List MF20. List of Formulae and Statistical Tables. Cambridge Pre-U Mathematics (9794) and Further Mathematics (9795)

List MF0 List of Formulae and Statistical Tables Cambridge Pre-U Mathematics (979) and Further Mathematics (979) For use from 07 in all aers for the above syllabuses. CST7

Mensuration Surface area of shere = πr Area of curved surface of cone = πr slant height Trigonometry a = b + c bc cos A Arithmetic series u n = a + (n )d S n = n(a + l) = n{a + (n )d} Geometric series u n = ar n n a( r ) S n = r S = a r for r < PURE MATHEMATICS Summations n r= n r= r = nn ( + )(n + ) = ( + ) r n n Binomial series n n n+ + = r r+ r+ n n n n n n n n r r n ( a+ b) = a + a b+ a b +... + a b +... + b, (n N), where r n n( n ) n( n )...( n r+ ) r ( + ) = + n + +... + +... ( <, n R).... r n! = n n Cr = r r!( n r)! Logarithms and eonentials e ln a = a Comle numbers {r(cos θ + i sin θ)} n = r n (cos nθ + i sin nθ) e iθ = cos θ + i sin θ The roots of z n = are given by z = e π ki n, for k = 0,,,..., n

Maclaurin s series r f() = f(0) + f (0) + f (0) +... + f (r) (0) +...! r! e r = e() = + + +... + +... for all! r! ln( + ) = +... + ( ) r + r +... ( < ) r sin = +... + ( ) r r+ +... for all!! (r + )! cos = +! tan = sinh = +!... + ( ) r r +... for all! ( r)! + +!... + ( ) r r+ +... ( ) r + r+ +... + +... for all (r + )! cosh = + + +... + +... for all!! ( r)! tanh = + + Hyerbolic functions cosh sinh = sinh = sinh cosh cosh = cosh + sinh r r+ +... + +... ( < < ) r + cosh = ln { + } ( ) sinh = ln { + + } tanh = ln + ( < ) Coordinate geometry The erendicular distance from (h, k) to a + by + c = 0 is ah + bk + c a + b The acute angle between lines with gradients m and m is tan m m + mm

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 B (A ± B (k + tan Atan B )π) For t = tan A : sin A = t t, cos A = + t + t A+ B A B sin A + sin B = sin cos A+ B A B sin A sin B = cos sin A+ B A B cos A + cos B = cos cos A+ B A B cos A cos B = sin sin Vectors The resolved art of a in the direction of b is a.b b The oint dividing AB in the ratio λ : μ is µ a+ λ b λ + µ i a b ab ab Vector roduct: a b = a b sin θ ˆ n= j a b = ab ab k a b ab ab If A is the oint with osition vector a = a i + a j + a k and the direction vector b is given by b = b i + b j + b k, then the straight line through A with direction vector b has cartesian equation a y a z a = = (= λ) b b b The lane through A with normal vector n = n i + n j + n k has cartesian equation n + n y + n z + d = 0 where d = a.n The lane through non-collinear oints A, B and C has vector equation r = a + λ(b a) + μ(c a) = ( λ μ)a + λb + μc The lane through the oint with osition vector a and arallel to b and c has equation r = a + sb + tc The erendicular distance of (α, β, γ) from n + n y + n z + d = 0 is nα + n β + n γ + d + + n n n Matri transformations cosθ sinθ Anticlockwise rotation through θ about O: sinθ cosθ cos θ sin θ Reflection in the line y = (tan θ): sin θ cos θ

Differentiation f() f () tan k k sec k sin cos tan + sec sec tan cot cosec cosec cosec cot sinh cosh cosh sinh tanh sech sinh + cosh tanh Integration (+ constant; a > 0 where relevant) f() f( ) d sec k tan k k tan ln sec cot ln sin cosec ln cosec + cot = ln tan( ) sec ln sec + tan = ln tan( + π) sinh cosh cosh sinh tanh ln cosh sin ( < a) a a a + a tan a cosh a a or ln{ + a } ( > a) sinh a + a or ln{ + + a } a+ ln = tanh ( < a) a a a a a a ln a a + a dv du u d = uv v d d d

Area of a sector A = r dθ (olar coordinates) A = dy d y dt (arametric form) dt dt Arc length s = s = s = dy + d (cartesian coordinates) d d dy + dt dt dt (arametric form) dr r + dθ (olar form) dθ Surface area of revolution S = π yds S =π y ds Numerical solution of equations The Newton-Rahson iteration for solving f() = 0: n + = n f( n ) f( ) n

MECHANICS Motion in a circle Transverse velocity: v= rθ Transverse acceleration: v = rθ v Radial acceleration: r θ = r PROBABILITY 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 )P( B A ) j ΣP( A)P( B A) i j i Discrete distributions For a discrete random variable X taking values i with robabilities i Eectation (mean): E(X) = μ = Σ i i Variance: Var(X) = σ = Σ( i μ) i = Σ i i μ For a function g(x) : E(g(X)) = Σ g( i ) i The robability generating function (P.G.F.) of X is G X (t) = E(t X ), and E(X) = G X (), Var(X) = G X() + G X() {G X()} For Z = X + Y, where X and Y are indeendent: G Z (t) = G X (t) G Y (t) The moment generating function (M.G.F.) of X is M X (t) = E(e tx ), and E(X) = M X (0), E(X n ) = M ( n) X (0), Var(X) = M X (0) { M X (0)} For Z = X + Y, where X and Y are indeendent: M Z (t) = M X (t) M Y (t) Standard discrete distributions Distribution of X P(X = ) Mean Variance P.G.F. M.G.F. n Binomial B(n, ) ( ) n n n( ) ( + t) n ( + e t ) n Poisson Po(λ) e λ λ! λ λ e λ(t ) t λ(e ) e Geometric Geo() on,, ( ) t ( t ) e ( )e t t 7

Continuous distributions For a continuous random variable X having robability density function (P.D.F.) f Eectation (mean): E(X) = μ = f( )d Variance: Var(X) = σ = ( µ ) f( )d= f( )d µ For a function g(x) : E(g(X)) = g( )f( )d Cumulative distribution function: F() = P(X ) = f( t)dt The moment generating function (M.G.F.) of X is M X (t) = E(e tx ), and E(X) = M X (0), E(X n ) = M ( n) X (0), Var(X) = M X (0) { M X (0)} For Z = X + Y, where X and Y are indeendent: M Z (t) = M X (t) M Y (t) Standard continuous distributions Distribution of X P.D.F Mean Variance M.G.F. Uniform (Rectangular) on [a, b] b a + b a ( a b) bt at ( ) e e ( b at ) Eonential λ e λ λ λ λ λ t Normal N(μ, σ ) e σ π µ σ μ σ e µ t+ σ t Eectation algebra For indeendent random variables X and Y E(XY) = E(X) E(Y), Var (ax ± by) = a Var(X) + b Var (Y) 8

