034 0.0004 0.000 0.0000 0.0000 0.0000 3.0000.0000.0000.0000 0.9999 0.999 0.9940 0.9836 0.9745 0.9 0.873 0.6550 0.4574 0.677 0.54 0.098 0.044 0.007 0.005 0.0003 0.000 0.0000 0.0000 4.0000.0000.0000.0000.0000 0.9998 0.998 0.9944 0.9907 0.9656 0.9040 0.7878 0.657 0.44 0.88 0.780 0.0978 0.097 0.0056 0.00 0.0005 0.0000 0.0000 5.0000.0000.0000.0000.0000.0000 0.9995 0.9984 0.997 0.9868 0.9560 0.885 0.7576 0.5754 0.3697 0.3044 0.894 0.073 0.073 0.0047 0.00 0.000 0.0000 6.0000.0000.0000.0000.0000.0000 0.9999 0.9996 0.999 0.9957 0.986 0.946 0.8660 0.765 0.533 0.464 0.33 0.494 0.0468 0.057 0.0080 0.0005 0.0000 7.0000.0000.0000.0000.0000.0000.0000 0.9999 0.9998 0.9988 0.994 0.9784 0.936 0.8464 0.6939 0.697 0.488 0.735 0.09 0.0447 0.055 0.003 0.0000 8.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9997 0.9984 0.997 0.974 0.964 0.866 0.7785 0.6593 0.4389 0.00 0.09 0.0695 0.0095 0.000 9.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9999 0.9996 0.9980 0.994 0.9706 0.974 0.8880 0.8065 0.67 0.3833 0.80 0.65 0.0334 0.00 0.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9999 0.9995 0.9977 0.9905 0.9680 0.9538 0.9095 0.7863 0.5793 0.4063 0.379 0.0980 0.007.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9999 0.9995 0.9976 0.9903 0.985 0.9668 0.9038 0.7660 0.684 0.589 0.364 0.034.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9999 0.9996 0.9979 0.9965 0.990 0.9679 0.908 0.83 0.7463 0.469 0.7 3.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9999 0.9997 0.9995 0.9984 0.9930 0.976 0.937 0.9069 0.788 0.3576 4.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9999 0.999 0.996 0.9895 0.988 0.98 0.76 5.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 4

CUMULATIVE BINOMIAL PROBABILITIES = 30 p 0.05 0. 0.5 /6 0. 0.5 0.3 /3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 /3 0.7 0.75 0.8 5/6 0.85 0.9 0.95 x = 0 0.46 0.044 0.0076 0.004 0.00 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.5535 0.837 0.0480 0.095 0.005 0.000 0.0003 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.8 0.44 0.54 0.08 0.044 0.006 0.00 0.0007 0.0003 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3 0.939 0.6474 0.37 0.396 0.7 0.0374 0.0093 0.0033 0.009 0.0003 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 4 0.9844 0.845 0.545 0.443 0.55 0.0979 0.030 0.0 0.0075 0.005 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 5 0.9967 0.968 0.706 0.664 0.475 0.06 0.0766 0.0355 0.033 0.0057 0.00 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 6 0.9994 0.974 0.8474 0.7765 0.6070 0.348 0.595 0.0838 0.0586 0.07 0.0040 0.0007 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 7 0.9999 0.99 0.930 0.8863 0.7608 0.543 0.84 0.668 0.38 0.0435 0.0 0.006 0.0004 