N38210a gce mathematical formulae statistical tables, Exams of Engineering Mathematics

Download N38210a gce mathematical formulae statistical tables and more Exams Engineering Mathematics in PDF only on Docsity! GCE Edexcel GCE in Mathematics Mathematical Formulae and Statistical Tables For use in Edexcel Advanced Subsidiary GCE and Advanced GCE examinations = Core Mathematics C1 – C4 Further Pure Mathematics FP1 – FP3 Mechanics M1 – M5 Statistics S1 – S4 For use from June 2009 This copy is the property of Edexcel. It is not to be removed from the examination room or marked in any way. = Edexcel AS/A level Mathematics Formulae List – Issue 1- September 2009 3 The formulae in this booklet have been arranged according to the unit in which they are first introduced. Thus a candidate sitting a unit may be required to use the formulae that were introduced in a preceding unit (e.g. candidates sitting C3 might be expected to use formulae first introduced in C1 or C2). It may also be the case that candidates sitting Mechanics and Statistics units need to use formulae introduced in appropriate Core Mathematics units, as outlined in the specification. 4 Edexcel AS/A level Mathematics Formulae List: Core Mathematics C1 – Issue 1 – September 2009 Core Mathematics C1 Mensuration Surface area of sphere = 4π r 2 Area of curved surface of cone = π r × slant height Arithmetic series un = a + (n – 1)d Sn = 2 1 n(a + l) = 2 1 n[2a + (n − 1)d] Edexcel AS/A level Mathematics Formulae List: Core Mathematics C2 – Issue 1 – September 2009 5 Core Mathematics C2 Candidates sitting C2 may also require those formulae listed under Core Mathematics C1. Cosine rule a2 = b2 + c2 – 2bc cos A Binomial series 2 1 )( 221 nrrnnnnn bba r n ba n ba n aba ++⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ++⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ +⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ +=+ −−− KK (n ∈ ℕ) where )!(! !C rnr n r n r n − ==⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∈<+ ××× +−−++ × −++=+ nxx r rnnnxnnnxx rn ,1( 21 )1()1( 21 )1(1)1( 2 K K K K ℝ) Logarithms and exponentials a x x b b a log log log = Geometric series un = arn − 1 Sn = r ra n − − 1 )1( S∞ = r a −1 for ⏐r⏐ < 1 Numerical integration The trapezium rule: ⎮⌡ ⌠ b a xy d ≈ 21 h{(y0 + yn) + 2(y1 + y2 + ... + yn – 1)}, where n abh −= 8 Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP1 – Issue 1 – September 2009 Further Pure Mathematics FP1 Candidates sitting FP1 may also require those formulae listed under Core Mathematics C1 and C2. Summations )12)(1(6 1 1 2 ++=∑ = nnnr n r 22 4 1 1 3 )1( +=∑ = nnr n r Numerical solution of equations The Newton-Raphson iteration for solving 0)f( =x : )(f )f( 1 n n nn x x xx ′ −=+ Conics Parabola Rectangular Hyperbola Standard Form axy 4 2 = xy = c2 Parametric Form (at 2, 2at) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ t cct, Foci )0 ,(a Not required Directrices ax −= Not required Matrix transformations Anticlockwise rotation through θ about O: ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − θθ θθ cos sin sincos Reflection in the line xy )(tanθ= : ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − θθ θθ 2cos2sin 2sin 2cos In FP1, θ will be a multiple of 45°. Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP2 – Issue 1 – September 2009 9 Further Pure Mathematics FP2 Candidates sitting FP2 may also require those formulae listed under Further Pure Mathematics FP1 and Core Mathematics C1–C4. Area of a sector A = ⎮⌡ ⌠ θd 2 1 2r (polar coordinates) Complex numbers θθθ sinicosei += )sini(cos)}sini(cos{ θθθθ nnrr nn +=+ The roots of 1=nz are given by n k z i2 e π = , for 1 , ,2 ,1 ,0 −= nk K Maclaurin’s and Taylor’s Series KK )0(f ! )0(f !2 )0(f)0f()f( )( 2 +++′′+′+= r r r xxxx KK )(f ! )( )(f !2 )()(f)()f()f( )( 2 + − ++′′ − +′−+= a r axaaxaaxax r r KK )(f ! )(f !2 )(f)f()f( )( 2 +++′′+′+=+ a r xaxaxaxa r r x r xxxx r x allfor ! !2 1)exp(e 2 KK +++++== )11( )1( 32 )1(ln 1 32 ≤<−+−+−+−=+ + x r xxxxx r r KK x r xxxxx r r allfor )!12( )1( !5!3 sin 1253 KK + + −+−+−= + x r xxxx r r allfor )!2( )1( !4!2 1cos 242 KK +−+−+−= )11( 12 )1( 53 arctan 1253 ≤≤−+ + −+−+−= + x r xxxxx r r KK 10 Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP3 – Issue 1 – September 2009 Further Pure Mathematics FP3 Candidates sitting FP3 may also require those formulae listed under Further Pure Mathematics FP1, and Core Mathematics C1–C4. Vectors The resolved part of a in the direction of b is b a.b The point dividing AB in the ratio μλ : is μλ λμ + + ba Vector product: ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − − − ===× 1221 3113 2332 321 321ˆ sin baba baba baba bbb aaa kji nbaba θ )()()( 321 321 321 bac.acb.cba. ×=×==× ccc bbb aaa If A is the point with position vector kjia 321 aaa ++= and the direction vector b is given by kjib 321 bbb ++= , then the straight line through A with direction vector b has cartesian equation )( 3 3 2 2 1 1 λ= − = − = − b az b ay b ax The plane through A with normal vector kjin 321 nnn ++= has cartesian equation a.n−==+++ ddznynxn where0321 The plane through non-collinear points A, B and C has vector equation cbaacabar μλμλμλ ++−−=−+−+= )1()()( The plane through the point with position vector a and parallel to b and c has equation cbar ts ++= The perpendicular distance of ) , ,( γβα from 0321 =+++ dznynxn is 2 3 2 2 2 1 321 nnn dnnn ++ +++ γβα . Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP3 – Issue 1–September 2009 13 Arc length x x ys d d d1 2 ⎮ ⌡ ⌠ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛+= (cartesian coordinates) t t y t xs d d d d d 22 ⎮ ⌡ ⌠ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛+⎟ ⎠ ⎞ ⎜ ⎝ ⎛= (parametric form) Surface area of revolution Sx = ⎮⌡ ⌠ sy d2π = ⎮ ⎮ ⌡ ⌠ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛+ x x yy d d d12 2 π = ⎮ ⌡ ⌠ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛+⎟ ⎠ ⎞ ⎜ ⎝ ⎛ t t y t xy d d d d d2 22 π 14 Edexcel AS/A level Mathematics Formulae List: Mechanics M1 – M3 – Issue 1 – September 2009 Mechanics M1 There are no formulae given for M1 in addition to those candidates are expected to know. Candidates sitting M1 may also require those formulae listed under Core Mathematics C1. Mechanics M2 Candidates sitting M2 may also require those formulae listed under Core Mathematics C1, C2 and C3. Centres of mass For uniform bodies: Triangular lamina: 32 along median from vertex Circular arc, radius r, angle at centre 2α : α αsinr from centre Sector of circle, radius r, angle at centre 2α : α α 3 sin2r from centre Mechanics M3 Candidates sitting M3 may also require those formulae listed under Mechanics M2, and also those formulae listed under Core Mathematics C1–C4. Motion in a circle Transverse velocity: θ&rv = Transverse acceleration: θ&&& rv = Radial acceleration: r vr 2 2 −=− θ& Centres of mass For uniform bodies: Solid hemisphere, radius r: r8 3 from centre Hemispherical shell, radius r: r21 from centre Solid cone or pyramid of height h: h41 above the base on the line from centre of base to vertex Conical shell of height h: h31 above the base on the line from centre of base to vertex Universal law of gravitation 2 21Force d mGm = Edexcel AS/A level Mathematics Formulae List: Mechanics M4–M5 – Issue 1 – September 2009 15 Mechanics M4 There are no formulae given for M4 in addition to those candidates are expected to know. Candidates sitting M4 may also require those formulae listed under Mechanics M2 and M3, and also those formulae listed under Core Mathematics C1–C4 and Further Pure Mathematics FP3. Mechanics M5 Candidates sitting M5 may also require those formulae listed under Mechanics M2 and M3, and also those formulae listed under Core Mathematics C1–C4 and Further Pure Mathematics FP3. Moments of inertia For uniform bodies of mass m: Thin rod, length 2l, about perpendicular axis through