Samling distributions For a random samle X, X,, X n of n indeendent observations from a distribution having mean μ and variance σ σ X is an unbiased estimator of μ, with Var( X ) = n S is an unbiased estimator of σ, where S Σ( Xi X) = n For a random samle of n observations from N(μ, σ ) X µ ~ N(0,) σ / n X µ ~ t n (also valid in matched-airs situations) S / n If X is the observed number of successes in n indeendent Bernoulli trials, in each of which the robability of success is, and Y = X n, then E(Y) = and Var(Y) = ( ) n For a random samle of n observations from N(μ, σ ) and, indeendently, a random samle of n y observations from N(μ y, σ y ) ( X Y) ( µ µ y) ~ N(0,) σ σ y + n n If σ = y ( X Y) ( µ ) σ y = σ µ y (unknown) then tn+ ny, where S + n n y Correlation and Regression For a set of n airs of values ( i, y i ) S = Σ( i ) ( Σ ) = Σ i i n S yy = Σ(y i y ) ( Σy ) = Σ y i i n S y = Σ( i )(y i y ) = Σ i y i ( Σi )( Σyi ) n The roduct-moment correlation coefficient is ( Σi)( Σyi) S Σy ( )( ) i i y Σ i yi y r = = = n S Syy { Σ( i ) }{ Σ( yi y) } ( Σi) ( Σyi) Σi Σyi n n Sy Σ( i )( yi y) The regression coefficient of y on is b = = S Σ( i ) Least squares regression line of y on is y = a + b where a = y b + y y ( n ) S ( n ) S S = n + n y 9

CUMULATIVE BINOMIAL PROBABILITIES n n r r P( Xø ) = Cr ( ) r= 0 n = 0.0 0. 0. / 0. 0. 0. / 0. 0. 0. 0. 0. 0. 0. / 0.7 0.7 0.8 / 0.8 0.9 0.9 = 0 0.778 0.90 0.7 0.09 0.77 0.7 0.8 0.7 0.0 0.0778 0.00 0.0 0.08 0.00 0.00 0.00 0.00 0.000 0.977 0.98 0.8 0.808 0.77 0.8 0.8 0.09 0.8 0.70 0. 0.87 0. 0.0870 0.00 0.0 0.008 0.0 0.007 0.00 0.00 0.000 0.9988 0.99 0.97 0.9 0.9 0.89 0.89 0.790 0.78 0.8 0.9 0.000 0.09 0.7 0. 0.099 0. 0.0 0.079 0.0 0.0 0.008 0.00 0.999 0.9978 0.997 0.99 0.98 0.99 0.97 0.90 0.90 0.888 0.8 0.78 0.0 0.7 0.9 0.78 0.7 0.7 0.9 0.8 0.08 0.0 0.9990 0.997 0.999 0.997 0.9898 0.98 0.988 0.997 0.9 0.880 0.88 0.89 0.77 0.7 0.98 0. 0.09 0. 0 n = 0.0 0. 0. / 0. 0. 0. / 0. 0. 0. 0. 0. 0. 0. / 0.7 0.7 0.8 / 0.8 0.9 0.9 = 0 0.7 0. 0.77 0.9 0. 0.780 0.7 0.0878 0.07 0.07 0.077 0.0 0.008 0.00 0.008 0.00 0.0007 0.000 0.97 0.887 0.77 0.78 0. 0.9 0.0 0. 0.9 0. 0. 0.09 0.09 0.00 0.0 0.078 0.009 0.00 0.00 0.0007 0.9978 0.98 0.97 0.977 0.90 0.80 0.7 0.80 0.7 0. 0. 0.8 0. 0.79 0.7 0.00 0.070 0.07 0.070 0.0087 0.009 0.00 0.9987 0.99 0.99 0.980 0.9 0.99 0.8999 0.88 0.808 0.77 0. 0.8 0.7 0.9 0.9 0.7 0.9 0.0989 0.0 0.07 0.09 0.00 0.999 0.998 0.99 0.989 0.98 0.9777 0.990 0.908 0.890 0.8 0.77 0.809 0.88 0.798 0. 0. 0. 0. 0. 0.08 0.999 0.998 0.998 0.999 0.997 0.98 0.97 0.9 0.9 0.9 0.88 0.80 0.779 0. 0.9 0.8 0.9 n = 7 0.0 0. 0. / 0. 0. 0. / 0. 0. 0. 0. 0. 0. 0. / 0.7 0.7 0.8 / 0.8 0.9 0.9 = 0 0.98 0.78 0.0 0.79 0.097 0. 0.08 0.08 0.090 0.080 0.0 0.0078 0.007 0.00 0.000 0.000 0.000 0.9 0.80 0.7 0.98 0.77 0.9 0.9 0. 0.8 0.8 0.0 0.0 0.07 0.088 0.0090 0.009 0.008 0.00 0.99 0.97 0.9 0.90 0.80 0.7 0.7 0.70 0. 0.99 0. 0. 0.9 0.09 0.0 0.0 0.088 0.09 0.007 0.000 0.00 0.000 0.997 0.9879 0.98 0.97 0.99 0.870 0.87 0.800 0.70 0.08 0.000 0.97 0.898 0.998 0.7 0.0 0.070 0.0 0.07 0.0 0.007 0.000 0.9988 0.9980 0.99 0.987 0.97 0.97 0.9 0.907 0.87 0.77 0.8 0.80 0.77 0.9 0.9 0. 0.80 0.098 0.078 0.07 0.008 0.9987 0.99 0.99 0.990 0.98 0.9 0.97 0.897 0.8 0.7 0.7 0.70 0. 0. 0.0 0.8 0.97 0.0 0.999 0.999 0.998 0.99 0.99 0.988 0.970 0.90 0.9 0.97 0.8 0.790 0.709 0.79 0.7 0.07 7 n = 8 0.0 0. 0. / 0. 0. 0. / 0. 0. 0. 0. 0. 0. 0. / 0.7 0.7 0.8 / 0.8 0.9 0.9 = 0 0. 0.0 0.7 0. 0.78 0.00 0.07 0.090 0.09 0.08 0.008 0.009 0.007 0.0007 0.000 0.000 0.98 0.8 0.7 0.07 0.0 0.7 0. 0.9 0.9 0.0 0.0 0.0 0.08 0.008 0.00 0.00 0.00 0.99 0.99 0.898 0.8 0.799 0.78 0.8 0.8 0.78 0. 0.0 0. 0.088 0.098 0.0 0.097 0.0 0.00 0.00 0.000 0.990 0.978 0.99 0.97 0.88 0.809 0.7 0.70 0.9 0.770 0. 0.0 0.77 0.0 0.0879 0.080 0.07 0.00 0.00 0.009 0.997 0.99 0.989 0.977 0.90 0.9 0.899 0.8 0.79 0.7 0.0 0.09 0.9 0.8 0.9 0.8 0.0 0.007 0.0 0.000 0.9988 0.998 0.9887 0.980 0.977 0.90 0.9 0.8 0.7799 0.8 0.7 0.8 0.8 0. 0.0 0.8 0.0 0.08 0.008 0.9987 0.997 0.99 0.99 0.989 0.98 0.98 0.89 0.809 0.809 0.77 0.9 0.97 0.9 0.8 0.89 0.07 7 0.999 0.998 0.99 0.99 0.98 0.98 0.90 0.9 0.8999 0.8 0.77 0.77 0.9 0. 8

