Aedi B: Mathematical Formulae ad Statistical Tables Pure Mathematics Mesuratio Surface area of shere = π r Area of curved surface of coe = π r slat height Trigoometry a = b + c bccosa Arithmetic Series u = a+ ( ) d S = ( a+ l) = { a+ ( ) d} Geometric Series u = ar S a( r ) = r a S = for r r < Summatios 6 r= r = ( + )(+ ) r = ( + ) r= OCR00 Aedi B: Mathematical Formulae ad Statistical Tables 95 Oford, Cambridge ad RSA Eamiatios GCE Mathematics
Biomial Series + + = r r+ r+ r r ( a+ b) = a + a b+ a b + + a b + + b r ( ),! where = Cr = r r!( r)! ( ) ( ) ( r+ ) r ( + ) = + + + + +..r ( <, ) Logarithms ad Eoetials l e a = a Comle Numbers { r(cosθ + isi θ)} = r (cos θ + isi θ) iθ e = cosθ + isiθ The roots of z = are give by π k i z = e, for k = 0,,,, Maclauri s Series ( r) f( ) = f(0) + f (0) + f (0) + + f (0) +! r! r = = + + +! e e( )! r + + for all r r+ l( + ) = + + ( ) + ( < ) r r 5 r+ r si = + + ( ) + for all! 5! ( r + )! r r cos = + + ( ) + for all!! ( r )! 96 Aedi B: Mathematical Formulae ad Statistical Tables OCR 00 GCE Mathematics Oford, Cambridge ad RSA Eamiatios
ta 5 r+ r = + + ( ) + ( ) 5 r + 5 r+ sih = + + + + + for all! 5! (r + )! r cosh = + + + + + for all!! ( r)! tah 5 r+ = + + + + + 5 r + ( < < ) Hyerbolic Fuctios cosh sih = sih = sih cosh cosh = cosh + sih { } cosh = l + ( ) { } sih = l + + tah l + = ( ) < Coordiate Geometry The eredicular distace from ( h, k ) to a + by + c = 0 is ah + bk + c a + b The acute agle betwee lies with gradiets m ad m is ta m m + mm OCR00 Aedi B: Mathematical Formulae ad Statistical Tables 97 Oford, Cambridge ad RSA Eamiatios GCE Mathematics
Trigoometric Idetities si( A± B) = si Acos B± cos Asi B cos( A± B) = cos Acos B si Asi B ta A± ta B ta( A± B) = ( A± B ( k + ) π ) taatab For t = ta A: t si A =, + t t cos A = + t A+ B A B si A+ si B= si cos A+ B A B si A si B= cos si A+ B A B cos A+ cos B= cos cos A+ B A B cos A cos B= si si Vectors The resolved art of a i the directio of b is a.b b The oit dividig AB i the ratio λ : µ is µ a+ λ b λ + µ Vector roduct: i a b ab ab ˆ a b= a b si θ = j a b = ab ab k a b ab ab If A is the oit with ositio vector a= ai+ aj+ ak ad the directio vector b is give by b= bi+ b j+ b k, the the straight lie through A with directio vector b has cartesia equatio a y a z a = = ( = λ) b b b 98 Aedi B: Mathematical Formulae ad Statistical Tables OCR 00 GCE Mathematics Oford, Cambridge ad RSA Eamiatios
The lae through A with ormal vector = i+ j+ k has cartesia equatio + y+ z + d= 0 where d= a. The lae through o-colliear oits A, B ad C has vector equatio r= a+ λ ( b a) + µ ( c a) = ( λ µ ) a+ λ b+ µ c The lae through the oit with ositio vector a ad arallel to b ad c has equatio r= a+ sb+ tc The eredicular distace of ( α, β, γ ) from + y+ z + d= 0 is α + β + γ + d + + Matri Trasformatios Aticlockwise rotatio through θ about O : cosθ siθ si θ cosθ Reflectio i the lie y = (ta θ ) : cos θ si θ si θ cosθ OCR00 Aedi B: Mathematical Formulae ad Statistical Tables 99 Oford, Cambridge ad RSA Eamiatios GCE Mathematics
Differetiatio f( ) f( ) ta k si ksec k cos ta + sec sec ta cot cosec sih cosh tah sih cosh tah cosec cosec cot cosh sih sech + 00 Aedi B: Mathematical Formulae ad Statistical Tables OCR 00 GCE Mathematics Oford, Cambridge ad RSA Eamiatios
Itegratio (+ costat; a > 0 where relevat) f( ) f( )d sec k ta cot ta k k l sec l si cosec sec sih cosh tah l cosec + cot = l ta( ) l sec + ta = l ta( + π ) cosh sih l cosh a si ( < a) a a + a ta a a cosh or l { + a } ( > a) a a + a sih or l{ + + a } a a+ l = tah ( < a) a a a a a a l a + a dv du u d= uv v d d d OCR00 Aedi B: Mathematical Formulae ad Statistical Tables 0 Oford, Cambridge ad RSA Eamiatios GCE Mathematics
Area of a Sector d A= r θ (olar coordiates) dy d A= y dt (arametric form) dt dt Numerical Mathematics Numerical Itegratio b The traezium rule: y d {( h y 0 + y ) + ( y a + y + + y ) }, where b a h = b Simso s rule: y d {( h y 0 + y ) + ( y a + y + + y ) + ( y + y + + y ) }, where b a h = ad is eve Numerical Solutio of Equatios f( ) The Newto-Rahso iteratio for solvig f( ) = 0 : + = f( ) Mechaics Motio i a Circle Trasverse velocity: v= rθ Trasverse acceleratio: v = rθ Radial acceleratio: v rθ = r 0 Aedi B: Mathematical Formulae ad Statistical Tables OCR 00 GCE Mathematics Oford, Cambridge ad RSA Eamiatios
Cetres of Mass For uiform bodies Triagular lamia: alog media from verte Solid hemishere, radius r: r from cetre Hemisherical shell, radius r: r from cetre 8 Circular arc, radius r, agle at cetre si α : r α from cetre α Sector of circle, radius r, agle at cetre r siα α : from cetre α Solid coe or yramid of height h: h above the base o the lie from cetre of base to verte Coical shell of height h: h above the base o the lie from cetre of base to verte Momets of Iertia For uiform bodies of mass m Thi rod, legth l, about eredicular ais through cetre: ml Rectagular lamia about ais i lae bisectig edges of legth l : ml Thi rod, legth l, about eredicular ais through ed: ml Rectagular lamia about edge eredicular to edges of legth l : ml Rectagular lamia, sides a ad b, about eredicular ais through cetre: ma ( + b ) Hoo or cylidrical shell of radius r about ais: mr Hoo of radius r about a diameter: mr Disc or solid cylider of radius r about ais: mr Disc of radius r about a diameter: mr OCR00 Aedi B: Mathematical Formulae ad Statistical Tables 0 Oford, Cambridge ad RSA Eamiatios GCE Mathematics