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 8.0000 0.9980 0.97 0.9494 0.873 0.6736 0.435 0.860 0.47 0.0940 0.03 0.008 0.006 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 9.0000 0.9995 0.9903 0.9803 0.9389 0.8034 0.5888 0.437 0.3575 0.763 0.0694 0.04 0.0050 0.0009 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9999 0.997 0.9933 0.9744 0.8943 0.7304 0.5848 0.5078 0.95 0.350 0.0494 0.038 0.009 0.0004 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000.0000.0000 0.999 0.9980 0.9905 0.9493 0.8407 0.739 0.6548 0.43 0.37 0.00 0.0334 0.0083 0.004 0.0007 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000.0000.0000 0.9998 0.9995 0.9969 0.9784 0.955 0.8340 0.780 0.5785 0.359 0.808 0.074 0.0 0.0045 0.005 0.0006 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 3.0000.0000.0000 0.9999 0.999 0.998 0.9599 0.90 0.8737 0.745 0.505 0.93 0.356 0.048 0.04 0.007 0.00 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 4.0000.0000.0000.0000 0.9998 0.9973 0.983 0.9565 0.9348 0.846 0.6448 0.478 0.309 0.097 0.030 0.088 0.0064 0.0008 0.000 0.0000 0.0000 0.0000 0.0000 5.0000.0000.0000.0000 0.9999 0.999 0.9936 0.98 0.9699 0.909 0.769 0.57 0.355 0.754 0.065 0.0435 0.069 0.007 0.000 0.0000 0.0000 0.0000 0.0000 6.0000.0000.0000.0000.0000 0.9998 0.9979 0.998 0.9876 0.959 0.8644 0.7077 0.4975 0.855 0.63 0.0898 0.040 0.008 0.0009 0.000 0.0000 0.0000 0.0000 7.0000.0000.0000.0000.0000 0.9999 0.9994 0.9975 0.9955 0.9788 0.986 0.89 0.6408 0.45 0.98 0.660 0.0845 0.06 0.003 0.0005 0.000 0.0000 0.0000 8.0000.0000.0000.0000.0000.0000 0.9998 0.9993 0.9986 0.997 0.9666 0.8998 0.7673 0.5689 0.345 0.76 0.593 0.0507 0.0095 0.000 0.0008 0.0000 0.0000 9.0000.0000.0000.0000.0000.0000.0000 0.9998 0.9996 0.997 0.986 0.9506 0.8650 0.7085 0.49 0.45 0.696 0.057 0.056 0.0067 0.009 0.000 0.0000 0.0000.0000.0000.0000.0000.0000.0000.0000 0.9999 0.999 0.9950 0.9786 0.9306 0.837 0.645 0.5683 0.4 0.966 0.06 0.097 0.0097 0.0005 0.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9998 0.9984 0.999 0.9688 0.9060 0.7753 0.740 0.5685 0.364 0.87 0.0506 0.078 0.000 0.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9996 0.9974 0.9879 0.9565 0.876 0.833 0.786 0.4857 0.39 0.37 0.0698 0.0078 0.000 3.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9999 0.9993 0.9960 0.988 0.944 0.96 0.8405 0.659 0.3930 0.35 0.56 0.058 0.0006 4.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9998 0.9989 0.9943 0.9767 0.9645 0.934 0.7974 0.575 0.3836 0.894 0.073 0.0033 5.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9998 0.9985 0.995 0.9878 0.9698 0.90 0.7448 0.5757 0.4755 0.755 0.056 6.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9997 0.998 0.9967 0.9907 0.966 0.8773 0.7604 0.6783 0.356 0.0608 7.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9997 0.9993 0.9979 0.9894 0.9558 0.897 0.8486 0.5886 0.878 8.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9999 0.9997 0.9980 0.9895 0.9705 0.950 0.863 0.4465 9.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9998 0.9988 0.9958 0.994 0.9576 0.7854 30.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 5