CUMULATIVE BINOMIAL PROBABILITIES 0.9 0.000 0.008 0.07 0.98 0.9 0.000 0.0 0.08 0.0 0.9 0.000 0.00 0.09 0.8 0.9 0.9 0.008 0.00 0. 0. 0.9 0.00 0.08 0.070 0.9 0. 0.9 0.000 0.00 0.0 0.09 0.0 0.77 0.8 0.000 0.00 0.09 0.09 0.00 0.78 0.8 0.00 0.0099 0.000 0.798 0.7 0.80 0.8 0.0007 0.00 0.09 0.09 0. 0. 0.878 / 0.00 0.0090 0.080 0.78 0.7 0.80 / 0.00 0.0 0.097 0.8 0. 0.88 / 0.000 0.00 0.0079 0.0 0. 0. 0.87 0.8878 0.8 0.00 0.09 0.08 0.8 0.8 0.88 0.8 0.00 0.08 0.09 0. 0. 0.89 0.8 0.000 0.009 0.09 0.07 0.0 0.7 0.7 0.9 0.7 0.00 0.000 0.089 0.7 0.99 0.997 0.99 0.7 0.00 0.097 0.078 0. 0.7 0.70 0.97 0.7 0.008 0.0 0.0 0.7 0. 0.09 0.8 0.98 0.7 0.00 0.0 0.0988 0.70 0.7 0.800 0.99 0.7 0.00 0.00 0.07 0.0 0.0 0.7 0.807 0.978 0.7 0.000 0.007 0.009 0.08 0.78 0.7 0.07 0.77 0.90 0.98 / 0.000 0.008 0.0 0.8 0.97 0.8 0.89 0.970 / 0.00 0.097 0.07 0. 0.07 0.7009 0.890 0.987 / 0.000 0.009 0.088 0.0 0.777 0.8 0.09 0.889 0.90 0.99 0. 0.00 0.0 0.0 0.77 0.9 0.7 0.8789 0.979 0. 0.000 0.008 0.00 0.099 0.8 0.8 0.78 0.90 0.98 0. 0.0008 0.00 0.0 0.08 0.7 0.7 0. 0.887 0.97 0.99 0. 0.008 0.00 0.099 0. 0.7 0.78 0.99 0.9899 0. 0.007 0.0 0.08 0. 0.9 0.77 0.87 0.9 0.990 0. 0.008 0.0 0.07 0.8 0.8 0.8 0.777 0.9 0.980 0.9978 0. 0.0008 0.009 0.098 0.8 0.78 0.8 0.80 0.9 0.99 0. 0.00 0.07 0.00 0. 0.9 0.70 0.900 0.977 0.997 0. 0.00 0.0079 0.0 0.7 0.07 0.7 0.9 0.8 0.979 0.997 0.999 0. 0.000 0.09 0.0898 0.9 0.000 0.7 0.90 0.980 0.9980 0. 0.000 0.007 0.07 0.79 0.770 0.0 0.88 0.9 0.989 0.9990 0. 0.000 0.00 0.09 0.070 0.98 0.87 0.8 0.80 0.970 0.9807 0.998 0. 0.00 0.08 0.9 0. 0. 0.8 0.90 0.9909 0.999 0. 0.00 0.0 0.099 0.0 0.0 0.78 0.8980 0.97 0.99 0. 0.0008 0.008 0.0 0. 0.0 0.9 0.79 0.888 0.9 0.99 0.9989 0. 0.00 0.070 0.8 0.8 0.7 0.900 0.970 0.99 0. 0.000 0.0 0.7 0.8 0. 0.88 0.9 0.9877 0.998 0. 0.00 0.09 0.08 0. 0.8 0. 0.88 0.97 0.987 0.997 0. 0.007 0. 0.7 0.089 0.88 0.9 0.9888 0.998 0. 0.0 0.080 0. 0.8 0.7 0.90 0.970 0.99 0.999 0. 0.007 0.0 0. 0.7 0.8 0.787 0.9 0.97 0.99 0.999 / 0.00 0. 0.77 0.0 0.8 0.97 0.997 0.9990 / 0.07 0.00 0.99 0.9 0.789 0.9 0.980 0.99 / 0.0077 0.00 0.8 0.9 0. 0.8 0.9 0.98 0.99 0.999 0. 0.00 0.90 0.8 0.797 0.90 0.977 0.997 0. 0.08 0.9 0.88 0.9 0.897 0.97 0.989 0.998 0. 0.08 0.080 0.8 0.9 0.77 0.88 0.9 0.990 0.998 0. 0.07 0.00 0.007 0.8 0.9 0.9900 0.9987 0. 0.0 0.0 0. 0.779 0.99 0.980 0.99 0. 0.07 0.8 0.907 0.88 0.8 0.9 0.987 0.997 0. 0. 0. 0.78 0.9 0.980 0.999 0. 0.07 0.78 0.778 0.879 0.97 0.99 0.999 0. 0.087 0.79 0.8 0.79 0.97 0.980 0.99 0.999 / 0.98 0.7 0.87 0.90 0.990 0.9989 / 0. 0.8 0.77 0.90 0.98 0.997 / 0. 0.8 0.77 0.878 0.9 0.99 0.9987 0. 0. 0.99 0.89 0.9 0.99 0.999 0. 0.99 0. 0.80 0.900 0.990 0.998 0. 0. 0. 0.78 0.9078 0.97 0.99 0.999 0. 0.87 0.778 0.970 0.997 0.999 0. 0.87 0.7 0.998 0.987 0.998 0. 0.8 0.90 0.889 0.97 0.997 0.999 0.0 0.0 0.988 0.99 0.999 0.0 0.987 0.99 0.988 0.9990 0.0 0.0 0.88 0.980 0.9978 n = 9 = 0 7 8 9 n = 0 = 0 7 8 9 0 n = = 0 7 8 9 0