Solid shere, radius r, about a diameter: 5 mr Sherical shell of radius r about a diameter: mr Parallel aes theorem: I = I + m( AG) A G Peredicular aes theorem: I z = I + Iy (for a lamia i the -y lae) Probability ad Statistics Probability P( A B) = P( A) + P( B) P( A B) P( A B) = P( A)P( B A) P( B A)P( A) P( A B) = P( B A)P( A) + P( B A )P( A ) Bayes Theorem: P( Aj)P( B Aj) P( Aj B) = Σ P( A)P( B A) i i Discrete Distributios For a discrete radom variable X takig values i with robabilities Eectatio (mea): E( X ) = µ =Σ i i i Variace: i i i i Var( X) = σ =Σ( µ ) =Σ µ For a fuctio g( X ) : E(g( X )) =Σg( ) The robability geeratig fuctio of X is G X ( t ) = E( t ), ad E( X ) = G (), X i i X Var( X ) = G () + G () {G ()} X X X For Z = X + Y, where X ad Y are ideedet: G ( t) = G ( t)g ( t) Z X Y 0 Aedi B: Mathematical Formulae ad Statistical Tables OCR 00 GCE Mathematics Oford, Cambridge ad RSA Eamiatios
Stadard Discrete Distributios: Distributio of X P(X = ) Mea Variace P.G.F. Biomial B(, ) ( ) ( ) ( + t) Poisso Po( λ) λ λ e λ λ ( ) e λ t! Geometric Geo () o,, ( ) t ( ) t Cotiuous Distributios For a cotiuous radom variable X havig robability desity fuctio f Eectatio (mea): E( X ) µ f( )d = = Variace: Var( X) = σ = ( µ ) f( )d= f( )d µ For a fuctio g( X ) : E(g( X )) = g( )f( )d Cumulative distributio fuctio: F( ) P( X ) f( t) dt = = tx The momet geeratig fuctio of X is M X ( t ) = E(e ) ad E( X ) = M (0), X ( ) X E( X ) = M (0), Var( X ) = M (0) {M (0)} X X For Z = X + Y, where X ad Y are ideedet: M ( t) = M ( t)m ( t) Z X Y OCR00 Aedi B: Mathematical Formulae ad Statistical Tables 05 Oford, Cambridge ad RSA Eamiatios GCE Mathematics
Stadard Cotiuous Distributios: Distributio of X P.D.F. Mea Variace M.G.F. Uiform (Rectagular) o [ a, b] b a ( a b) ( b a) e e ( b a) t + bt at Eoetial λ λ e λ λ λ λ t Normal µ µ σ ( ) σ N(, ) e σ π µ σ e µ t+ σ t Eectatio Algebra Covariace: Cov( X, Y) = E(( X µ )( Y µ )) = E( XY) µ µ X Y X Y Var( ax ± by ) = a Var( X ) + b Var( Y ) ± abcov( X, Y ) Product momet correlatio coefficiet: Cov( X, Y ) ρ = σ σ X Y If X = ax + b ad Y = cy + d, the Cov( X, Y) = accov( X, Y ) For ideedet radom variables X ad Y E( XY) = E( X)E( Y) Var( ax ± by ) = a Var( X ) + b Var( Y ) Samlig Distributios For a radom samle X, X,, X of ideedet 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, where Σ( X i X ) S = 06 Aedi B: Mathematical Formulae ad Statistical Tables OCR 00 GCE Mathematics Oford, Cambridge ad RSA Eamiatios
For a radom samle of observatios from X µ ~N(0, ) σ / N( µ, σ ) X µ ~ t S/ (also valid i matched-airs situatios) If X is the observed umber of successes i ideedet Beroulli trials i each of which the X robability of success is, ad Y =, the E( Y) = ad Var( Y ) = ( ) For a radom samle of observatios from y observatios from y N( µ, σ ) y N( µ, σ ) ad, ideedetly, a radom samle of ( X Y) ( µ µ y) ~N(0, ) σ σ y + y y If σ = σ = σ (ukow) the ( X Y) ( µ µ y) ~ t S + y + y, where + y y ( ) S ( ) S S = + y Correlatio ad Regressio For a set of airs of values (, y ) i i ( Σi ) S =Σ( i ) =Σi ( Σyi ) Syy =Σ( yi y) =Σyi ( Σi)( Σyi) Sy =Σ( i )( yi y) =Σiyi OCR00 Aedi B: Mathematical Formulae ad Statistical Tables 07 Oford, Cambridge ad RSA Eamiatios GCE Mathematics
The roduct momet correlatio coefficiet is ( Σi)( Σyi) S Σy ( )( ) i i y Σ i yi y r = = = S Syy { Σ( i ) }{ Σ( yi y) } ( Σi) ( Σyi) Σi Σyi Searma s rak correlatio coefficiet is 6Σd rs = ( ) The regressio coefficiet of y o is S Σ( )( y y) b = = S y i i Σ( i ) Least squares regressio lie of y o is y = a+ b where a= y b Distributio-free (No-arametric) Tests Goodess-of-fit test ad cotigecy tables: ( O i E i ) E i ν ~ χ Aroimate distributios for large samles: Wilcoo Siged Rak test: T ~ N ( ( + ), ( + )(+ ) ) Wilcoo Rak Sum test (samles of sizes m ad, with m ): ( + + + + ) W ~ N m( m ), m( m ) 08 Aedi B: Mathematical Formulae ad Statistical Tables OCR 00 GCE Mathematics Oford, Cambridge ad RSA Eamiatios