CUMULATIVE POISSON PROBABILITIES λ 0.0 0.0 0.03 0.04 0.05 0.06 0.07 0.08 0.09 x = 0 0.9900 0.980 0.9704 0.9608 0.95 0.948 0.934 0.93 0.939.0000 0.9998 0.9996 0.999 0.9988 0.9983 0.9977 0.9970 0.996.0000.0000.0000.0000.0000.0000 0.9999 0.9999 0.9999 3.0000.0000.0000.0000.0000.0000.0000.0000.0000 λ 0.0 0.0 0.30 0.40 0.50 0.60 0.70 0.80 0.90 x = 0 0.9048 0.887 0.7408 0.6703 0.6065 0.5488 0.4966 0.4493 0.4066 0.9953 0.985 0.963 0.9384 0.9098 0.878 0.844 0.8088 0.775 0.9998 0.9989 0.9964 0.99 0.9856 0.9769 0.9659 0.956 0.937 3.0000 0.9999 0.9997 0.999 0.998 0.9966 0.994 0.9909 0.9865 4.0000.0000.0000 0.9999 0.9998 0.9996 0.999 0.9986 0.9977 5.0000.0000.0000.0000.0000.0000 0.9999 0.9998 0.9997 6.0000.0000.0000.0000.0000.0000.0000.0000.0000 λ.00.0.0.30.40.50.60.70.80.90 x = 0 0.3679 0.339 0.30 0.75 0.466 0.3 0.09 0.87 0.653 0.496 0.7358 0.6990 0.666 0.668 0.598 0.5578 0.549 0.493 0.468 0.4337 0.997 0.9004 0.8795 0.857 0.8335 0.8088 0.7834 0.757 0.7306 0.7037 3 0.980 0.9743 0.966 0.9569 0.9463 0.9344 0.9 0.9068 0.893 0.8747 4 0.9963 0.9946 0.993 0.9893 0.9857 0.984 0.9763 0.9704 0.9636 0.9559 5 0.9994 0.9990 0.9985 0.9978 0.9968 0.9955 0.9940 0.990 0.9896 0.9868 6 0.9999 0.9999 0.9997 0.9996 0.9994 0.999 0.9987 0.998 0.9974 0.9966 7.0000.0000.0000 0.9999 0.9999 0.9998 0.9997 0.9996 0.9994 0.999 8.0000.0000.0000.0000.0000.0000.0000 0.9999 0.9999 0.9998 9.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 λ.00.0.0.30.40.50.60.70.80.90 x = 0 0.353 0.5 0.08 0.003 0.0907 0.08 0.0743 0.067 0.0608 0.0550 0.4060 0.3796 0.3546 0.3309 0.3084 0.873 0.674 0.487 0.3 0.46 0.6767 0.6496 0.67 0.5960 0.5697 0.5438 0.584 0.4936 0.4695 0.4460 3 0.857 0.8386 0.894 0.7993 0.7787 0.7576 0.7360 0.74 0.699 0.6696 4 0.9473 0.9379 0.975 0.96 0.904 0.89 0.8774 0.869 0.8477 0.838 5 0.9834 0.9796 0.975 0.9700 0.9643 0.9580 0.950 0.9433 0.9349 0.958 6 0.9955 0.994 0.995 0.9906 0.9884 0.9858 0.988 0.9794 0.9756 0.973 7 0.9989 0.9985 0.9980 0.9974 0.9967 0.9958 0.9947 0.9934 0.999 0.990 8 0.9998 0.9997 0.9995 0.9994 0.999 0.9989 0.9985 0.998 0.9976 0.9969 9.0000 0.9999 0.9999 0.9999 0.9998 0.9997 0.9996 0.9995 0.9993 0.999 0.0000.0000.0000.0000.0000 0.9999 0.9999 0.9999 0.9998 0.9998.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9999.