CUMULATIVE BINOMIAL PROBABILITIES 0.9 0.00 0.00 0.0 0. 0.9 0.0070 0.09 0.89 0.99 0.9 0.000 0.00 0.009 0.0 0.8 0. 0.77 0.9 0.000 0.00 0.070 0.08 0.08 0.8 0.87 0.8 0.00 0.0 0.07 0. 0. 0. 0.897 0.8 0.000 0.00 0.00 0.0 0.079 0.0 0.8 0.7 0.97 / 0.0007 0.00 0.09 0.090 0.97 0.0 0.700 0.9 / 0.00 0.00 0.078 0. 0.709 0. 0.778 0.99 0.8 0.00 0.0 0.09 0.98 0.08 0.9 0.80 0.90 0.8 0.000 0.00 0.0070 0.07 0.087 0.08 0.09 0.8 0.89 0.979 0.7 0.00 0.00 0.08 0.7 0.8 0.787 0.789 0.8990 0.98 0.7 0.00 0.007 0.07 0.079 0.897 0.98 0.90 0.809 0.9 0.9900 0.7 0.000 0.007 0.008 0.0 0.09 0.9 0.8 0.8 0.89 0.9 0.99 0.7 0.00 0.007 0.07 0.07 0.7 0.0 0.0 0.7 0.900 0.979 0.997 / 0.0007 0.000 0.07 0.07 0.9 0.0 0. 0.788 0.897 0.97 0.99 / 0.0008 0.000 0.09 0.000 0. 0. 0. 0.09 0.8 0.90 0.98 0.998 0. 0.00 0.000 0.0 0.07 0.8 0.9 0.77 0.779 0.9 0.979 0.997 0. 0.000 0.00 0.00 0.09 0.07 0.9 0.9 0.00 0.708 0.8 0.99 0.990 0.9990 0. 0.000 0.009 0.07 0.08 0.0 0.07 0. 0.707 0.877 0.90 0.999 0.999 0. 0.009 0.09 0.08 0. 0.89 0.78 0.7 0.8 0.99 0.987 0.997 0. 0.00 0.0 0.0 0.89 0.8 0.9 0.7 0.88 0.98 0.980 0.997 0. 0.000 0.00 0.09 0.08 0. 0.9 0.7 0.0 0.80 0.97 0.979 0.99 0.9990 0. 0.00 0.087 0.0898 0.0 0.9 0.07 0.7880 0.90 0.97 0.99 0.999 0. 0.00 0.00 0.08 0.0 0.7 0.08 0.98 0.778 0.899 0.9 0.989 0.9979 0. 0.000 0.009 0.070 0.0 0.7 0.7 0. 0.7 0.88 0.97 0.988 0.9978 0. 0.000 0.00 0.08 0.08 0.97 0.0 0.9 0.7 0.879 0.9 0.98 0.99 0.999 0. 0.0008 0.008 0.098 0. 0.79 0.89 0.9 0.899 0.97 0.98 0.99 0.999 0. 0.00 0.08 0.0 0. 0.88 0.7 0.7 0.877 0.97 0.9809 0.99 0.999 0. 0.00 0.00 0.089 0.0 0.7 0.0 0.8 0.97 0.977 0.990 0.9989 0. 0.000 0.0098 0.0 0.9 0.89 0.900 0.88 0.80 0.99 0.977 0.998 0.9987 / 0.00 0.07 0.0 0. 0.7 0.898 0.80 0.9 0.98 0.990 0.999 / 0.00 0.07 0.09 0.9 0.9 0.9 0.77 0.87 0.900 0.98 0.990 0.999 0. 0.008 0.07 0.08 0. 0.8 0.780 0.907 0.98 0.997 0.998 0. 0.00 0.0 0.099 0.9 0.99 0.98 0.87 0.9 0.97 0.999 0.998 0. 0.078 0.00 0.8 0. 0.7 0.888 0.97 0.9897 0.9978 0. 0.000 0.0 0.97 0.00 0.0 0.80 0.90 0.979 0.99 0.998 0. 0.00 0.979 0.8 0.98 0.870 0.9 0.988 0.997 0. 0.08 0.07 0.8 0.98 0.798 0.98 0.97 0.990 0.998 / 0.0779 0.90 0.79 0.80 0.90 0.9809 0.999 0.999 / 0.0 0.7 0.88 0.79 0.88 0.9 0.9899 0.9979 0. 0.08 0.7 0.79 0.8 0.9 0.988 0.9978 0. 0.07 0.89 0. 0.7899 0.909 0.97 0.99 0.9989 0. 0.88 0.8 0.8 0.99 0.9908 0.998 0. 0.8 0.7 0.789 0.9 0.980 0.997 0.999 0.0 0.877 0.870 0.999 0.998 0.0 0.0 0.808 0.97 0.990 0.999 n = = 0 7 8 9 0 n = = 0 7 8 9 0