OCR00 Aedi B: Mathematical Formulae ad Statistical Tables 09 Oford, Cambridge ad RSA Eamiatios GCE Mathematics 0.95 0.00 0.06 0.6 0.95 0.00 0.08 0.69 0.95 0.000 0.008 0.0 0.07 0.95 0.0058 0.057 0.66 0.9 0.0086 0.085 0.095 0.9 0.00 0.059 0. 0.686 0.9 0.000 0.007 0.057 0.97 0.57 0.9 0.0050 0.08 0.869 0.5695 0.85 0.00 0.066 0.68 0.556 0.85 0.0059 0.07 0.5 0.69 0.85 0.00 0.0 0.078 0.8 0.679 0.85 0.000 0.009 0.0 0.05 0.8 0.775 5/6 0.00 0.055 0.96 0.598 5/6 0.0007 0.0087 0.06 0.6 0.665 5/6 0.000 0.076 0.0958 0.0 0.709 5/6 0.006 0.007 0.8 0.95 0.767 0.8 0.0067 0.0579 0.67 0.67 0.8 0.006 0.070 0.0989 0.6 0.779 0.8 0.007 0.0 0.80 0. 0.790 0.8 0.00 0.00 0.056 0.0 0.967 0.8 0.75 0.000 0.056 0.05 0.67 0.767 0.75 0.000 0.006 0.076 0.69 0.66 0.80 0.75 0.00 0.09 0.0706 0.6 0.555 0.8665 0.75 0.00 0.07 0.8 0.5 0.69 0.8999 0.7 0.00 0.008 0.6 0.78 0.89 0.7 0.0007 0.009 0.0705 0.557 0.5798 0.88 0.7 0.000 0.008 0.088 0.60 0.59 0.6706 0.976 0.7 0.00 0.0 0.0580 0.9 0.8 0.77 0.9 / 0.00 0.05 0.099 0.59 0.868 / 0.00 0.078 0.00 0.96 0.688 0.9 / 0.0069 0.05 0.7 0.9 0.766 0.95 / 0.000 0.006 0.097 0.0879 0.586 0.58 0.809 0.960 0.65 0.005 0.050 0.5 0.576 0.880 0.65 0.008 0.0 0.7 0.59 0.6809 0.96 0.65 0.0006 0.0090 0.0556 0.998 0.677 0.766 0.950 0.65 0.000 0.006 0.05 0.06 0.96 0.57 0.809 0.968 0.6 0.00 0.0870 0.7 0.660 0.9 0.6 0.00 0.00 0.79 0.557 0.7667 0.95 0.6 0.006 0.088 0.096 0.898 0.580 0.8 0.970 0.6 0.0007 0.0085 0.098 0.77 0.059 0.686 0.896 0.98 0.55 0.085 0. 0.069 0.78 0.997 0.55 0.008 0.069 0.55 0.5585 0.86 0.97 0.55 0.007 0.057 0.59 0.97 0.686 0.8976 0.988 0.55 0.007 0.08 0.0885 0.60 0.50 0.7799 0.968 0.996 0.5 0.0 0.875 0.5000 0.85 0.9688 0.5 0.056 0.09 0.8 0.656 0.8906 0.98 0.5 0.0078 0.065 0.66 0.5000 0.77 0.975 0.99 0.5 0.009 0.05 0.5 0.6 0.667 0.8555 0.968 0.996 0.5 0.050 0.56 0.59 0.8688 0.985 0.5 0.077 0.66 0.5 0.77 0.908 0.997 0.5 0.05 0.0 0.6 0.608 0.87 0.96 0.996 0.5 0.008 0.06 0.0 0.770 0.796 0.95 0.989 0.998 0. 0.0778 0.70 0.686 0.90 0.9898 0. 0.067 0. 0.5 0.808 0.9590 0.9959 0. 0.080 0.586 0.99 0.70 0.907 0.98 0.998 0. 0.068 0.06 0.5 0.59 0.86 0.950 0.995 0.999 0.5 0.60 0.8 0.768 0.960 0.997 0.5 0.075 0.9 0.67 0.886 0.9777 0.998 0.5 0.090 0.8 0.5 0.800 0.9 0.990 0.999 0.5 0.09 0.69 0.78 0.706 0.899 0.977 0.996 / 0.7 0.609 0.790 0.957 0.9959 / 0.0878 0.5 0.680 0.8999 0.98 0.9986 / 0.0585 0.6 0.5706 0.867 0.957 0.99 / 0.090 0.95 0.68 0.7 0.9 0.980 0.997 0. 0.68 0.58 0.869 0.969 0.9976 0. 0.76 0.0 0.7 0.995 0.989 0.999 0. 0.08 0.9 0.67 0.870 0.97 0.996 0. 0.0576 0.55 0.558 0.8059 0.90 0.9887 0.9987 0.5 0.7 0.68 0.8965 0.98 0.9990 0.5 0.780 0.59 0.806 0.96 0.995 0.5 0.5 0.9 0.756 0.99 0.987 0.9987 0.5 0.00 0.67 0.6785 0.886 0.977 0.9958 0. 0.77 0.77 0.9 0.99 0. 0.6 0.655 0.90 0.980 0.998 0. 0.097 0.5767 0.850 0.9667 0.995 0. 0.678 0.50 0.7969 0.97 0.9896 0.9988 /6 0.09 0.808 0.965 0.9967 /6 0.9 0.768 0.977 0.99 0.999 /6 0.79 0.6698 0.90 0.98 0.9980 /6 0.6 0.607 0.865 0.969 0.995 0.5 0.7 0.85 0.97 0.9978 0.5 0.77 0.7765 0.957 0.99 0.5 0.06 0.766 0.96 0.9879 0.9988 0.5 0.75 0.657 0.898 0.9786 0.997 0. 0.5905 0.985 0.99 0. 0.5 0.8857 0.98 0.9987 0. 0.78 0.850 0.97 0.997 0. 0.05 0.8 0.969 0.9950 0.05 0.778 0.977 0.9988 0.05 0.75 0.967 0.9978 0.05 0.698 0.9556 0.996 0.05 0.66 0.98 0.99 CUMULATIVE BINOMIAL PROBABILITIES = 5 = 0 5 = 6 = 0 5 6 = 7 = 0 5 6 7 = 8 = 0 5 6 7 8
0 Aedi B: Mathematical Formulae ad Statistical Tables OCR 00 GCE Mathematics Oford, Cambridge ad RSA Eamiatios 0.95 0.0006 0.008 0.07 0.698 0.95 0.000 0.05 0.086 0.0 0.95 0.000 0.00 0.096 0.8 0.596 0.9 0.0009 0.008 0.050 0.5 0.66 0.9 0.006 0.08 0.070 0.69 0.65 0.9 0.00 0.056 0.09 0.0 0.776 0.85 0.0006 0.0056 0.09 0.09 0.005 0.768 0.85 0.00 0.0099 0.0500 0.798 0.557 0.80 0.85 0.0007 0.006 0.09 0.09 0.6 0.5565 0.8578 5/6 0.00 0.0090 0.080 0.78 0.57 0.806 5/6 0.00 0.055 0.0697 0.8 0.555 0.885 5/6 0.000 0.00 0.0079 0.06 0.5 0.6 0.687 0.8878 0.8 0.00 0.096 0.0856 0.68 0.568 0.8658 0.8 0.0009 0.006 0.08 0.09 0. 0.6 0.896 0.8 0.0006 0.009 0.09 0.076 0.05 0.7 0.75 0.9 0.75 0.00 0.000 0.089 0.657 0.99 0.6997 0.99 0.75 0.005 0.097 0.078 0. 0.7 0.7560 0.97 0.75 0.008 0.0 0.05 0.576 0.5 0.609 0.86 0.968 0.7 0.00 0.05 0.0988 0.70 0.57 0.800 0.9596 0.7 0.006 0.006 0.07 0.50 0.50 0.67 0.8507 0.978 0.7 0.000 0.007 0.0095 0.086 0.78 0.76 0.5075 0.77 0.950 0.986 / 0.000 0.008 0.0 0.8 0.97 0.68 0.8569 0.970 / 0.00 0.097 0.0766 0. 0.07 0.7009 0.8960 0.987 / 0.009 0.088 0.066 0.777 0.685 0.6069 0.889 0.960 0.99 0.65 0.00 0.0 0.056 0.77 0.9 0.667 0.8789 0.979 0.65 0.008 0.060 0.099 0.85 0.86 0.78 0.90 0.9865 0.65 0.0008 0.0056 0.055 0.086 0.7 0.67 0.65 0.887 0.9576 0.99 0.6 0.008 0.050 0.099 0.666 0.57 0.768 0.995 0.9899 0.6 0.007 0.0 0.058 0.66 0.669 0.677 0.87 0.956 0.990 0.6 0.008 0.05 0.057 0.58 0.8 0.568 0.777 0.966 0.980 0.9978 0.55 0.0008 0.009 0.098 0.658 0.786 0.686 0.8505 0.965 0.995 0.55 0.005 0.07 0.00 0.66 0.956 0.70 0.900 0.9767 0.9975 0.55 0.00 0.0079 0.056 0.7 0.607 0.7 0.6956 0.8655 0.9579 0.997 0.999 0.5 0.000 0.095 0.0898 0.59 0.5000 0.76 0.90 0.9805 0.9980 0.5 0.000 0.007 0.057 0.79 0.770 0.60 0.88 0.95 0.989 0.9990 0.5 0.000 0.00 0.09 0.070 0.98 0.87 0.68 0.806 0.970 0.9807 0.9968 0.5 0.006 0.085 0.95 0.6 0.6 0.8 0.950 0.9909 0.999 0.5 0.005 0.0 0.0996 0.660 0.50 0.78 0.8980 0.976 0.9955 0.5 0.0008 0.008 0.0 0.5 0.0 0.569 0.79 0.888 0.96 0.99 0.9989 0. 0.00 0.0705 0.8 0.86 0.7 0.9006 0.9750 0.996 0. 0.0060 0.06 0.67 0.8 0.6 0.88 0.95 0.9877 0.998 0. 0.00 0.096 0.08 0.5 0.8 0.665 0.88 0.97 0.987 0.997 0.5 0.007 0. 