Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE Mathematics and Further Mathematics

Transcription

1 Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE Mathematics and Further Mathematics Mathematical formulae and statistical tables First certification from 08 Advanced Subsidiary GCE in Mathematics (8MA0) Advanced GCE in Mathematics (9MA0) Advanced Subsidiary GCE in Further Mathematics (8FM0) First certification from 09 Advanced GCE in Further Mathematics (9FM0) This copy is the property of Pearson. It is not to be removed from the eamination room or marked in any way. P54458A 07 Pearson Education Ltd. //////

2 Edecel, BTEC and LCCI qualifications Edecel, BTEC and LCCI qualifications are awarded by Pearson, the UK s largest awarding body offering academic and vocational qualifications that are globally recognised and benchmarked. For further information, please visit our qualification website at qualifications.pearson.com. Alternatively, you can get in touch with us using the details on our contact us page at qualifications.pearson.com/contactus About Pearson Pearson is the world s leading learning company, with 35,000 employees in more than 70 countries working to help people of all ages to make measurable progress in their lives through learning. We put the learner at the centre of everything we do, because wherever learning flourishes, so do people. Find out more about how we can help you and your learners at qualifications.pearson.com References to third party material made in this sample assessment materials are made in good faith. Pearson does not endorse, approve or accept responsibility for the content of materals, which may be subject to change, or any opinions epressed therein. (Material may include tetbooks, journals, magazines and other publications and websites.) All information in this document is correct at time of publication. ISBN Pearson Education Limited 07

3 Contents Introduction AS Mathematics 3 Pure Mathematics 3 Statistics 3 Mechanics 4 A Level Mathematics 5 Pure Mathematics 5 Statistics 7 Mechanics 8 3 AS Further Mathematics 9 Pure Mathematics 9 Statistics 3 Mechanics 5 4 A Level Further Mathematics 7 Pure Mathematics 7 Statistics 3 Mechanics 7 5 Statistical Tables 9 Binomial Cumulative Distribution Function 9 Percentage Points of The Normal Distribution 34 Poisson Cumulative Distribution Function 35 Percentage Points of the χ Distribution 36 Critical Values for Correlation Coefficients 37 Random Numbers 38 Percentage Points of Student s t Distribution 39 Percentage Points of the F Distribution 40

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5 Introduction The formulae in this booklet have been arranged by qualification. Students sitting AS or A Level Further Mathematics papers may be required to use the formulae that were introduced in AS or A Level Mathematics papers. It may also be the case that students sitting Mechanics and Statistics papers will need to use formulae introduced in the appropriate Pure Mathematics papers for the qualification they are sitting. Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07

6 Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07

7 AS Mathematics Pure Mathematics Mensuration Surface area of sphere = 4πr Area of curved surface of cone = πr slant height Binomial series n (a + b) n = a n + an b + n an b n r an r b r b n (n ) where n r = n C r = n! r!( n r)! Logarithms and eponentials log a = log log e ln a = a b b a Differentiation First Principles f + f f ( ) = lim ( h ) ( ) h 0 h Statistics Probability P( A ) = P( A) Standard deviation Standard deviation = (Variance) Interquartile range = IQR = Q 3 Q For a set of n values,,... i,... n S = Σ( i ) = Σ i (Σ i ) n Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07 3

8 Standard deviation = S n or n Statistical tables The following statistical tables are required for A Level Mathematics: Binomial Cumulative Distribution Function (see page 9) Random Numbers (see page 38) Mechanics Kinematics For motion in a straight line with constant acceleration: v = u + at s = ut + at s = vt at v = u + as s = (u + v)t 4 Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07

9 A Level Mathematics Pure Mathematics Mensuration Surface area of sphere = 4πr Area of curved surface of cone = πr slant height Arithmetic series S n = n(a + l) = n[a + (n )d] Binomial series n (a + b) n = a n + an b + n an b n r an r b r b n (n ) where n r = n C r = n! r!( n r)! ( + ) n = + n + nn ( ) nn ( )...( n r + ) r +... (½ ½ <, n )... r Logarithms and eponentials log a = log log e ln a = a b b a Geometric series S n = a r n ( ) r S = a r for ½ r ½ < Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07 5