CUMULATIVE BINOMIAL PROBABILITIES 0.9 0.000 0.00 0.009 0.08 0. 0.08 0.9 0.00 0.09 0.07 0. 0. 0.9 0.000 0.00 0.00 0.08 0.098 0. 0.97 0.899 0.9 0.00 0.0 0.0 0.0 0. 0.08 0.878 0.8 0.000 0.007 0.08 0.09 0.0 0.798 0.0 0.779 0.9 0.8 0.000 0.00 0.009 0.09 0.07 0.70 0. 0.9 0.8 0.9 / 0.000 0.00 0.00 0.00 0.0 0.8 0. 0.97 0.87 0.9 / 0.000 0.008 0.0 0.07 0.08 0. 0. 0.7 0.89 0.979 0.8 0.000 0.00 0.0 0.0 0.9 0.8 0.990 0.787 0.9009 0.980 0.8 0.000 0.00 0.000 0.0 0.087 0.98 0.70 0.88 0.799 0.908 0.988 0.7 0.000 0.00 0.00 0.09 0.09 0.90 0.8 0.8 0.9 0.87 0.90 0.99 0.7 0.000 0.009 0.09 0.009 0.08 0. 0.88 0.8 0.778 0.9087 0.977 0.998 0.7 0.00 0.00 0.00 0.09 0.07 0.78 0. 0.7 0.8 0.900 0.988 0.998 0.7 0.00 0.00 0.07 0.080 0. 0.77 0.90 0.8 0.7 0.899 0.9 0.99 0.999 / 0.009 0.0 0.0 0.07 0. 0.9 0.878 0.789 0.898 0.97 0.99 0.999 / 0.000 0.007 0.00 0.07 0.099 0.90 0.8 0.07 0.708 0.88 0.99 0.98 0.997 0. 0.00 0.00 0.0 0.097 0.9 0.77 0.09 0.0 0.8 0.97 0.97 0.99 0. 0.00 0.000 0.09 0.0 0.8 0.7 0.990 0.8 0.7 0.888 0.9 0.9879 0.9979 0. 0.000 0.00 0.008 0.00 0.07 0.7 0. 0. 0.7 0.79 0.908 0.97 0.998 0.9987 0. 0.00 0.00 0.00 0.0 0.7 0.7 0.0 0.8 0.700 0.87 0.990 0.980 0.99 0.999 0. 0.000 0.009 0.08 0.07 0.80 0.7 0. 0.08 0.77 0.89 0.989 0.9880 0.997 0. 0.00 0.00 0.0 0.080 0.08 0.9 0.08 0.87 0.780 0.870 0.97 0.98 0.99 0.999 0. 0.0007 0.008 0.0 0.08 0.89 0.0 0.07 0.97 0.797 0.88 0.99 0.98 0.99 0.999 0. 0.000 0.00 0.009 0.007 0.077 0. 0.7 0.9 0.88 0.78 0.88 0.9 0.979 0.99 0.9987 0. 0.00 0.00 0.0 0.077 0.8 0.9 0.778 0.77 0.870 0.9 0.987 0.99 0.9990 0. 0.009 0.089 0.0 0.99 0.0 0. 0.9 0.707 0.89 0.90 0.978 0.99 0.998 0. 0.00 0.008 0.08 0.09 0.088 0.7 0. 0.78 0.8 0.9 0.9797 0.99 0.9987 0. 0.000 0.00 0.00 0.00 0. 0.00 0.9 0.9 0.7 0.87 0.9 0.9790 0.99 0.998 0. 0.00 0.0 0.078 0.88 0.0 0.9 0.78 0.809 0.90 0.9788 0.998 0.998 0. 0.000 0.00 0.0 0.0 0.8 0. 0. 0.00 0.7 0.878 0.98 0.980 0.990 0.998 / 0.0007 0.008 0.0 0.07 0. 0. 0.08 0.777 0.89 0.97 0.98 0.99 0.999 / 0.00 0.07 0.00 0. 0.97 0.79 0. 0.809 0.908 0.9 0.9870 0.99 0.999 0. 0.00 0.0 0.000 0. 0.7 0. 0.77 0.89 0.90 0.9790 0.999 0.998 0. 0.0008 0.007 0.0 0.07 0.7 0. 0.080 0.77 0.887 0.90 0.989 0.999 0.9987 0. 0.00 0.09 0. 0.07 0.87 0.77 0.80 0.9 0.9807 0.99 0.9988 0. 0.00 0.0 0.09 0. 0.8 0.7 0.788 0.898 0.99 0.98 0.99 0.999 0. 0.080 0.099 0.7 0.00 0.7 0.87 0.987 0.987 0.997 0.999 0. 0.0 0.09 0.0 0. 0.9 0.80 0.9 0.979 0.9900 0.997 0.999 / 0.07 0.78 0.07 0.79 0.88 0.97 0.979 0.997 0.9989 / 0.0 0.0 0.87 0. 0.787 0.898 0.99 0.9887 0.997 0.999 0. 0.0 0. 0.797 0.70 0.879 0.98 0.988 0.997 0.999 0. 0.088 0.7 0.09 0.77 0.898 0.97 0.978 0.99 0.9987 0. 0.0 0.0 0.78 0.908 0.978 0.99 0.9988 0. 0. 0.97 0.79 0.870 0.98 0.9887 0.997 0.0 0.97 0.77 0.99 0.989 0.998 0.0 0.8 0.78 0.9 0.98 0.997 n = 8 = 0 7 8 9 0 7 8 n = 0 = 0 7 8 9 0 7 8 9 0

CUMULATIVE BINOMIAL PROBABILITIES 0.9 0.000 0.00 0.007 0.0 0.7 0.7 0.7 0.9 0.000 0.00 0.009 0.0 0.0980 0. 0.9 0.788 0.98 0.8 0.000 0.00 0.0080 0.0 0.09 0. 0.79 0.89 0.7 0.909 0.988 / 0.00 0.007 0.07 0.07 0.09 0.80 0.0 0.8 0.8 0.97 0.989 0.8 0.00 0.00 0.07 0.08 0.09 0.00 0.8 0.79 0.70 0.908 0.97 0.99 0.7 0.000 0.00 0.007 0.097 0.07 0.9 0.7 0.89 0.7 0.78 0.908 0.979 0.990 0.999 0.7 0.000 0.008 0.000 0.07 0.0 0.0978 0.89 0. 0.88 0.9 0.80 0.909 0.98 0.990 0.998 / 0.00 0.00 0.0 0.0 0.098 0.780 0.0 0. 0.97 0.778 0.8880 0.98 0.98 0.99 0.999 0. 0.000 0.0008 0.009 0.009 0.0 0.00 0. 0.88 0.97 0. 0.99 0.8 0.97 0.980 0.990 0.9979 0. 0.00 0.00 0.0 0.0 0.0778 0.8 0.77 0. 0.7 0.7 0.8 0.9 0.970 0.990 0.997 0. 0.00 0.008 0.07 0.00 0.090 0.87 0.0 0.7 0.7 0.77 0.80 0.9 0.97 0.99 0.9977 0.999 0. 0.000 0.000 0.007 0.0 0.09 0.8 0. 0.0 0.000 0.0 0.7878 0.88 0.9 0.978 0.997 0.9980 0.999 0. 0.000 0.00 0.008 0.08 0.09 0.0 0. 0.8 0. 0.97 0.87 0.900 0.90 0.98 0.99 0.998 0. 0.00 0.009 0.09 0.07 0. 0.7 0. 0.88 0.7 0.8 0.9 0.9 0.988 0.997 0.9988 0. 0.00 0.0097 0.00 0.08 0.7 0.0 0.8 0.0 0.77 0.87 0.99 0.97 0.9907 0.997 0.999 / 0.000 0.00 0.09 0.0 0.0 0. 0.70 0.7 0.9 0.80 0.908 0.98 0.98 0.99 0.998 0. 0.00 0.0090 0.0 0.090 0.9 0.07 0.8 0.79 0.80 0.90 0.98 0.98 0.990 0.998 0.999 0. 0.0008 0.0070 0.0 0.09 0.7 0.78 0. 0.7 0.80 0.987 0.970 0.989 0.99 0.999 0. 0.008 0.07 0.098 0.0 0.07 0.7 0.7800 0.8909 0.9 0.987 0.99 0.998 / 0.00 0.09 0.887 0.8 0.97 0.770 0.8908 0.9 0.98 0.99 0.9988 0. 0.07 0.09 0.7 0.7 0.8 0.88 0.90 0.97 0.990 0.9979 0.999 0. 0.078 0.7 0.7 0.7 0.900 0.9 0.990 0.9977 0.999 0.0 0.77 0. 0.879 0.99 0.998 0.9988 n = = 0 7 8 9 0 7 8 9 0