0.7 0.6089 0.88 0.96 0.9888 0.9986 0.5 0.05 0.0860 0.66 0.58 0.755 0.905 0.970 0.995 0.5 0.0057 0.0 0.5 0.67 0.58 0.787 0.95 0.975 0.99 0.999 / 0.060 0. 0.77 0.650 0.855 0.9576 0.997 0.9990 / 0.07 0.00 0.99 0.559 0.7869 0.9 0.980 0.9966 / 0.0077 0.050 0.8 0.9 0.65 0.8 0.96 0.98 0.996 0. 0.00 0.960 0.68 0.797 0.90 0.977 0.9957 0. 0.08 0.9 0.88 0.696 0.897 0.957 0.989 0.998 0. 0.08 0.0850 0.58 0.95 0.77 0.88 0.96 0.9905 0.998 0.5 0.075 0.00 0.6007 0.8 0.95 0.9900 0.9987 0.5 0.056 0.0 0.556 0.7759 0.99 0.980 0.9965 0.5 0.07 0.58 0.907 0.688 0.8 0.956 0.9857 0.997 0. 0. 0.6 0.78 0.9 0.980 0.9969 0. 0.07 0.758 0.6778 0.879 0.967 0.996 0.999 0. 0.0687 0.79 0.558 0.796 0.97 0.9806 0.996 0.999 /6 0.98 0.57 0.87 0.950 0.990 0.9989 /6 0.65 0.85 0.775 0.90 0.985 0.9976 /6 0. 0.8 0.677 0.878 0.966 0.99 0.9987 0.5 0.6 0.5995 0.859 0.966 0.99 0.999 0.5 0.969 0.5 0.80 0.9500 0.990 0.9986 0.5 0. 0.5 0.758 0.9078 0.976 0.995 0.999 0. 0.87 0.778 0.970 0.997 0.999 0. 0.87 0.76 0.998 0.987 0.998 0. 0.8 0.6590 0.889 0.97 0.9957 0.05 0.60 0.988 0.996 0.999 0.05 0.5987 0.99 0.9885 0.9990 0.05 0.50 0.886 0.980 0.9978 CUMULATIVE BINOMIAL PROBABILITIES = 9 = 0 5 6 7 8 9 = 0 = 0 5 6 7 8 9 0 = = 0 5 6 7 8 9 0
OCR00 Aedi B: Mathematical Formulae ad Statistical Tables Oford, Cambridge ad RSA Eamiatios GCE Mathematics 0.95 0.00 0.00 0.50 0.5 0.95 0.0009 0.0070 0.09 0.89 0.5599 0.9 0.000 0.005 0.009 0.0 0.58 0.5 0.77 0.9 0.00 0.070 0.068 0.08 0.85 0.87 0.85 0.00 0.05 0.067 0.65 0.5 0.6 0.897 0.85 0.000 0.00 0.0056 0.05 0.079 0.0 0.86 0.76 0.957 5/6 0.0007 0.00 0.09 0.0690 0.97 0.05 0.700 0.9 5/6 0.00 0.00 0.078 0. 0.709 0.5 0.778 0.959 0.8 0.00 0.06 0.09 0.98 0.08 0.559 0.80 0.9560 0.8 0.000 0.005 0.0070 0.067 0.087 0.08 0.09 0.68 0.859 0.979 0.75 0.00 0.00 0.08 0.7 0.585 0.787 0.789 0.8990 0.98 0.75 0.006 0.0075 0.07 0.0796 0.897 0.698 0.5950 0.809 0.965 0.9900 0.7 0.000 0.007 0.008 0.05 0.09 0.95 0.58 0.68 0.89 0.955 0.99 0.7 0.006 0.007 0.057 0.07 0.75 0.0 0.550 0.75 0.9006 0.979 0.9967 / 0.0007 0.000 0.07 0.0576 0.95 0.0 0.55 0.788 0.897 0.976 0.9966 / 0.0008 0.000 0.059 0.0500 0.65 0.66 0.5 0.6609 0.8 0.906 0.986 0.9985 0.65 0.00 0.0060 0.0 0.075 0.86 0.595 0.577 0.7795 0.96 0.9795 0.9976 0.65 0.000 0.00 0.006 0.09 0.067 0.59 0.9 0.500 0.708 0.866 0.959 0.990 0.9990 0.6 0.0006 0.009 0.075 0.058 0.50 0.075 0.5 0.707 0.8757 0.960 0.999 0.999 0.6 0.0009 0.009 0.09 0.058 0. 0.89 0.78 0.67 0.8 0.99 0.987 0.9967 0.55 0.00 0.0 0.06 0.89 0.586 0.59 0.667 0.88 0.968 0.980 0.997 0.55 0.0006 0.005 0.09 0.086 0. 0.559 0.7 0.60 0.80 0.97 0.979 0.99 0.9990 0.5 0.0009 0.0065 0.087 0.0898 0.0 0.95 0.607 0.7880 0.90 0.97 0.995 0.999 0.5 0.00 0.006 0.08 0.05 0.7 0.08 0.598 0.778 0.899 0.966 0.989 0.9979 0.5 0.000 0.009 0.070 0.06 0.67 0.7 0.56 0.7 0.88 0.957 0.9886 0.9978 0.5 0.000 0.0066 0.08 0.085 0.976 0.660 0.569 0.7 0.8759 0.95 0.985 0.9965 0.999 0. 0.0008 0.008 0.098 0. 0.79 0.859 0.695 0.899 0.97 0.985 0.996 0.999 0. 0.00 0.08 0.065 0.666 0.88 0.57 0.76 0.8577 0.97 0.9809 0.995 0.999 0.5 0.00 0.005 0.089 0.05 0.7 0.605 0.86 0.97 0.9757 0.990 0.9989 0.5 0.000 0.0098 0.05 0.9 0.89 0.900 0.688 0.806 0.99 0.977 0.998 0.9987 / 0.00 0.07 0.05 0.6 0.755 0.6898 0.8505 0.9 0.986 0.9960 0.999 / 0.005 0.07 0.059 0.659 0.9 0.569 0.77 0.875 0.9500 0.98 0.9960 0.999 0. 0.0068 0.075 0.608 0.55 0.58 0.7805 0.9067 0.9685 0.997 0.998 0. 0.00 0.06 0.099 0.59 0.99 0.6598 0.87 0.956 0.97 0.999 0.998 0.5 0.078 0.00 0.8 0.5 0.75 0.888 0.967 0.9897 0.9978 0.5 0.000 0.065 0.97 0.050 0.60 0.80 0.90 0.979 0.995 0.998 0. 0.00 0.979 0.8 0.698 0.870 0.956 0.988 0.9976 0. 0.08 0.07 0.58 0.598 0.798 0.98 0.97 0.990 0.9985 /6 0.0779 0.960 0.5795 0.806 0.90 0.9809 0.9959 0.999 /6 0.05 0.7 0.868 0.79 0.8866 0.96 0.9899 0.9979 0.5 0.08 0.567 0.679 0.855 0.95 0.9885 0.9978 0.5 0.07 0.89 0.56 0.7899 0.909 0.9765 0.99 0.9989 0. 0.88 0.586 0.86 0.9559 0.9908 0.9985 0. 0.85 0.57 0.789 0.96 0.980 0.9967 0.05 0.877 0.870 0.9699 0.9958 0.05 0.0 0.808 0.957 0.990 0.999 CUMULATIVE BINOMIAL PROBABILITIES = = 0 5 6 7 8 9 0 = 6 = 0 5 6 7 8 9 0 5 6
Aedi B: Mathematical Formulae ad Statistical Tables OCR 00 GCE Mathematics Oford, Cambridge ad RSA Eamiatios 0.95 0.000 0.005 0.009 0.058 0.65 0.608 0.95 0.006 0.059 0.0755 0.6 0.65 0.9 0.000 0.00 0.006 0.08 0.098 0.66 0.597 0.899 0.9 0.00 0.0 0.0 0.0 0. 0.608 0.878 0.85 0.007 0.08 0.09 0.06 0.798 0.50 0.7759 0.96 0.85 0.000 0.00 0.0059 0.09 0.067 0.70 0.5 0.595 0.8 0.96 5/6 0.000 0.00 0.005 0.006 0.065 0.68 0.5 0.597 0.87 0.96 5/6 0.0006 0.008 0.0 0.07 0.08 0. 0.5 0.67 0.8696 0.979 0.8 0.000 0.0009 0.00 0.06 0.05 0.9 0.86 0.990 0.787 0.9009 0.980 0.8 0.0006 0.006 0.000 0.0 0.0867 0.958 0.70 0.5886 0.799 0.908 0.9885 0.75 0.000 0.00 0.005 0.09 0.0569 0.90 0.85 0.8 0.69 0.867 0.9605 0.99 0.75 0.000 0.0009 0.009 0.09 0.009 0.08 0. 0.88 0.585 0.778 0.9087 0.9757 0.9968 0.7 0.00 0.006 0.00 0.0596 0.07 0.78 0.656 0.667 0.85 0.900 0.9858 0.998 0.7 0.00 0.005 0.07 0.080 0. 0.77 0.90 0.586 0.765 0.899 0.965 0.99 0.999 / 0.0009 0.009 0.0 0.0 0.076 0. 