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 λ 3.00 3.0 3.0 3.30 3.40 3.50 3.60 3.70 3.80 3.90 x = 0 0.0498 0.0450 0.0408 0.0369 0.0334 0.030 0.073 0.047 0.04 0.00 0.99 0.847 0.7 0.586 0.468 0.359 0.57 0.6 0.074 0.099 0.43 0.40 0.3799 0.3594 0.3397 0.308 0.307 0.854 0.689 0.53 3 0.647 0.648 0.605 0.5803 0.5584 0.5366 0.55 0.494 0.4735 0.453 4 0.853 0.798 0.7806 0.766 0.744 0.754 0.7064 0.687 0.6678 0.6484 5 0.96 0.9057 0.8946 0.889 0.8705 0.8576 0.844 0.830 0.856 0.8006 6 0.9665 0.96 0.9554 0.9490 0.94 0.9347 0.967 0.98 0.909 0.8995 7 0.988 0.9858 0.983 0.980 0.9769 0.9733 0.969 0.9648 0.9599 0.9546 8 0.996 0.9953 0.9943 0.993 0.997 0.990 0.9883 0.9863 0.9840 0.985 9 0.9989 0.9986 0.998 0.9978 0.9973 0.9967 0.9960 0.995 0.994 0.993 0 0.9997 0.9996 0.9995 0.9994 0.999 0.9990 0.9987 0.9984 0.998 0.9977 0.9999 0.9999 0.9999 0.9998 0.9998 0.9997 0.9996 0.9995 0.9994 0.9993.0000.0000.0000.0000 0.9999 0.9999 0.9999 0.9999 0.9998 0.9998 3.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9999 4.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 6

CUMULATIVE POISSON PROBABILITIES λ 4.00 4.0 4.0 4.30 4.40 4.50 4.60 4.70 4.80 4.90 x = 0 0.083 0.066 0.050 0.036 0.03 0.0 0.00 0.009 0.008 0.0074 0.096 0.0845 0.0780 0.079 0.0663 0.06 0.0563 0.058 0.0477 0.0439 0.38 0.38 0.0 0.974 0.85 0.736 0.66 0.53 0.45 0.333 3 0.4335 0.44 0.3954 0.377 0.3594 0.343 0.357 0.3097 0.94 0.793 4 0.688 0.6093 0.5898 0.5704 0.55 0.53 0.53 0.4946 0.4763 0.458 5 0.785 0.7693 0.753 0.7367 0.799 0.709 0.6858 0.6684 0.650 0.6335 6 0.8893 0.8786 0.8675 0.8558 0.8436 0.83 0.880 0.8046 0.7908 0.7767 7 0.9489 0.947 0.936 0.990 0.94 0.934 0.9049 0.8960 0.8867 0.8769 8 0.9786 0.9755 0.97 0.9683 0.964 0.9597 0.9549 0.9497 0.944 0.938 9 0.999 0.9905 0.9889 0.987 0.985 0.989 0.9805 0.9778 0.9749 0.977 0 0.997 0.9966 0.9959 0.995 0.9943 0.9933 0.99 0.990 0.9896 0.9880 0.999 0.9989 0.9986 0.9983 0.9980 0.9976 0.997 0.9966 0.9960 0.9953 0.9997 0.9997 0.9996 0.9995 0.9993 0.999 0.9990 0.9988 0.9986 0.9983 3 0.9999 0.9999 0.9999 0.9998 0.9998 0.9997 0.9997 0.9996 0.9995 0.9994 4.0000.0000.0000.0000 0.9999 0.9999 0.9999 0.9999 0.9999 0.9998 5.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9999 6.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 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 0.004 0.005 0.005 0.0009 0.0006 0.0003 0.000 0.000 0.000 0.0404 0.066 0.074 0.03 0.0073 0.0047 0.0030 0.009 0.00 0.0008 0.47 0.0884 0.060 0.0430 0.096 0.003 0.038 0.0093 0.006 0.004 3 0.650 0.07 0.5 0.8 0.088 0.059 0.044 0.030 0.0 0.049 4 0.4405 0.3575 0.85 0.37 0.730 0.3 0.0996 0.0744 0.0550 0.0403 5 0.660 0.589 0.4457 0.3690 0.3007 0.44 0.9 0.496 0.57 0.0885 6 0.76 0.6860 0.6063 0.565 0.4497 0.378 0.334 0.56 0.068 0.649 7 0.8666 0.8095 0.7440 0.678 0.5987 0.546 0.4530 0.3856 0.339 0.687 8 0.939 0.8944 0.847 0.796 0.79 0.660 