CUMULATIVE BINOMIAL PROBABILITIES 0.9 0.000 0.00 0.0 0.008 0.878 0. 0.78 0.9 0.000 0.000 0.0078 0.08 0.07 0.7 0. 0.88 0.8 0.97 0.8 0.000 0.0008 0.009 0.0097 0.078 0.098 0. 0.89 0.7 0.78 0.88 0.90 0.99 / 0.000 0.000 0.007 0.097 0.00 0.7 0. 0.8 0.77 0.70 0.897 0.970 0.998 0.8 0.000 0.00 0.009 0.0 0.0 0.87 0.9 0.90 0.7 0.78 0.877 0.98 0.989 0.9988 0.7 0.000 0.0008 0.007 0.008 0.0 0.007 0.07 0.9 0. 0.87 0.9 0.797 0.90 0.9 0.989 0.9980 0.7 0.000 0.000 0.00 0.00 0.09 0.00 0.08 0.9 0.9 0. 0.8 0.78 0.80 0.9 0.998 0.9907 0.9979 / 0.000 0.0007 0.00 0.007 0.088 0.0 0.0898 0.0 0.7 0. 0.8 0.70 0.8 0.9 0.9 0.9878 0.997 0.999 0. 0.00 0.00 0.0 0.00 0.0 0. 0.98 0. 0.9 0. 0.77 0.87 0.9 0.977 0.99 0.998 0. 0.000 0.009 0.008 0.0 0.08 0.097 0.7 0.8 0. 0.89 0.708 0.87 0.900 0.9 0.988 0.99 0.998 0. 0.00 0.000 0.08 0.0 0.07 0. 0.09 0. 0.97 0.08 0.77 0.80 0.90 0.988 0.9879 0.990 0.9989 0. 0.000 0.0007 0.00 0.008 0.0 0.09 0.00 0.808 0.9 0.78 0.7 0.7077 0.89 0.8998 0.90 0.978 0.999 0.997 0.999 0. 0.000 0.00 0.000 0.0 0.0 0.09 0.0 0.7 0.9 0.0 0.8 0.79 0.8 0.98 0.9 0.98 0.990 0.998 0. 0.00 0.007 0.07 0.0 0.090 0.7 0.9 0. 0.78 0.7 0.8 0.909 0.99 0.9788 0.997 0.997 0.999 0. 0.009 0.007 0.0 0.08 0.8 0.7 0.7 0.078 0.8 0.780 0.877 0.98 0.999 0.987 0.99 0.998 / 0.0007 0.00 0.0 0.0 0.088 0.8 0.80 0.7 0.88 0.79 0.80 0.90 0.9 0.98 0.998 0.997 0.999 0. 0.00 0.009 0.00 0.07 0.9 0.8 0. 0.888 0.70 0.807 0.9 0.999 0.98 0.99 0.9979 0.999 0. 0.000 0.000 0.00 0.07 0.0979 0.0 0.8 0. 0.7 0.80 0.89 0.99 0.978 0.998 0.997 0.999 0. 0.00 0.00 0.0 0.7 0. 0.7 0.070 0.708 0.87 0.989 0.97 0.990 0.999 0.999 / 0.00 0.09 0.08 0.9 0. 0. 0.77 0.88 0.99 0.980 0.99 0.9980 0.999 0. 0.007 0.080 0. 0.7 0. 0.70 0.87 0.90 0.97 0.990 0.997 0.999 0. 0.0 0.87 0. 0.7 0.8 0.98 0.97 0.99 0.9980 0.999 0.0 0. 0. 0.8 0.99 0.98 0.997 0.999 n = 0 = 0 7 8 9 0 7 8 9 0 7 8 9 0

CUMULATIVE POISSON PROBABILITIES P( Xø ) = e r= 0 r λ λ λ 0.0 0.0 0.0 0.0 0.0 0.0 0.07 0.08 0.09 = 0 0.9900 0.980 0.970 0.908 0.9 0.98 0.9 0.9 0.99 0.999 0.9988 0.998 0.9977 0.9970 0.99 λ 0.0 0.0 0.0 0.0 0.0 0.0 0.70 0.80 0.90 = 0 0.908 0.887 0.708 0.70 0.0 0.88 0.9 0.9 0.0 0.99 0.98 0.9 0.98 0.9098 0.878 0.8 0.8088 0.77 0.9989 0.99 0.99 0.98 0.979 0.99 0.9 0.97 0.999 0.998 0.99 0.99 0.9909 0.98 0.999 0.998 0.9977 λ.00.0.0.0.0.0.0.70.80.90 = 0 0.79 0.9 0.0 0.7 0. 0. 0.09 0.87 0. 0.9 0.78 0.990 0. 0.8 0.98 0.78 0.9 0.9 0.8 0.7 0.997 0.900 0.879 0.87 0.8 0.8088 0.78 0.77 0.70 0.707 0.980 0.97 0.9 0.99 0.9 0.9 0.9 0.908 0.89 0.877 0.99 0.99 0.99 0.989 0.987 0.98 0.97 0.970 0.9 0.99 0.999 0.9990 0.998 0.9978 0.998 0.99 0.990 0.990 0.989 0.988 0.999 0.999 0.9987 0.998 0.997 0.99 7 0.999 0.999 8 9 λ.00.0.0.0.0.0.0.70.80.90 = 0 0. 0. 0.08 0.00 0.0907 0.08 0.07 0.07 0.008 0.00 0.00 0.79 0. 0.09 0.08 0.87 0.7 0.87 0. 0. 0.77 0.9 0.7 0.90 0.97 0.8 0.8 0.9 0.9 0.0 0.87 0.88 0.89 0.799 0.7787 0.77 0.70 0.7 0.99 0.9 0.97 0.979 0.97 0.9 0.90 0.89 0.877 0.89 0.877 0.88 0.98 0.979 0.97 0.9700 0.9 0.980 0.90 0.9 0.99 0.98 0.99 0.99 0.99 0.990 0.988 0.988 0.988 0.979 0.97 0.97 7 0.9989 0.998 0.9980 0.997 0.997 0.998 0.997 0.99 0.999 0.990 8 0.999 0.999 0.999 0.9989 0.998 0.998 0.997 0.999 9 0.999 0.999 0.999 0 λ.00.0.0.0.0.0.0.70.80.90 = 0 0.098 0.00 0.008 0.09 0.0 0.00 0.07 0.07 0.0 0.00 0.99 0.87 0.7 0.8 0.8 0.9 0.7 0. 0.07 0.099 0. 0.0 0.799 0.9 0.97 0.08 0.07 0.8 0.89 0. 0.7 0.8 0.0 0.80 0.8 0. 0. 0.9 0.7 0. 0.8 0.798 0.780 0.7 0.7 0.7 0.70 0.87 0.78 0.8 0.9 0.907 0.89 0.889 0.870 0.87 0.8 0.80 0.8 0.800 0.9 0.9 0.9 0.990 0.9 0.97 0.97 0.98 0.909 0.899 7 0.988 0.988 0.98 0.980 0.979 0.97 0.99 0.98 0.999 0.9 8 0.99 0.99 0.99 0.99 0.997 0.990 0.988 0.98 0.980 0.98 9 0.9989 0.998 0.998 0.9978 0.997 0.997 0.990 0.99 0.99 0.99 0 0.999 0.999 0.999 0.9990 0.9987 0.998 0.998 0.9977 0.999 0.999 0.999 r!