0.95 0.5878 0.7689 0.898 0.967 0.99 0.999 / 0.000 0.0009 0.007 0.00 0.076 0.099 0.905 0.85 0.507 0.708 0.885 0.996 0.98 0.9967 0.65 0.00 0.006 0.0 0.0597 0.9 0.77 0.509 0.650 0.8 0.97 0.976 0.995 0.65 0.005 0.0060 0.096 0.05 0.8 0.76 0.990 0.58 0.756 0.888 0.9556 0.9879 0.9979 0.6 0.000 0.00 0.0058 0.00 0.0576 0.7 0.6 0.66 0.657 0.79 0.9058 0.967 0.998 0.9987 0.6 0.006 0.0065 0.00 0.0565 0.75 0.7 0.0 0.58 0.7500 0.87 0.990 0.980 0.996 0.55 0.000 0.009 0.08 0.057 0.80 0.57 0. 0.6085 0.77 0.89 0.9589 0.9880 0.9975 0.55 0.005 0.006 0.0 0.0580 0.08 0.9 0.086 0.5857 0.780 0.870 0.97 0.98 0.995 0.999 0.5 0.0007 0.008 0.05 0.08 0.89 0.0 0.07 0.597 0.7597 0.88 0.959 0.986 0.996 0.999 0.5 0.000 0.00 0.0059 0.007 0.0577 0.6 0.57 0.9 0.588 0.78 0.868 0.9 0.979 0.99 0.9987 0.5 0.005 0.00 0.0 0.077 0.58 0.95 0.5778 0.77 0.870 0.96 0.987 0.995 0.9990 0.5 0.0009 0.009 0.089 0.055 0.99 0.50 0. 0.59 0.7507 0.869 0.90 0.9786 0.996 0.9985 0. 0.00 0.008 0.08 0.09 0.088 0.7 0.56 0.768 0.865 0.9 0.9797 0.99 0.9987 0. 0.006 0.060 0.050 0.56 0.500 0.59 0.5956 0.755 0.875 0.95 0.9790 0.995 0.998 0.5 0.006 0.06 0.078 0.886 0.550 0.59 0.78 0.8609 0.90 0.9788 0.998 0.9986 0.5 0.000 0.00 0.0 0.0 0.8 0.5 0.66 0.600 0.76 0.878 0.968 0.980 0.990 0.9985 / 0.0007 0.0068 0.06 0.07 0. 0. 0.6085 0.7767 0.89 0.9567 0.9856 0.996 0.999 / 0.00 0.076 0.060 0.55 0.97 0.79 0.665 0.8095 0.908 0.96 0.9870 0.996 0.999 0. 0.006 0.0 0.0600 0.66 0.7 0.5 0.77 0.859 0.90 0.9790 0.999 0.9986 0. 0.0008 0.0076 0.055 0.07 0.75 0.6 0.6080 0.77 0.8867 0.950 0.989 0.999 0.9987 0.5 0.0056 0.095 0.5 0.057 0.587 0.775 0.860 0.9 0.9807 0.996 0.9988 0.5 0.00 0.0 0.09 0.5 0.8 0.67 0.7858 0.898 0.959 0.986 0.996 0.999 0. 0.080 0.099 0.7 0.500 0.76 0.867 0.987 0.987 0.9957 0.999 0. 0.05 0.069 0.06 0. 0.696 0.80 0.9 0.9679 0.9900 0.997 0.999 /6 0.076 0.78 0.07 0.679 0.88 0.97 0.979 0.997 0.9989 /6 0.06 0.0 0.87 0.5665 0.7687 0.898 0.969 0.9887 0.997 0.999 0.5 0.056 0. 0.797 0.70 0.879 0.958 0.988 0.997 0.5 0.088 0.756 0.09 0.677 0.898 0.97 0.978 0.99 0.9987 0. 0.50 0.50 0.78 0.908 0.978 0.996 0.9988 0. 0.6 0.97 0.6769 0.8670 0.9568 0.9887 0.9976 0.05 0.97 0.775 0.99 0.989 0.9985 0.05 0.585 0.758 0.95 0.98 0.997 CUMULATIVE BINOMIAL PROBABILITIES = 8 = 0 5 6 7 8 9 0 5 6 7 8 = 0 = 0 5 6 7 8 9 0 5 6 7 8 9 0
OCR00 Aedi B: Mathematical Formulae ad Statistical Tables Oford, Cambridge ad RSA Eamiatios GCE Mathematics 0.95 0.000 0.00 0.007 0.0 0.7 0.576 0.76 0.9 0.00 0.0095 0.0 0.0980 0.6 0.69 0.788 0.98 0.85 0.00 0.0080 0.055 0.0695 0.65 0.79 0.589 0.76 0.9069 0.988 5/6 0.00 0.007 0.057 0.07 0.09 0.80 0.06 0.68 0.8 0.97 0.9895 0.8 0.005 0.0056 0.07 0.068 0.09 0.00 0.8 0.579 0.7660 0.908 0.976 0.996 0.75 0.000 0.0009 0.00 0.007 0.097 0.07 0.9 0.75 0.89 0.67 0.786 0.908 0.9679 0.990 0.999 0.7 0.008 0.0060 0.075 0.0 0.0978 0.89 0. 0.88 0.659 0.8065 0.9095 0.9668 0.990 0.998 / 0.006 0.0056 0.06 0.05 0.098 0.780 0.0 0.6 0.697 0.7785 0.8880 0.958 0.985 0.9965 0.65 0.000 0.0008 0.009 0.009 0.055 0.060 0.5 0.88 0.697 0.5 0.699 0.866 0.97 0.9680 0.990 0.9979 0.6 0.00 0.00 0.0 0.0 0.0778 0.58 0.677 0. 0.575 0.765 0.86 0.96 0.9706 0.9905 0.9976 0.55 0.006 0.0058 0.07 0.00 0.0960 0.87 0.06 0.57 0.657 0.7576 0.8660 0.96 0.97 0.99 0.9977 0.5 0.000 0.007 0.06 0.059 0.8 0. 0.50 0.5000 0.6550 0.7878 0.885 0.96 0.978 0.997 0.9980 0.5 0.00 0.0086 0.058 0.069 0.0 0. 0.8 0.56 0.697 0.87 0.900 0.9560 0.986 0.99 0.998 0. 0.00 0.0095 0.09 0.076 0.56 0.75 0.6 0.5858 0.7 0.86 0.9 0.9656 0.9868 0.9957 0.9988 0.5 0.00 0.0097 0.00 0.086 0.7 0.06 0.668 0.60 0.77 0.876 0.996 0.975 0.9907 0.997 0.999 / 0.005 0.09 0.06 0.0 0.5 0.70 0.576 0.6956 0.80 0.908 0.9585 0.986 0.99 0.998 0. 0.006 0.0090 0.0 0.0905 0.95 0.07 0.58 0.6769 0.806 0.90 0.9558 0.985 0.990 0.998 0.5 0.0008 0.0070 0.0 0.096 0.7 0.78 0.56 0.765 0.8506 0.987 0.970 0.989 0.9966 0.999 0. 0.008 0.07 0.098 0.0 0.07 0.667 0.7800 0.8909 0.95 0.987 0.99 0.9985 /6 0.005 0.069 0.887 0.86 0.597 0.770 0.8908 0.955 0.98 0.995 0.9988 0.5 0.07 0.09 0.57 0.7 0.68 0.885 0.905 0.975 0.990 0.9979 0. 0.078 0.7 0.57 0.766 0.900 0.9666 0.9905 0.9977 0.05 0.77 0.6 0.879 0.9659 0.998 0.9988 CUMULATIVE BINOMIAL PROBABILITIES = 5 = 0 5 6 7 8 9 0 5 6 7 8 9 0 5
Aedi B: Mathematical Formulae ad Statistical Tables OCR 00 GCE Mathematics Oford, Cambridge ad RSA Eamiatios 0.95 0.0006 0.00 0.056 0.0608 0.878 0.65 0.785 0.9 0.000 0.0078 0.058 0.07 0.755 0.56 0.5886 0.86 0.9576 0.85 0.000 0.0008 0.009 0.0097 0.078 0.0698 0.56 0.89 0.755 0.678 0.886 0.950 0.99 5/6 0.000 0.0067 0.097 0.0506 0.7 0.5 0.86 0.5757 0.760 0.897 0.9705 0.9958 0.8 0.000 0.0009 0.00 0.0095 0.056 0.06 0.87 0.9 0.90 0.575 0.78 0.877 0.9558 0.9895 0.9988 0.75 0.000 0.0008 0.007 0.008 0.06 0.0507 0.057 0.966 0.6 0.857 0.659 0.797 0.90 0.966 0.989 0.9980 0.7 0.000 0.0006 0.00 0.006 0.069 0.00 0.085 0.59 0.696 0. 