10 Trigonometric identities sin(a ± B) = sin A cosb ± cosa sinb cos(a ± B) = cosa cosb sin A sinb tan(a ± B) = tan A ± tan B (A ± B (k + tan Atan B )π) sin A + sinb = sin A + B cos A B sin A sinb = cos A + B sin A B cosa + cosb = cos A + B cos A B cosa cosb = sin A + B sin A B Small angle approimations sin θ θ cos θ tan θ θ θ where θ is measured in radians Differentiation First Principles f + f f ( ) = lim ( h ) ( ) h 0 h f() tan k sec k cot k cosec k f ( ) g( ) f () k sec k k sec k tan k k cosec k k cosec k cot k f ( ) g( ) f( ) g ( ) (g ( )) 6 Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07

11 Integration (+ constant) f() f() d sec k tan k cot k k tan k ln½sec k½ k ln½sin k½ k cosec k k ln½cosec k + cot k½, k ln½tan ( k)½ sec k u d v d d = uv v d u d d k ln½sec k + tan k½, k ln½tan ( k + π)½ 4 Numerical Methods The trapezium rule: b y d h{( y + y ) + (y + y y b a )}, where h = a 0 n n n The Newton-Raphson iteration for solving f() = 0 : n+ = n f ( n ) f ( ) Statistics n Probability P( A ) = P( A) P(A B) = P(A) + P(B) P(A B) P(A B) = P(A)P(B ½ A) P( B A) P( A) P(A ½ B) = P( B A) P( A) + P( B A ) P( A ) For independent events A and B, P(B ½ A) = P(B) P(A ½ B) = P(A) P(A B) = P(A) P(B) Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07 7

12 Standard deviation Standard deviation = (Variance) Interquartile range = IQR = Q 3 Q For a set of n values,,... i,... n S = Σ( i ) = Σ i (Σ i ) n Standard deviation = S n or n Discrete distributions Distribution of X P(X = ) Mean Variance Binomial B(n, p) n p ( p) n np np( p) Sampling distributions For a random sample of n observations from N(μ, σ ) X σ / μ n ~ N(0, ) Statistical tables The following statistical tables are required for A Level Mathematics: Binomial Cumulative Distribution Function (see page 9) Percentage Points of The Normal Distribution (see page 34) Critical Values for Correlation Coefficients: Product Moment Coefficient (see page 37) Random Numbers (see page 38) Mechanics Kinematics For motion in a straight line with constant acceleration: v = u + at s = ut + at s = vt at v = u + as s = (u + v)t 8 Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07

13 3 AS Further Mathematics Students sitting an AS Level Further Mathematics paper may also require those formulae listed for A Level Mathematics in Section. Pure Mathematics Summations n r = 6 n(n + )(n + ) r = n 3 r = 4 n (n + ) r = Matri transformations Anticlockwise rotation through θ about O: cosθ sinθ sin θ cosθ cosθ Reflection in the line y = (tanθ): sinθ sinθ cosθ Area of a sector A = r dθ (polar coordinates) 3 Comple numbers {r(cosθ + isinθ)} n = r n (cosnθ + isinnθ) The roots of z n = are given by z = e πki n, for k = 0,,,..., n Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07 9

14 Maclaurin s and Taylor s Series f() = f(0) + f (0) +! f (0) r! f (r) (0) +... r e = ep() = + +! r! r +... for all ln( + ) = r ( )r+ r sin = 3 3! + 5 5!... + ( )r r + ( r + )! +... ( < ) +... for all cos =! + 4 4!... + ( )r r ( r)! +... for all arctan = ( )r r + r ( ) Vectors Vector product: a b = ½a½½b½sinθ n = i j k a a a 3 b b b 3 = ab ab ab ab ab ab a.(b c) = a a a 3 b b b 3 c c c 3 = b.(c a) = c.(a b) If A is the point with position vector a = a i + a j + a 3 k and the direction vector b is given by b = b i + b j + b 3 k, then the straight line through A with direction vector b has cartesian equation a y a z a b b b 3 = = = 3 ( λ) The plane through A with normal vector n = n i + n j + n 3 k has cartesian equation n + n y + n 3 z + d = 0 where d = a.n The plane through non-collinear points A, B and C has vector equation r = a + λ(b a) + μ(c a) = ( λ μ)a + λb + μc The plane through the point with position vector a and parallel to b and c has equation r = a + sb + tc The perpendicular distance of (α, β, γ) from n + n y + n 3 z + d = 0 is nα+ nβ+ nγ+ d 3 n + n + n 3. 0 Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07

15 Hyperbolic functions cosh sinh = sinh = sinh cosh cosh = cosh + sinh arcosh = ln{ + } ( ) arsinh = ln{ + + } artanh = ln + (½½ < ) Differentiation f() arcsin arccos arctan sinh cosh tanh arsinh arcosh artanh f () + cosh sinh sech + 3 Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07