0.595 0.53 0.4557 0.398 9 0.968 0.946 0.96 0.8774 0.8305 0.7764 0.766 0.6530 0.5874 0.58 0 0.9863 0.9747 0.9574 0.933 0.905 0.86 0.859 0.7634 0.7060 0.6453 0.9945 0.9890 0.9799 0.966 0.9467 0.908 0.888 0.8487 0.8030 0.750 0.9980 0.9955 0.99 0.9840 0.9730 0.9573 0.936 0.909 0.8758 0.8364 3 0.9993 0.9983 0.9964 0.999 0.987 0.9784 0.9658 0.9486 0.96 0.898 4 0.9998 0.9994 0.9986 0.9970 0.9943 0.9897 0.987 0.976 0.9585 0.9400 5 0.9999 0.9998 0.9995 0.9988 0.9976 0.9954 0.998 0.986 0.9780 0.9665 6.0000 0.9999 0.9998 0.9996 0.9990 0.9980 0.9963 0.9934 0.9889 0.983 7.0000.0000 0.9999 0.9998 0.9996 0.999 0.9984 0.9970 0.9947 0.99 8.0000.0000.0000 0.9999 0.9999 0.9997 0.9993 0.9987 0.9976 0.9957 9.0000.0000.0000.0000.0000 0.9999 0.9997 0.9995 0.9989 0.9980 0.0000.0000.0000.0000.0000.0000 0.9999 0.9998 0.9996 0.999.0000.0000.0000.0000.0000.0000.0000 0.9999 0.9998 0.9996.0000.0000.0000.0000.0000.0000.0000.0000 0.9999 0.9999 3.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9999 4.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 7

CUMULATIVE POISSON PROBABILITIES λ 0.00.00.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 x = 0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0005 0.000 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.008 0.00 0.0005 0.000 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 3 0.003 0.0049 0.003 0.00 0.0005 0.000 0.000 0.0000 0.0000 0.0000 4 0.093 0.05 0.0076 0.0037 0.008 0.0009 0.0004 0.000 0.000 0.0000 5 0.067 0.0375 0.003 0.007 0.0055 0.008 0.004 0.0007 0.0003 0.000 6 0.30 0.0786 0.0458 0.059 0.04 0.0076 0.0040 0.00 0.000 0.0005 7 0.0 0.43 0.0895 0.0540 0.036 0.080 0.000 0.0054 0.009 0.005 8 0.338 0.30 0.550 0.0998 0.06 0.0374 0.00 0.06 0.007 0.0039 9 0.4579 0.3405 0.44 0.658 0.094 0.0699 0.0433 0.06 0.054 0.0089 0 0.5830 0.4599 0.347 0.57 0.757 0.85 0.0774 0.049 0.0304 0.083 0.6968 0.5793 0.466 0.353 0.600 0.848 0.70 0.0847 0.0549 0.0347 0.796 0.6887 0.5760 0.463 0.3585 0.676 0.93 0.350 0.097 0.0606 3 0.8645 0.783 0.685 0.5730 0.4644 0.363 0.745 0.009 0.46 0.0984 4 0.965 0.8540 0.770 0.675 0.5704 0.4657 0.3675 0.808 0.08 0.497 5 0.953 0.9074 0.8444 0.7636 0.6694 0.568 0.4667 0.375 0.867 0.48 6 0.9730 0.944 0.8987 0.8355 0.7559 0.664 0.5660 0.4677 0.375 0.90 7 0.9857 0.9678 0.9370 0.8905 0.87 0.7489 0.6593 0.5640 0.4686 0.3784 8 0.998 0.983 0.966 0.930 0.886 0.895 0.743 0.6550 0.56 0.4695 9 0.9965 0.9907 0.9787 0.9573 0.935 0.875 0.8 0.7363 0.6509 0.5606 0 0.9984 0.9953 0.9884 0.9750 0.95 0.970 0.868 0.8055 0.7307 0.647 0.9993 0.9977 0.9939 0.9859 0.97 0.9469 0.908 0.865 0.799 0.755 0.9997 0.9990 0.9970 0.994 0.9833 0.9673 0.948 0.9047 0.855 0.793 3 0.9999 0.9995 0.9985 0.9960 0.9907 0.9805 0.9633 0.9367 0.8989 0.8490 4.0000 0.9998 