0.5685 0.786 0.805 0.9 0.9698 0.9907 0.9979 / 0.000 0.0007 0.005 0.007 0.088 0.05 0.0898 0.660 0.76 0.5 0.568 0.70 0.8 0.96 0.965 0.9878 0.9967 0.999 0.65 0.00 0.005 0.0 0.00 0.065 0.6 0.98 0.5 0.9 0.65 0.775 0.876 0.9 0.9767 0.995 0.998 0.6 0.000 0.0009 0.009 0.008 0.0 0.08 0.097 0.75 0.855 0.5 0.5689 0.7085 0.87 0.9060 0.9565 0.988 0.99 0.9985 0.55 0.006 0.0050 0.08 0.0 0.07 0.56 0.09 0.55 0.975 0.608 0.767 0.8650 0.906 0.9688 0.9879 0.9960 0.9989 0.5 0.000 0.0007 0.006 0.008 0.0 0.09 0.00 0.808 0.9 0.78 0.57 0.7077 0.89 0.8998 0.9506 0.9786 0.999 0.997 0.999 0.5 0.000 0.00 0.000 0.0 0.0 0.069 0.50 0.7 0.59 0.505 0.68 0.769 0.86 0.986 0.9666 0.986 0.9950 0.998 0. 0.005 0.0057 0.07 0.05 0.090 0.76 0.95 0. 0.5785 0.75 0.86 0.909 0.959 0.9788 0.997 0.997 0.999 0.5 0.009 0.0075 0.0 0.0586 0.8 0.7 0.575 0.5078 0.658 0.780 0.877 0.98 0.9699 0.9876 0.9955 0.9986 / 0.0007 0.00 0.0 0.055 0.088 0.668 0.860 0.7 0.588 0.79 0.80 0.90 0.9565 0.98 0.998 0.9975 0.999 0. 0.00 0.009 0.00 0.0766 0.595 0.8 0.5 0.5888 0.70 0.807 0.955 0.9599 0.98 0.996 0.9979 0.999 0.5 0.000 0.000 0.006 0.07 0.0979 0.06 0.8 0.5 0.676 0.80 0.89 0.99 0.978 0.998 0.997 0.999 0. 0.00 0.005 0.0 0.7 0.55 0.75 0.6070 0.7608 0.87 0.989 0.97 0.9905 0.9969 0.999 /6 0.00 0.095 0.08 0.96 0. 0.66 0.7765 0.886 0.99 0.980 0.99 0.9980 0.5 0.0076 0.080 0.5 0.7 0.55 0.706 0.87 0.90 0.97 0.990 0.997 0.999 0. 0.0 0.87 0. 0.67 0.85 0.968 0.97 0.99 0.9980 0.05 0.6 0.555 0.8 0.99 0.98 0.9967 0.999 CUMULATIVE BINOMIAL PROBABILITIES = 0 = 0 5 6 7 8 9 0 5 6 7 8 9 0 5 6 7 8 9 0
CUMULATIVE POISSON PROBABILITIES λ 0.0 0.0 0.0 0.0 0.05 0.06 0.07 0.08 0.09 = 0 0.9900 0.980 0.970 0.9608 0.95 0.98 0.9 0.9 0.99 0.999 0.9988 0.998 0.9977 0.9970 0.996 λ 0.0 0.0 0.0 0.0 0.50 0.60 0.70 0.80 0.90 = 0 0.908 0.887 0.708 0.670 0.6065 0.588 0.966 0.9 0.066 0.995 0.985 0.96 0.98 0.9098 0.878 0.8 0.8088 0.775 0.9989 0.996 0.99 0.9856 0.9769 0.9659 0.956 0.97 0.999 0.998 0.9966 0.99 0.9909 0.9865 0.999 0.9986 0.9977 5 6 λ.00.0.0.0.0.50.60.70.80.90 = 0 0.679 0.9 0.0 0.75 0.66 0. 0.09 0.87 0.65 0.96 0.758 0.6990 0.666 0.668 0.598 0.5578 0.59 0.9 0.68 0.7 0.997 0.900 0.8795 0.857 0.85 0.8088 0.78 0.757 0.706 0.707 0.980 0.97 0.966 0.9569 0.96 0.9 0.9 0.9068 0.89 0.877 0.996 0.996 0.99 0.989 0.9857 0.98 0.976 0.970 0.966 0.9559 5 0.999 0.9990 0.9985 0.9978 0.9968 0.9955 0.990 0.990 0.9896 0.9868 6 0.999 0.999 0.9987 0.998 0.997 0.9966 7 0.999 0.999 8 9 λ.00.0.0.0.0.50.60.70.80.90 = 0 0.5 0.5 0.08 0.00 0.0907 0.08 0.07 0.067 0.0608 0.0550 0.060 0.796 0.56 0.09 0.08 0.87 0.67 0.87 0. 0.6 0.6767 0.696 0.67 0.5960 0.5697 0.58 0.58 0.96 0.695 0.60 0.857 0.886 0.89 0.799 0.7787 0.7576 0.760 0.7 0.699 0.6696 0.97 0.979 0.975 0.96 0.90 0.89 0.877 0.869 0.877 0.88 5 0.98 0.9796 0.975 0.9700 0.96 0.9580 0.950 0.9 0.99 0.958 6 0.9955 0.99 0.995 0.9906 0.988 0.9858 0.988 0.979 0.9756 0.97 7 0.9989 0.9985 0.9980 0.997 0.9967 0.9958 0.997 0.99 0.999 0.990 8 0.999 0.999 0.9989 0.9985 0.998 0.9976 0.9969 9 0.999 0.999 0 OCR00 Aedi B: Mathematical Formulae ad Statistical Tables 5 Oford, Cambridge ad RSA Eamiatios GCE Mathematics
CUMULATIVE POISSON PROBABILITIES λ.00.0.0.0.0.50.60.70.80.90 = 0 0.098 0.050 0.008 0.069 0.0 0.00 0.07 0.07 0.0 0.00 0.99 0.87 0.7 0.586 0.68 0.59 0.57 0.6 0.07 0.099 0. 0.0 0.799 0.59 0.97 0.08 0.07 0.85 0.689 0.5 0.67 0.68 0.605 0.580 0.558 0.566 0.55 0.9 0.75 0.5 0.85 0.798 0.7806 0.766 0.7 0.75 0.706 0.687 0.6678 0.68 5 0.96 0.9057 0.896 0.889 0.8705 0.8576 0.8 0.80 0.856 0.8006 6 0.9665 0.96 0.955 0.990 0.9 0.97 0.967 0.98 0.909 0.8995 7 0.988 0.9858 0.98 0.980 0.9769 0.97 0.969 0.968 0.9599 0.956 8 0.996 0.995 0.99 0.99 0.997 0.990 0.988 0.986 0.980 0.985 9 0.9989 0.9986 0.998 0.9978 0.997 0.9967 0.9960 0.995 0.99 0.99 0 0.999 0.999 0.9990 0.9987 0.998 0.998 0.9977 0.999 0.999 λ.00.0.0.0.0.50.60.70.80.90 = 0 0.08 0.066 0.050 0.06 0.0 0.0 0.00 0.009 0.008 0.007 0.096 0.085 0.0780 0.079 0.066 0.06 0.056 0.058 0.077 0.09 0.8 0.8 0.0 0.97 0.85 0.76 0.66 0.5 0.5 0. 0.5 0. 0.95 0.77 0.59 0. 0.57 0.097 0.9 0.79 0.688 0.609 0.5898 0.570 0.55 0.5 0.5 0.96 0.76 0.58 5 0.785 0.769 0.75 0.767 0.799 0.709 0.6858 0.668 0.650 0.65 6 0.889 0.8786 0.8675 0.8558 0.86 0.8 0.880 0.806 0.7908 0.7767 7 0.989 0.97 0.96 0.990 0.9 0.9 0.909 0.8960 0.8867 0.8769 8 0.9786 0.9755 0.97 0.968 0.96 0.9597 0.959 0.997 0.9 0.98 9 0.999 0.9905 0.9889 0.987 0.985 0.989 0.9805 0.9778 0.979 0.977 0 0.997 0.9966 0.9959 0.995 0.99 0.99 0.99 0.990 0.9896 0.9880 0.999 0.9989 0.9986 0.998 0.9980 0.9976 0.997 0.9966 0.9960 0.995 0.999 0.999 0.9990 0.9988 0.9986 0.998 0.999 5 6 6 Aedi B: Mathematical Formulae ad Statistical Tables OCR 00 GCE Mathematics Oford, Cambridge ad RSA Eamiatios