16 Integration (+ constant; a > 0 where relevant) f() sinh cosh tanh a a a + a a + f() d cosh sinh lncosh arcsin a a arctan arcosh arsinh a ln a a a (½½ < a) a, ln{ + a } ( > a) a, ln{ + + a } + = a artanh a (½½ < a) a a ln + a a Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07

17 Statistics Discrete distributions For a discrete random variable X taking values i with probabilities P(X = i ) Epectation (mean): E(X ) = μ = Σ i P(X = i ) Variance: Var(X ) = σ = Σ( i μ) P(X = i ) = Σ P(X = ) μ i i Discrete distributions Standard discrete distributions: Distribution of X P(X = ) Mean Variance Binomial B(n, p) n p ( p) n np np( p) Poisson Po(λ) e λ λ! λ λ Continuous distributions For a continuous random variable X having probability density function f Epectation (mean): E(X ) = μ = f() d 3 Variance: Var(X ) = σ = ( μ) f() d = f() d μ For a function g(x ): E(g(X )) = g() f() d 0 Cumulative distribution function: F( 0 ) = P(X 0 ) = # f(t) dt Standard continuous distribution: Distribution of X P.D.F. Mean Variance Normal N(μ, σ ) σ π e μ σ μ σ Uniform (Rectangular) on [a, b] b a (a + b) (b a) Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07 3

18 Correlation and regression For a set of n pairs of values ( i, y i ) S = Σ( i ) = Σ i S yy = Σ( y i y) = Σy i (Σ i ) n (Σ y i ) n S y = Σ( i )( y i y) = Σ i y i (Σ i)(σ yi) n The product moment correlation coefficient is (Σ i)(σ yi) S Σy Σ( )( ) i i y i yi y r = = = n S S yy { Σ( ) }{ Σ( ) } (Σ ) (Σ ) i yi y i yi Σi Σyi n n Sy Σ( i )( yi y) The regression coefficient of y on is b = = S Σ( ) Least squares regression line of y on is y = a + b where a = y b i Residual Sum of Squares (RSS) = S yy S S ( y ) = S yy ( r ) Spearman s rank correlation coefficient is r S = 6Σd nn ( ) Non-parametric tests Goodness-of-fit test and contingency tables: ( Oi Ei) E i ~ χ v Statistical tables The following statistical tables are required for AS Level Further Mathematics: Binomial Cumulative Distribution Function (see page 9) Poisson Cumulative Distribution Function (see page 35) Percentage Points of the χ Distribution (see page 36) Critical Values for Correlation Coefficients: Product Moment Coefficient and Spearman s Coefficient (see page 37) Random Numbers (see page 38) 4 Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07

19 Mechanics Centres of mass For uniform bodies: Triangular lamina: 3 along median from verte rsin α Circular arc, radius r, angle at centre α : α Sector of circle, radius r, angle at centre α r sin α : 3α from centre from centre 3 Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07 5

20 6 Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07

21 4 A Level Further Mathematics Students sitting an A Level Further Mathematics paper may also require those formulae listed for A Level Mathematics in Section. Pure Mathematics Summations n r = 6 n(n + )(n + ) r = n 3 r = 4 n (n + ) r = Matri transformations Anticlockwise rotation through θ about O: cosθ sinθ sin θ cosθ cosθ Reflection in the line y = (tanθ): sinθ sinθ cosθ Area of a sector A = r dθ (polar coordinates) Comple numbers {r(cosθ + isinθ)} n = r n (cosnθ + isinnθ) The roots of z n = are given by z = e πki n, for k = 0,,,..., n 4 Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07 7

22 Maclaurin s and Taylor s Series f() = f(0) + f (0) +! f (0) r! f (r) (0) +... r e = ep() = + +! r! r +... for all ln( + ) = r ( )r+ r sin = 3 3! + 5 5!... + ( )r r + ( r + )! +... ( < ) +... for all cos =! + 4 4!... + ( )r r ( r)! +... for all arctan = ( )r r + r ( ) Vectors Vector product: a b = ½a½½b½sinθ n = i j k a a a 3 b b b 3 = ab ab ab ab ab ab a.(b c) = a a a 3 b b b 3 c c c 3 = b.(c a) = c.(a b) If A is the point with position vector a = a i + a j + a 3 k and the direction vector b is given by b = b i + b j + b 3 k, then the straight line through A with direction vector b has cartesian equation a y a z a b b b 3 = = = 3 ( λ) The plane through A with normal vector n = n i + n j + n 3 k has cartesian equation n + n y + n 3 z + d = 0 where d = a.n The plane through non-collinear points A, B and C has vector equation r = a + λ(b a) + μ(c a) = ( λ μ)a + λb + μc The plane through the point with position vector a and parallel to b and c has equation r = a + sb + tc The perpendicular distance of (α, β, γ) from n + n y + n 3 z + d = 0 is nα+ nβ+ nγ+ d 3 n + n + n 3. 8 Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07