0.9993 0.9980 0.9950 0.9888 0.9777 0.9594 0.937 0.8933 5.0000 0.9999 0.9997 0.9990 0.9974 0.9938 0.9869 0.9748 0.9554 0.969 6.0000.0000 0.9999 0.9995 0.9987 0.9967 0.995 0.9848 0.978 0.954 7.0000.0000 0.9999 0.9998 0.9994 0.9983 0.9959 0.99 0.987 0.9687 8.0000.0000.0000 0.9999 0.9997 0.999 0.9978 0.9950 0.9897 0.9805 9.0000.0000.0000.0000 0.9999 0.9996 0.9989 0.9973 0.994 0.988 30.0000.0000.0000.0000 0.9999 0.9998 0.9994 0.9986 0.9967 0.9930 3.0000.0000.0000.0000.0000 0.9999 0.9997 0.9993 0.998 0.9960 3.0000.0000.0000.0000.0000.0000 0.9999 0.9996 0.9990 0.9978 33.0000.0000.0000.0000.0000.0000 0.9999 0.9998 0.9995 0.9988 34.0000.0000.0000.0000.0000.0000.0000 0.9999 0.9998 0.9994 35.0000.0000.0000.0000.0000.0000.0000.0000 0.9999 0.9997 36.0000.0000.0000.0000.0000.0000.0000.0000 0.9999 0.9998 37.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.9999 38.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 8

THE NORMAL DISTRIBUTION FUNCTION If Z has a ormal distributio with mea 0 ad variace the, for each value of, the table gives the value of Φ( ), where Φ( ) = P(Z ). For egative values of use Φ( ) = Φ( ). 0 3 4 5 6 7 8 9 3 4 5 6 7 8 9 ADD 0.0 0.5000 0.5040 0.5080 0.50 0.560 0.599 0.539 0.579 0.539 0.5359 4 8 6 0 4 8 3 36 0. 0.5398 0.5438 0.5478 0.557 0.5557 0.5596 0.5636 0.5675 0.574 0.5753 4 8 6 0 4 8 3 36 0. 0.5793 0.583 0.587 0.590 0.5948 0.5987 0.606 0.6064 0.603 0.64 4 8 5 9 3 7 3 35 0.3 0.679 0.67 0.655 0.693 0.633 0.6368 0.6406 0.6443 0.6480 0.657 4 7 5 9 6 30 34 0.4 0.6554 0.659 0.668 0.6664 0.6700 0.6736 0.677 0.6808 0.6844 0.6879 4 7 4 8 5 9 3 0.5 0.695 0.6950 0.6985 0.709 0.7054 0.7088 0.73 0.757 0.790 0.74 3 7 0 4 7 0 4 7 3 0.6 0.757 0.79 0.734 0.7357 0.7389 0.74 0.7454 0.7486 0.757 0.7549 3 7 0 3 6 9 3 6 9 0.7 0.7580 0.76 0.764 0.7673 0.7704 0.7734 0.7764 0.7794 0.783 0.785 3 6 9 5 8 4 7 0.8 0.788 0.790 0.7939 0.7967 0.7995 0.803 0.805 0.8078 0.806 0.833 3 5 8 4 6 9 5 0.9 0.859 0.886 0.8 0.838 0.864 0.889 0.835 0.8340 0.8365 0.8389 3 5 8 0 3 5 8 0 3.0 0.843 0.8438 0.846 0.8485 0.8508 0.853 0.8554 0.8577 0.8599 0.86 5 7 9 4 6 9. 0.8643 0.8665 0.8686 0.8708 0.879 0.8749 0.8770 0.8790 0.880 0.8830 4 6 8 0 4 6 8. 0.8849 0.8869 0.8888 0.8907 0.895 0.8944 0.896 0.8980 0.8997 0.905 4 6 7 9 3 5 7.3 0.903 0.9049 0.9066 0.908 0.9099 0.95 0.93 0.947 0.96 0.977 3 5 6 8 0 3 4.4 0.99 0.907 0.9 0.936 0.95 0.965 0.979 0.99 0.9306 0.939 3 4 6 7 8 0 3.5 0.933 0.9345 0.9357 0.9370 0.938 0.9394 0.9406 0.948 0.949 0.944 4 5 6 7 80.6 0.945 0.9463 0.9474 0.9484 0.9495 0.9505 0.955 0.955 0.9535 0.9545 3 4 5 6 7 8 9.7 0.9554 0.9564 0.9573 0.958 0.959 0.9599 0.9608 0.966 0.965 0.9633 3 4 4 5 6 7 8.8 0.964 0.9649 0.9656 0.9664 0.967 0.9678 0.9686 0.9693 0.9699 0.9706 3 4 4 5 6 6.9 0.973 0.979 0.976 0.973 0.9738 0.9744 0.9750 0.9756 0.976 0.9767 3 4 4 5 5.0 0.977 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.98 0.987 0 3 3 4 4. 