CUMULATIVE POISSON PROBABILITIES λ 5.00 5.50 6.00 6.50 7.00 7.50 8.00 8.50 9.00 9.50 = 0 0.0067 0.00 0.005 0.005 0.0009 0.0006 0.000 0.00 0.066 0.07 0.0 0.007 0.007 0.000 0.009 0.00 0.0008 0.7 0.088 0.060 0.00 0.096 0.00 0.08 0.009 0.006 0.00 0.650 0.07 0.5 0.8 0.088 0.059 0.0 0.00 0.0 0.09 0.05 0.575 0.85 0.7 0.70 0. 0.0996 0.07 0.0550 0.00 5 0.660 0.589 0.57 0.690 0.007 0. 0.9 0.96 0.57 0.0885 6 0.76 0.6860 0.606 0.565 0.97 0.78 0. 0.56 0.068 0.69 7 0.8666 0.8095 0.70 0.678 0.5987 0.56 0.50 0.856 0.9 0.687 8 0.99 0.89 0.87 0.796 0.79 0.660 0.595 0.5 0.557 0.98 9 0.968 0.96 0.96 0.877 0.805 0.776 0.766 0.650 0.587 0.58 0 0.986 0.977 0.957 0.9 0.905 0.86 0.859 0.76 0.7060 0.65 0.995 0.9890 0.9799 0.966 0.967 0.908 0.888 0.887 0.800 0.750 0.9980 0.9955 0.99 0.980 0.970 0.957 0.96 0.909 0.8758 0.86 0.999 0.998 0.996 0.999 0.987 0.978 0.9658 0.986 0.96 0.898 0.999 0.9986 0.9970 0.99 0.9897 0.987 0.976 0.9585 0.900 5 0.9988 0.9976 0.995 0.998 0.986 0.9780 0.9665 6 0.9990 0.9980 0.996 0.99 0.9889 0.98 7 0.999 0.998 0.9970 0.997 0.99 8 0.999 0.9987 0.9976 0.9957 9 0.9989 0.9980 0 0.999 OCR00 Aedi B: Mathematical Formulae ad Statistical Tables 7 Oford, Cambridge ad RSA Eamiatios GCE Mathematics
CUMULATIVE POISSON PROBABILITIES λ 0.00.00.00.00.00 5.00 6.00 7.00 8.00 9.00 = 0 0.000 0.008 0.00 0.000 0.00 0.009 0.00 0.00 0.000 0.09 0.05 0.0076 0.007 0.008 0.0009 0.000 5 0.067 0.075 0.00 0.007 0.0055 0.008 0.00 0.0007 0.000 6 0.0 0.0786 0.058 0.059 0.0 0.0076 0.000 0.00 0.000 7 0.0 0. 0.0895 0.050 0.06 0.080 0.000 0.005 0.009 0.005 8 0.8 0.0 0.550 0.0998 0.06 0.07 0.00 0.06 0.007 0.009 9 0.579 0.05 0. 0.658 0.09 0.0699 0.0 0.06 0.05 0.0089 0 0.580 0.599 0.7 0.57 0.757 0.85 0.077 0.09 0.00 0.08 0.6968 0.579 0.66 0.5 0.600 0.88 0.70 0.087 0.059 0.07 0.796 0.6887 0.5760 0.6 0.585 0.676 0.9 0.50 0.097 0.0606 0.865 0.78 0.685 0.570 0.6 0.6 0.75 0.009 0.6 0.098 0.965 0.850 0.770 0.675 0.570 0.657 0.675 0.808 0.08 0.97 5 0.95 0.907 0.8 0.766 0.669 0.568 0.667 0.75 0.867 0.8 6 0.970 0.9 0.8987 0.855 0.7559 0.66 0.5660 0.677 0.75 0.90 7 0.9857 0.9678 0.970 0.8905 0.87 0.789 0.659 0.560 0.686 0.78 8 0.998 0.98 0.966 0.90 0.886 0.895 0.7 0.6550 0.56 0.695 9 0.9965 0.9907 0.9787 0.957 0.95 0.875 0.8 0.76 0.6509 0.5606 0 0.998 0.995 0.988 0.9750 0.95 0.970 0.868 0.8055 0.707 0.67 0.999 0.9977 0.999 0.9859 0.97 0.969 0.908 0.865 0.799 0.755 0.9990 0.9970 0.99 0.98 0.967 0.98 0.907 0.855 0.79 0.9985 0.9960 0.9907 0.9805 0.96 0.967 0.8989 0.890 0.999 0.9980 0.9950 0.9888 0.9777 0.959 0.97 0.89 5 0.9990 0.997 0.998 0.9869 0.978 0.955 0.969 6 0.9987 0.9967 0.995 0.988 0.978 0.95 7 0.999 0.998 0.9959 0.99 0.987 0.9687 8 0.999 0.9978 0.9950 0.9897 0.9805 9 0.9989 0.997 0.99 0.988 0 0.999 0.9986 0.9967 0.990 0.999 0.998 0.9960 0.9990 0.9978 0.9988 0.999 5 6 7 8 8 Aedi B: Mathematical Formulae ad Statistical Tables OCR 00 GCE Mathematics Oford, Cambridge ad RSA Eamiatios
THE NORMAL DISTRIBUTION FUNCTION If Z has a ormal distributio with mea 0 ad variace the, for each value of z, the table gives the value of Φ ( z), where: Φ ( z) = P( Z z) For egative values of z use Φ ( z) = Φ ( z). z 0 5 6 7 8 9 5 6 7 8 9 ADD 0.0 0.5000 0.500 0.5080 0.50 0.560 0.599 0.59 0.579 0.59 0.559 8 6 0 8 6 0. 0.598 0.58 0.578 0.557 0.5557 0.5596 0.566 0.5675 0.57 0.575 8 6 0 8 6 0. 0.579 0.58 0.587 0.590 0.598 0.5987 0.606 0.606 0.60 0.6 8 5 9 7 5 0. 0.679 0.67 0.655 0.69 0.6 0.668 0.606 0.6 0.680 0.657 7 5 9 6 0 0. 0.655 0.659 0.668 0.666 0.6700 0.676 0.677 0.6808 0.68 0.6879 7 8 5 9 0.5 0.695 0.6950 0.6985 0.709 0.705 0.7088 0.7 0.757 0.790 0.7 7 0 7 0 7 0.6 0.757 0.79 0.7 0.757 0.789 0.7 0.75 0.786 0.757 0.759 7 0 6 9 6 9 0.7 0.7580 0.76 0.76 0.767 0.770 0.77 0.776 0.779 0.78 0.785 6 9 5 8 7 0.8 0.788 0.790 0.799 0.7967 0.7995 0.80 0.805 0.8078 0.806 0.8 5 8 6 9 5 0.9 0.859 0.886 0.8 0.88 0.86 0.889 0.85 0.80 0.865 0.889 5 8 0 5 8 0.0 0.8 0.88 0.86 0.885 0.8508 0.85 0.855 0.8577 0.8599 0.86 5 7 9 6 9. 0.86 0.8665 0.8686 0.8708 0.879 0.879 0.8770 0.8790 0.880 0.880 6 8 0 6 8. 0.889 0.8869 0.8888 0.8907 0.895 0.89 0.896 0.8980 0.8997 0.905 6 7 9 5 7. 0.90 0.909 0.9066 0.908 0.9099 0.95 0.9 0.97 0.96 0.977 5 6 8 0. 0.99 0.907 0.9 0.96 0.95 0.965 0.979 0.99 0.906 0.99 6 7 8 0.5 0.9 0.95 0.957 0.970 0.98 0.99 0.906 0.98 0.99 0.9 5 6 7 8 0.6 0.95 0.96 0.97 0.98 0.995 0.9505 0.955 0.955 0.955 0.955 5 6 7 8 9.7 0.955 0.956 0.957 0.958 0.959 0.9599 0.9608 0.966 0.965 0.96 5 6 7 8.8 0.96 0.969 0.9656 0.966 0.967 0.9678 0.9686 0.969 0.9699 0.9706 5 6 6.9 0.97 0.979 0.976 0.97 0.978 0.97 0.9750 0.9756 0.976 0.9767 5 5.0 0.977 0.9778 0.978 0.9788 0.979 0.9798 0.980 0.9808 0.98 0.987 0. 0.98 0.986 0.980 0.98 0.988 0.98 0.986 0.9850 0.985 0.9857 0. 0.986 0.986 0.9868 0.987 0.9875 0.9878 0.988 0.988 0.9887 0.9890 0. 0.989 0.9896 0.9898 0.990 0.990 0.9906 0.9909 0.99 0.99 0.996 0. 0.998 0.990 0.99 0.995 0.997 0.999 0.99 0.99 0.99 0.996 0 0.5 0.998 0.990 0.99 0.99 0.995 0.996 0.998 0.999 0.995 0.995 0 0 0.6 0.995 0.9955 0.9956 0.9957 0.9959 0.9960 0.996 0.996 0.996 0.996 0 0 0 0.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.997 0.997 0.997 0.997 0 0 0 0 0.8 0.997 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 0.998 0.998 0.998 0.9985 0.9985 0.9986 0.9986 0 0 0 0 0 0 0 0 0 Critical values for the ormal distributio If Z has a ormal distributio with mea 0 ad variace the, for each value of, the table gives the value of z such that: P( Z z) =. 0.75 0.90 0.95 0.975 0.99 0.995 0.9975 0.999 z 0.67.8.65.960.6.576.807.090.9 OCR00 Aedi B: Mathematical Formulae ad Statistical Tables 9 Oford, Cambridge ad RSA Eamiatios GCE Mathematics