23 Hyperbolic functions cosh sinh = sinh = sinh cosh cosh = cosh + sinh arcosh = ln{ + } ( ) arsinh = ln{ + + } artanh = ln + (½½ < ) Conics Ellipse Parabola Hyperbola Rectangular Hyperbola Standard Form a y + = y = 4a b a y = y = c b Parametric Form (a cosθ, b sinθ) (at, at) (a secθ, b tanθ) (±a coshθ, b sinhθ) ct, c t Eccentricity e < b = a ( e ) e = e > b = a (e ) e = Foci (±ae, 0) (a, 0) (±ae, 0) (± c, ± c) Directrices = ± a e = a = ± a e + y = ± c Asymptotes none none a y =± = 0, y = 0 b 4 Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07 9

24 Differentiation f() arcsin arccos arctan sinh cosh tanh arsinh arcosh artanh f () + cosh sinh sech + 0 Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07

25 Integration (+ constant; a > 0 where relevant) f() sinh cosh tanh a a a + a a + f() d cosh sinh lncosh arcsin a a arctan arcosh arsinh a ln a a a (½½ < a) a, ln{ + a } ( > a) a, ln{ + + a } + = a artanh a (½½ < a) a a ln + a a Arc length s y = + # d d d (cartesian coordinates) y s = t + # d d t d d dt (parametric form) dr # d (polar form) 4 s = r + θ dθ Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07

26 Surface area of revolution dy # (cartesian coordinates) s = π y + d d d dy (parametric form) # s = π y + dt dt dt dr (polar form) # s = π rsin θ r + dθ dθ Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07

27 Statistics Discrete distributions For a discrete random variable X taking values i with probabilities P(X = i ) Epectation (mean): E(X ) = μ = Σ i P(X = i ) Variance: Var(X ) = σ = Σ( i μ) P(X = i ) = Σ P(X = ) μ i i For a function g(x ): E(g(X )) = Σg( i ) P(X = i ) The probability generating function of X is G X (t) = E(t X ) and E(X ) = G X () and Var(X ) = G X () + G X () [G X ()] For Z = X + Y, where X and Y are independent: G Z (t) = G X (t) G Y (t) Discrete distributions Standard discrete distributions: Distribution of X P(X = ) Mean Variance P.G.F. Binomial B(n, p) n p ( p) n np np( p) ( p + pt) n Poisson Po(λ) e λ λ! λ(t ) λ λ e Geometric Geo( p) on,,... p( p) p p p pt ( pt ) Negative binomial on r, r +,... r r pr ( p) r p r( p) p pt ( pt ) r Continuous distributions For a continuous random variable X having probability density function f Epectation (mean): E(X ) = μ = f() d 4 Variance: Var(X ) = σ = ( μ) f() d = f() d μ For a function g(x ): E(g(X )) = g() f() d 0 Cumulative distribution function: F( 0 ) = P(X 0 ) = # f(t) dt Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07 3

28 Standard continuous distribution: Distribution of X P.D.F. Mean Variance Normal N(μ, σ ) σ π e μ σ μ σ Uniform (Rectangular) on [a, b] b a (a + b) (b a) Correlation and regression For a set of n pairs of values ( i, y i ) S = Σ( i ) = Σ i S yy = Σ( y i y) = Σy i (Σ i ) n (Σ y i ) n S y = Σ( i )( y i y) = Σ i y i (Σ i)(σ yi) n The product moment correlation coefficient is (Σ i)(σ yi) S Σy Σ( )( ) i i y i yi y r = = = n S S yy { Σ( ) }{ Σ( ) } (Σ ) (Σ ) i yi y i yi Σi Σyi n n Sy Σ( i )( yi y) The regression coefficient of y on is b = = S Σ( ) Least squares regression line of y on is y = a + b where a = y b i Residual Sum of Squares (RSS) = S yy S S ( y ) = S yy ( r ) Spearman s rank correlation coefficient is r s = 6Σd nn ( ) Epectation algebra For independent random variables X and Y E(XY ) = E(X )E(Y ), Var(aX ± by ) = a Var(X ) + b Var(Y ) 4 Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07