0.98 0.986 0.9830 0.9834 0.9838 0.984 0.9846 0.9850 0.9854 0.9857 0 3 3 4. 0.986 0.9864 0.9868 0.987 0.9875 0.9878 0.988 0.9884 0.9887 0.9890 0 3 3.3 0.9893 0.9896 0.9898 0.990 0.9904 0.9906 0.9909 0.99 0.993 0.996 0.4 0.998 0.990 0.99 0.995 0.997 0.999 0.993 0.993 0.9934 0.9936 0 0.5 0.9938 0.9940 0.994 0.9943 0.9945 0.9946 0.9948 0.9949 0.995 0.995 0 0 0.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.996 0.996 0.9963 0.9964 0 0 0 0.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.997 0.997 0.9973 0.9974 0 0 0 0 0.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.998 0 0 0 0 0 0 0.9 0.998 0.998 0.998 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986 0 0 0 0 0 0 0 0 0 If Z has a ormal distributio with mea 0 ad variace the, for each value of p,thetablegives the value of such that P(Z )=p. Critical values for the ormal distributio p 0.75 0.90 0.95 0.975 0.99 0.995 0.9975 0.999 0.9995 0.674.8.645.960.36.576.807 3.090 3.9 9

CRITICAL VALUES FOR THE t DISTRIBUTION If T has a t distributio with v degrees of freedom the, for each pair of values of p ad v, the table gives 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 =.000 3.078 6.34.7 3.8 63.66 7.3 38.3 636.6 0.86.886.90 4.303 6.965 9.95 4.09.33 3.60 3 0.765.638.353 3.8 4.54 5.84 7.453 0..9 4 0.74.533.3.776 3.747 4.604 5.598 7.73 8.60 5 0.77.476.05.57 3.365 4.03 4.773 5.894 6.869 6 0.78.440.943.447 3.43 3.707 4.37 5.08 5.959 7 0.7.45.895.365.998 3.499 4.09 4.785 5.408 8 0.706.397.860.306.896 3.355 3.833 4.50 5.04 9 0.703.383.833.6.8 3.50 3.690 4.97 4.78 0 0.700.37.8.8.764 3.69 3.58 4.44 4.587 0.697.363.796.0.78 3.06 3.497 4.05 4.437 0.695.356.78.79.68 3.055 3.48 3.930 4.38 3 0.694.350.77.60.650 3.0 3.37 3.85 4. 4 0.69.345.76.45.64.977 3.36 3.787 4.40 5 0.69.34.753.3.60.947 3.86 3.733 4.073 6 0.690.337.746.0.583.9 3.5 3.686 4.05 7 0.689.333.740.0.567.898 3. 3.646 3.965 8 0.688.330.734.0.55.878 3.97 3.60 3.9 9 0.688.38.79.093.539.86 3.74 3.579 3.883 0 0.687.35.75.086.58.845 3.53 3.55 3.850 0.686.33.7.080.58.83 3.35 3.57 3.89 0.686.3.77.074.508.89 3.9 3.505 3.79 3 0.685.39.74.069.500.807 3.04 3.485 3.768 4 0.685.38.7.064.49.797 3.09 3.467 3.745 5 0.684.36.708.060.485.787 3.078 3.450 3.75 6 0.684.35.706.056.479.779 3.067 3.435 3.707 7 0.684.34.703.05.473.77 3.057 3.4 3.689 8 0.683.33.70.048.467.763 3.047 3.408 3.674 9 0.683.3.699.045.46.756 3.038 3.396 3.660 30 0.683.30.697.04.457.750 3.030 3.385 3.646 40 0.68.303.684.0.43.704.97 3.307 3.55 60 0.679.96.67.000.390.660.95 3.3 3.460 0 0.677.89.658.980.358.67.860 3.60 3.373 0.674.8.645.960.36.576.807 3.090 3.9 0