CUMULATIVE POISSON PROBABILITIES λ.00.0.0.0.0.0.0.70.80.90 = 0 0.08 0.0 0.00 0.0 0.0 0.0 0.00 0.009 0.008 0.007 0.09 0.08 0.0780 0.079 0.0 0.0 0.0 0.08 0.077 0.09 0.8 0.8 0.0 0.97 0.8 0.7 0. 0. 0. 0. 0. 0. 0.9 0.77 0.9 0. 0.7 0.097 0.9 0.79 0.88 0.09 0.898 0.70 0. 0. 0. 0.9 0.7 0.8 0.78 0.79 0.7 0.77 0.799 0.709 0.88 0.8 0.0 0. 0.889 0.878 0.87 0.88 0.8 0.8 0.880 0.80 0.7908 0.777 7 0.989 0.97 0.9 0.990 0.9 0.9 0.909 0.890 0.887 0.879 8 0.978 0.97 0.97 0.98 0.9 0.997 0.99 0.997 0.9 0.98 9 0.999 0.990 0.9889 0.987 0.98 0.989 0.980 0.9778 0.979 0.977 0 0.997 0.99 0.999 0.99 0.99 0.99 0.99 0.990 0.989 0.9880 0.999 0.9989 0.998 0.998 0.9980 0.997 0.997 0.99 0.990 0.99 0.999 0.999 0.999 0.9990 0.9988 0.998 0.998 0.999 0.999 λ.00.0.00.0 7.00 7.0 8.00 8.0 9.00 9.0 = 0 0.007 0.00 0.00 0.00 0.000 0.000 0.00 0.0 0.07 0.0 0.007 0.007 0.000 0.009 0.00 0.0008 0.7 0.088 0.00 0.00 0.09 0.00 0.08 0.009 0.00 0.00 0.0 0.07 0. 0.8 0.088 0.09 0.0 0.00 0.0 0.09 0.0 0.7 0.8 0.7 0.70 0. 0.099 0.07 0.00 0.00 0.0 0.89 0.7 0.90 0.007 0. 0.9 0.9 0.7 0.088 0.7 0.80 0.0 0. 0.97 0.78 0. 0. 0.08 0.9 7 0.8 0.809 0.70 0.78 0.987 0. 0.0 0.8 0.9 0.87 8 0.99 0.89 0.87 0.79 0.79 0.0 0.9 0. 0.7 0.98 9 0.98 0.9 0.9 0.877 0.80 0.77 0.7 0.0 0.87 0.8 0 0.98 0.977 0.97 0.9 0.90 0.8 0.89 0.7 0.700 0. 0.99 0.9890 0.9799 0.9 0.97 0.908 0.888 0.887 0.800 0.70 0.9980 0.99 0.99 0.980 0.970 0.97 0.9 0.909 0.878 0.8 0.999 0.998 0.99 0.999 0.987 0.978 0.98 0.98 0.9 0.898 0.999 0.998 0.9970 0.99 0.9897 0.987 0.97 0.98 0.900 0.999 0.9988 0.997 0.99 0.998 0.98 0.9780 0.9 0.9990 0.9980 0.99 0.99 0.9889 0.98 7 0.999 0.998 0.9970 0.997 0.99 8 0.999 0.9987 0.997 0.997 9 0.999 0.9989 0.9980 0 0.999 7

CUMULATIVE POISSON PROBABILITIES λ 0.00.00.00.00.00.00.00 7.00 8.00 9.00 = 0 0.000 0.000 0.008 0.00 0.000 0.000 0.00 0.009 0.00 0.00 0.000 0.000 0.09 0.0 0.007 0.007 0.008 0.000 0.07 0.07 0.00 0.007 0.00 0.008 0.00 0.0007 0.000 0.0 0.078 0.08 0.09 0.0 0.007 0.000 0.00 0.000 0.000 7 0.0 0. 0.089 0.00 0.0 0.080 0.000 0.00 0.009 0.00 8 0.8 0.0 0.0 0.0998 0.0 0.07 0.00 0.0 0.007 0.009 9 0.79 0.0 0. 0.8 0.09 0.099 0.0 0.0 0.0 0.0089 0 0.80 0.99 0.7 0.7 0.77 0.8 0.077 0.09 0.00 0.08 0.98 0.79 0. 0. 0.00 0.88 0.70 0.087 0.09 0.07 0.79 0.887 0.70 0. 0.8 0.7 0.9 0.0 0.097 0.00 0.8 0.78 0.8 0.70 0. 0. 0.7 0.009 0. 0.098 0.9 0.80 0.770 0.7 0.70 0.7 0.7 0.808 0.08 0.97 0.9 0.907 0.8 0.7 0.9 0.8 0.7 0.7 0.87 0.8 0.970 0.9 0.8987 0.8 0.79 0. 0.0 0.77 0.7 0.90 7 0.987 0.978 0.970 0.890 0.87 0.789 0.9 0.0 0.8 0.78 8 0.998 0.98 0.9 0.90 0.88 0.89 0.7 0.0 0. 0.9 9 0.99 0.9907 0.9787 0.97 0.9 0.87 0.8 0.7 0.09 0.0 0 0.998 0.99 0.988 0.970 0.9 0.970 0.88 0.80 0.707 0.7 0.999 0.9977 0.999 0.989 0.97 0.99 0.908 0.8 0.799 0.7 0.9990 0.9970 0.99 0.98 0.97 0.98 0.907 0.8 0.79 0.999 0.998 0.990 0.9907 0.980 0.9 0.97 0.8989 0.890 0.999 0.9980 0.990 0.9888 0.9777 0.99 0.97 0.89 0.9990 0.997 0.998 0.989 0.978 0.9 0.99 0.999 0.9987 0.997 0.99 0.988 0.978 0.9 7 0.999 0.998 0.999 0.99 0.987 0.987 8 0.999 0.9978 0.990 0.9897 0.980 9 0.9989 0.997 0.99 0.988 0 0.999 0.998 0.997 0.990 0.999 0.998 0.990 0.9990 0.9978 0.999 0.9988 0.999 7 8 8