29 Sampling distributions (i) Tests for mean when σ is known For a random sample X, X,..., X n of n independent observations from a distribution having mean μ and variance σ : X is an unbiased estimator of μ, with Var(X ) = σ n S is an unbiased estimator of σ, where S = Σ( Xi X) n For a random sample of n observations from N(μ, σ ), X σ / μ n ~ N(0, ) For a random sample of n observations from N(μ, σ ) and, independently, a random ( X Y) ( μ ) sample of n y observations from N(μ y, σ y ), μy ~ N(0, ) σ σ y + n n y (ii) Tests for variance and mean when σ is not known For a random sample of n observations from N(μ, σ ) ( n ) S σ ~ χ n X S / μ n ~ t n (also valid in matched-pairs situations) For a random sample of n observations from N(μ, σ ) and, independently, a random sample of n y observations from N(μ y, σ ) y S / σ S / σ ~ F n, n y y y If σ = σ y = σ (unknown) then 4 ( X Y) ( μ μy) ~ S p + n n y t n+ ny where S p = ( n ) S + ( n ) S n y y + n y Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07 5

30 Non-parametric tests Goodness-of-fit test and contingency tables: ( Oi Ei) E i ~ χ v Statistical tables The following statistical tables are required for A Level Further Mathematics: Binomial Cumulative Distribution Function (see page 9) Poisson Cumulative Distribution Function (see page 35) Percentage Points of the χ Distribution (see page 36) Critical Values for Correlation Coefficients: Product Moment Coefficient and Spearman s Coefficient (see page 37) Random Numbers (see page 38) Percentage Points of Student s t Distribution (see page 39) Percentage Points of the F Distribution (see page 40) 6 Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07

31 Mechanics Centres of mass For uniform bodies: Triangular lamina: 3 along median from verte Circular arc, radius r, angle at centre α : rsin α α from centre Sector of circle, radius r, angle at centre α r sin α : 3α Solid hemisphere, radius r: 3 r from centre 8 from centre Hemispherical shell, radius r: r from centre Solid cone or pyramid of height h: h above the base on the line from centre of base to verte 4 Conical shell of height h: h above the base on the line from centre of base to verte 3 Motion in a circle Transverse velocity: v = rθ. Transverse acceleration: v. = rθ.. Radial acceleration: rθ. = v r 4 Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07 7

32 8 Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07

33 5 Statistical Tables Binomial Cumulative Distribution Function The tabulated value is P(X ), where X has a binomial distribution with inde n and parameter p. p = n = 5, = n = 6, = n = 7, = n = 8, = n = 9, = n = 0, = Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07 9

34 p = n =, = n = 5, = n = 0, = Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07

35 p = n = 5, = n = 30, = Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07 3

36 p = n = 40, = Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07

37 p = n = 50, = Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07 33

38 Percentage Points of The Normal Distribution The values z in the table are those which a random variable Z N(0, ) eceeds with probability p; that is, P(Z > z) = Φ(z) = p. p z p z Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07

39 Poisson Cumulative Distribution Function The tabulated value is P(X ), where X has a Poisson distribution with parameter λ. λ = = λ = = Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07 35

40 Percentage Points of the χ Distribution The values in the table are those which a random variable with the χ distribution on v degrees of freedom eceeds with the probability shown. v Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07

41 Critical Values for Correlation Coefficients These tables concern tests of the hypothesis that a population correlation coefficient ρ is 0. The values in the tables are the minimum values which need to be reached by a sample correlation coefficient in order to be significant at the level shown, on a one tailed test. Product Moment Coefficient Spearman s Coefficient Level Sample size, n Level Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07 37

42 Random Numbers Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07

43 Percentage Points of Student s t Distribution The values in the table are those which a random variable with Student s t distribution on v degrees of freedom eceeds with the probability shown. v Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07 39

44 Percentage Points of the F Distribution The values in the table are those which a random variable with the F distribution on v and v degrees of freedom eceeds with probability 0.05 or 0.0. Probability v v If an upper percentage point of the F distribution on v and v degrees of freedom is f, then the corresponding lower percentage point of the F distribution on v and v degrees of freedom is / f. 40 Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue July 07 Pearson Education Limited 07