0//0 DRAFT MATHEMATICS DRAFT 0//0 CRITICAL VALUES FOR THE t-distribution If T has a t-distributio with ν degrees of freedom the, for each air of values of ad ν, the table gives the value of t such that: P( T t) =. 0.75 0.90 0.95 0.975 0.99 0.995 0.9975 0.999 ν =.000.078 6..7.8 6.66 7. 8. 66.6 0.86.886.90.0 6.965 9.95.09..60 0.765.68.5.8.5 5.8 7.5 0..9 0.7.5..776.77.60 5.598 7.7 8.60 5 0.77.76.05.57.65.0.77 5.89 6.869 6 0.78.0.9.7..707.7 5.08 5.959 7 0.7.5.895.65.998.99.09.785 5.08 8 0.706.97.860.06.896.55.8.50 5.0 9 0.70.8.8.6.8.50.690.97.78 0 0.700.7.8.8.76.69.58..587 0.697.6.796.0.78.06.97.05.7 0.695.56.78.79.68.055.8.90.8 0.69.50.77.60.650.0.7.85. 0.69.5.76.5.6.977.6.787.0 5 0.69..75..60.97.86.7.07 6 0.690.7.76.0.58.9.5.686.05 7 0.689..70.0.567.898..66.965 8 0.688.0.7.0.55.878.97.60.9 9 0.688.8.79.09.59.86.7.579.88 0 0.687.5.75.086.58.85.5.55.850 0.686..7.080.58.8.5.57.89 0.686..77.07.508.89.9.505.79 0.685.9.7.069.500.807.0.85.768 0.685.8.7.06.9.797.09.67.75 5 0.68.6.708.060.85.787.078.50.75 6 0.68.5.706.056.79.779.067.5.707 7 0.68..70.05.7.77.057..689 8 0.68..70.08.67.76.07.08.67 9 0.68..699.05.6.756.08.96.660 0 0.68.0.697.0.57.750.00.85.66 0 0.68.0.68.0..70.97.07.55 60 0.679.96.67.000.90.660.95..60 0 0.677.89.658.980.58.67.860.60.7 0.67.8.65.960.6.576.807.090.9 0 Aedi B: Mathematical Formulae ad Statistical Tables OCR 00 GCE Mathematics Oford, Cambridge ad RSA Eamiatios
CRITICAL VALUES FOR THE χ -DISTRIBUTION If X has a χ -distributio with ν degrees of freedom the, for each air of values of ad ν, the table gives the value of such that: P( X ) = 0.0 0.05 0.05 0.9 0.95 0.975 0.99 0.995 0.999 ν = 0.0 57 0.0 98 0.0 9.706.8 5.0 6.65 7.879 0.8 0.000 0.0506 0.06.605 5.99 7.78 9.0 0.60.8 0.8 0.58 0.58 6.5 7.85 9.8..8 6.7 0.97 0.8 0.707 7.779 9.88..8.86 8.7 5 0.55 0.8.5 9.6.07.8 5.09 6.75 0.5 6 0.87.7.65 0.6.59.5 6.8 8.55.6 7.9.690.67.0.07 6.0 8.8 0.8. 8.67.80.7.6 5.5 7.5 0.09.95 6. 9.088.700.5.68 6.9 9.0.67.59 7.88 0.558.7.90 5.99 8. 0.8. 5.9 9.59.05.86.575 7.8 9.68.9.7 6.76.6.57.0 5.6 8.55.0. 6. 8.0.9.07 5.009 5.89 9.8.6.7 7.69 9.8.5.660 5.69 6.57.06.68 6. 9.. 6. 5 5.9 6.6 7.6. 5.00 7.9 0.58.80 7.70 6 5.8 6.908 7.96.5 6.0 8.85.00.7 9.5 7 6.08 7.56 8.67.77 7.59 0.9. 5.7 0.79 8 7.05 8. 9.90 5.99 8.87.5.8 7.6. 9 7.6 8.907 0. 7.0 0..85 6.9 8.58.8 0 8.60 9.59 0.85 8...7 7.57 0.00 5. 8.897 0.8.59 9.6.67 5.8 8.9.0 6.80 9.5 0.98. 0.8.9 6.78 0.9.80 8.7 0.0.69.09.0 5.7 8.08.6.8 9.7 0.86.0.85.0 6. 9.6.98 5.56 5.8 5.5..6.8 7.65 0.65. 6.9 5.6 0.95 6.79 8.9 0.6.77 6.98 50.89 5.67 59.70 0.6. 6.5 5.8 55.76 59. 6.69 66.77 7.0 50 9.7.6.76 6.7 67.50 7. 76.5 79.9 86.66 60 7.8 0.8.9 7.0 79.08 8.0 88.8 9.95 99.6 70 5. 8.76 5.7 85.5 90.5 95.0 00. 0.. 80 5.5 57.5 60.9 96.58 0.9 06.6. 6..8 90 6.75 65.65 69. 07.6. 8.. 8. 7. 00 70.06 7. 77.9 8.5. 9.6 5.8 0. 9. OCR00 Aedi B: Mathematical Formulae ad Statistical Tables Oford, Cambridge ad RSA Eamiatios GCE Mathematics
WILCOXON SIGNED RANK TEST P is the sum of the raks corresodig to the ositive differeces, Q is the sum of the raks corresodig to the egative differeces, T is the smaller of P ad Q. For each value of the table gives the largest value of T which will lead to rejectio of the ull hyothesis at the level of sigificace idicated. Critical values of T Level of sigificace Oe Tail 0.05 0.05 0.0 0.005 Two Tail 0. 0.05 0.0 0.0 = 6 0 7 0 8 5 0 9 8 5 0 0 8 5 0 7 5 7 9 7 7 9 5 5 5 0 5 9 5 6 5 9 9 7 7 8 7 0 7 9 5 6 7 0 60 5 7 For larger values of, each of P ad Q ca be aroimated by the ormal distributio with mea + ( ) ad variace ( + )(+ ). Aedi B: Mathematical Formulae ad Statistical Tables OCR 00 GCE Mathematics Oford, Cambridge ad RSA Eamiatios
WILCOXON RANK SUM TEST The two samles have sizes m ad, where m. R m is the sum of the raks of the items i the samle of size m. W is the smaller of R m ad m ( + m+ ) Rm. For each air of values of m ad, the table gives the largest value of W which will lead to rejectio of the ull hyothesis at the level of sigificace idicated. Critical values of W Level of sigificace Oe Tail 0.05 0.05 0.0 0.05 0.05 0.0 0.05 0.05 0.0 0.05 0.05 0.0 Two Tail 0. 0.05 0.0 0. 0.05 0.0 0. 0.05 0.0 0. 0.05 0.0 m = m = m = 5 m = 6 6 6 0 5 7 6 0 9 7 6 6 8 7 0 8 7 8 6 7 8 7 6 0 8 9 7 5 8 9 8 6 5 9 9 7 9 0 8 7 6 0 8 0 0 9 7 7 5 6 5 9 Level of sigificace Oe Tail 0.05 0.05 0.0 0.05 0.05 0.0 0.05 0.05 0.0 0.05 0.05 0.0 Two Tail 0. 0.05 0.0 0. 0.05 0.0 0. 0.05 0.0 0. 0.05 0.0 m = 7 m = 8 m = 9 m = 0 7 9 6 8 8 5 5 9 5 9 0 7 5 5 7 66 6 59 0 5 9 56 5 9 69 65 6 8 78 7 For larger values of m ad, the ormal distributio with mea mm ( + + ) ad variace m( m + + ) should be used as a aroimatio to the distributio of R m. OCR00 Aedi B: Mathematical Formulae ad Statistical Tables Oford, Cambridge ad RSA Eamiatios GCE Mathematics