If Z has a normal distribution with mean 0 and variance, then, for each value of z, the table gives the value of Φ(z), where Φ(z) = P(Z z). For negative values of z, use Φ( z) = Φ(z). THE NORMAL DISTRIBUTION FUNCTION z 0 7 8 9 7 8 9 ADD 0.0 0.000 0.00 0.080 0.0 0.0 0.99 0.9 0.79 0.9 0.9 8 0 8 0. 0.98 0.8 0.78 0.7 0.7 0.9 0. 0.7 0.7 0.7 8 0 8 0. 0.79 0.8 0.87 0.90 0.98 0.987 0.0 0.0 0.0 0. 8 9 7 0. 0.79 0.7 0. 0.9 0. 0.8 0.0 0. 0.80 0.7 7 9 0 0. 0. 0.9 0.8 0. 0.700 0.7 0.77 0.808 0.8 0.879 7 8 9 0. 0.9 0.90 0.98 0.709 0.70 0.7088 0.7 0.77 0.790 0.7 7 0 7 0 7 0. 0.77 0.79 0.7 0.77 0.789 0.7 0.7 0.78 0.77 0.79 7 0 9 9 0.7 0.780 0.7 0.7 0.77 0.770 0.77 0.77 0.779 0.78 0.78 9 8 7 0.8 0.788 0.790 0.799 0.797 0.799 0.80 0.80 0.8078 0.80 0.8 8 9 0.9 0.89 0.88 0.8 0.88 0.8 0.889 0.8 0.80 0.8 0.889 8 0 8 0.0 0.8 0.88 0.8 0.88 0.808 0.8 0.8 0.877 0.899 0.8 7 9 9. 0.8 0.8 0.88 0.8708 0.879 0.879 0.8770 0.8790 0.880 0.880 8 0 8. 0.889 0.889 0.8888 0.8907 0.89 0.89 0.89 0.8980 0.8997 0.90 7 9 7. 0.90 0.909 0.90 0.908 0.9099 0.9 0.9 0.97 0.9 0.977 8 0. 0.99 0.907 0.9 0.9 0.9 0.9 0.979 0.99 0.90 0.99 7 8 0. 0.9 0.9 0.97 0.970 0.98 0.99 0.90 0.98 0.99 0.9 7 8 0. 0.9 0.9 0.97 0.98 0.99 0.90 0.9 0.9 0.9 0.9 7 8 9.7 0.9 0.9 0.97 0.98 0.99 0.999 0.908 0.9 0.9 0.9 7 8.8 0.9 0.99 0.9 0.9 0.97 0.978 0.98 0.99 0.999 0.970.9 0.97 0.979 0.97 0.97 0.978 0.97 0.970 0.97 0.97 0.977.0 0.977 0.9778 0.978 0.9788 0.979 0.9798 0.980 0.9808 0.98 0.987 0. 0.98 0.98 0.980 0.98 0.988 0.98 0.98 0.980 0.98 0.987 0. 0.98 0.98 0.988 0.987 0.987 0.9878 0.988 0.988 0.9887 0.9890 0. 0.989 0.989 0.9898 0.990 0.990 0.990 0.9909 0.99 0.99 0.99 0. 0.998 0.990 0.99 0.99 0.997 0.999 0.99 0.99 0.99 0.99 0 0. 0.998 0.990 0.99 0.99 0.99 0.99 0.998 0.999 0.99 0.99 0 0 0. 0.99 0.99 0.99 0.997 0.999 0.990 0.99 0.99 0.99 0.99 0 0 0 0.7 0.99 0.99 0.997 0.998 0.999 0.9970 0.997 0.997 0.997 0.997 0 0 0 0 0.8 0.997 0.997 0.997 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 0.998 0.998 0.998 0.998 0.998 0.998 0.998 0 0 0 0 0 0 0 0 0 If Z has a normal distribution with mean 0 and variance, then, for each value of, the table gives the value of z such that P(Z z) =. Critical values for the normal distribution 0.7 0.90 0.9 0.97 0.99 0.99 0.997 0.999 0.999 z 0.7.8..90..7.807.090.9 9

CRITICAL VALUES FOR THE t-distribution If T has a t-distribution with ν degrees of freedom, then, for each air of values of and ν, the table gives the value of t such that: P(T t) =. 0.7 0.90 0.9 0.97 0.99 0.99 0.997 0.999 0.999 ν =.000.078..7.8. 7. 8.. 0.8.88.90.0.9 9.9.09..0 0.7.8..8..8 7. 0..9 0.7...77.77.0.98 7.7 8.0 0.77.7.0.7..0.77.89.89 0.78.0.9.7..707.7.08.99 7 0.7..89..998.99.09.78.08 8 0.70.97.80.0.89..8.0.0 9 0.70.8.8..8.0.90.97.78 0 0.700.7.8.8.7.9.8..87 0.97..79.0.78.0.97.0.7 0.9..78.79.8.0.8.90.8 0.9.0.77.0.0.0.7.8. 0.9..7...977..787.0 0.9..7..0.97.8.7.07 0.90.7.7.0.8.9..8.0 7 0.89..70.0.7.898...9 8 0.88.0.7.0..878.97.0.9 9 0.88.8.79.09.9.8.7.79.88 0 0.87..7.08.8.8...80 0.8..7.080.8.8..7.89 0.8..77.07.08.89.9.0.79 0.8.9.7.09.00.807.0.8.78 0.8.8.7.0.9.797.09.7.7 0.8..708.00.8.787.078.0.7 0.8..70.0.79.779.07..707 7 0.8..70.0.7.77.07..89 8 0.8..70.08.7.7.07.08.7 9 0.8..99.0..7.08.9.0 0 0.8.0.97.0.7.70.00.8. 0 0.8.0.8.0..70.97.07. 0 0.79.9.7.000.90.0.9..0 0 0.77.89.8.980.8.7.80.0.7 0.7.8..90..7.807.090.9 0