centre: 23 1 ml Rectangular lamina about axis in plane bisecting edges of length 2l: 23 1 ml Thin rod, length 2l, about perpendicular axis through end: 23 4 ml Rectangular lamina about edge perpendicular to edges of length 2l: 23 4 ml Rectangular lamina, sides 2a and 2b, about perpendicular axis through centre: )( 223 1 bam + Hoop or cylindrical shell of radius r about axis through centre: 2mr Hoop of radius r about a diameter: 221 mr Disc or solid cylinder of radius r about axis through centre: 22 1 mr Disc of radius r about a diameter: 241 mr Solid sphere, radius r, about diameter: 25 2 mr Spherical shell of radius r about a diameter: 232 mr Parallel axes theorem: 2)(AGmII GA += Perpendicular axes theorem: yxz III += (for a lamina in the x-y plane) Moments as vectors The moment about O of F acting at r is Fr × 18 Edexcel AS/A level Mathematics Formulae List: Statistics S1 – Issue 1 – September 2009 THE NORMAL DISTRIBUTION FUNCTION The function tabulated below is Φ(z), defined as Φ(z) = te z t d 2 1 2 2 1 ⎮⌡ ⌠ ∞− − π . z Φ(z) z Φ(z) z Φ(z) z Φ(z) z Φ(z) 0.00 0.5000 0.50 0.6915 1.00 0.8413 1.50 0.9332 2.00 0.9772 0.01 0.5040 0.51 0.6950 1.01 0.8438 1.51 0.9345 2.02 0.9783 0.02 0.5080 0.52 0.6985 1.02 0.8461 1.52 0.9357 2.04 0.9793 0.03 0.5120 0.53 0.7019 1.03 0.8485 1.53 0.9370 2.06 0.9803 0.04 0.5160 0.54 0.7054 1.04 0.8508 1.54 0.9382 2.08 0.9812 0.05 0.5199 0.55 0.7088 1.05 0.8531 1.55 0.9394 2.10 0.9821 0.06 0.5239 0.56 0.7123 1.06 0.8554 1.56 0.9406 2.12 0.9830 0.07 0.5279 0.57 0.7157 1.07 0.8577 1.57 0.9418 2.14 0.9838 0.08 0.5319 0.58 0.7190 1.08 0.8599 1.58 0.9429 2.16 0.9846 0.09 0.5359 0.59 0.7224 1.09 0.8621 1.59 0.9441 2.18 0.9854 0.10 0.5398 0.60 0.7257 1.10 0.8643 1.60 0.9452 2.20 0.9861 0.11 0.5438 0.61 0.7291 1.11 0.8665 1.61 0.9463 2.22 0.9868 0.12 0.5478 0.62 0.7324 1.12 0.8686 1.62 0.9474 2.24 0.9875 0.13 0.5517 0.63 0.7357 1.13 0.8708 1.63 0.9484 2.26 0.9881 0.14 0.5557 0.64 0.7389 1.14 0.8729 1.64 0.9495 2.28 0.9887 0.15 0.5596 0.65 0.7422 1.15 0.8749 1.65 0.9505 2.30 0.9893 0.16 0.5636 0.66 0.7454 1.16 0.8770 1.66 0.9515 2.32 0.9898 0.17 0.5675 0.67 0.7486 1.17 0.8790 1.67 0.9525 2.34 0.9904 0.18 0.5714 0.68 0.7517 1.18 0.8810 1.68 0.9535 2.36 0.9909 0.19 0.5753 0.69 0.7549 1.19 0.8830 1.69 0.9545 2.38 0.9913 0.20 0.5793 0.70 0.7580 1.20 0.8849 1.70 0.9554 2.40 0.9918 0.21 0.5832 0.71 0.7611 1.21 0.8869 1.71 0.9564 2.42 0.9922 0.22 0.5871 0.72 0.7642 1.22 0.8888 1.72 0.9573 2.44 0.9927 0.23 0.5910 0.73 0.7673 1.23 0.8907 1.73 0.9582 2.46 0.9931 0.24 0.5948 0.74 0.7704 1.24 0.8925 1.74 0.9591 2.48 0.9934 0.25 0.5987 0.75 0.7734 1.25 0.8944 1.75 0.9599 2.50 0.9938 0.26 0.6026 0.76 0.7764 1.26 0.8962 1.76 0.9608 2.55 0.9946 0.27 0.6064 0.77 0.7794 1.27 0.8980 1.77 0.9616 2.60 0.9953 0.28 0.6103 0.78 0.7823 1.28 0.8997 1.78 0.9625 2.65 0.9960 0.29 0.6141 0.79 0.7852 1.29 0.9015 1.79 0.9633 2.70 0.9965 0.30 0.6179 0.80 0.7881 1.30 0.9032 1.80 0.9641 2.75 0.9970 0.31 0.6217 0.81 0.7910 1.31 0.9049 1.81 0.9649 2.80 0.9974 0.32 0.6255 0.82 0.7939 1.32 0.9066 1.82 0.9656 2.85 0.9978 0.33 0.6293 0.83 0.7967 1.33 0.9082 1.83 0.9664 2.90 0.9981 0.34 0.6331 0.84 0.7995 1.34 0.9099 1.84 0.9671 2.95 0.9984 0.35 0.6368 0.85 0.8023 1.35 0.9115 1.85 0.9678 3.00 0.9987 0.36 0.6406 0.86 0.8051 1.36 0.9131 1.86 0.9686 3.05 0.9989 0.37 0.6443 0.87 0.8078 1.37 0.9147 1.87 0.9693 3.10 0.9990 0.38 0.6480 0.88 0.8106 1.38 0.9162 1.88 0.9699 3.15 0.9992 0.39 0.6517 0.89 0.8133 1.39 0.9177 1.89 0.9706 3.20 0.9993 0.40 0.6554 0.90 0.8159 1.40 0.9192 1.90 0.9713 3.25 0.9994 0.41 0.6591 0.91 0.8186 1.41 0.9207 1.91 0.9719 3.30 0.9995 0.42 0.6628 0.92 0.8212 1.42 0.9222 1.92 0.9726 3.35 0.9996 0.43 0.6664 0.93 0.8238 1.43 0.9236 1.93 0.9732 3.40 0.9997 0.44 0.6700 0.94 0.8264 1.44 0.9251 1.94 0.9738 3.50 0.9998 0.45 0.6736 0.95 0.8289 1.45 0.9265 1.95 0.9744 3.60 0.9998 0.46 0.6772 0.96 0.8315 1.46 0.9279 1.96 0.9750 3.70 0.9999 0.47 0.6808 0.97 0.8340 1.47 0.9292 1.97 0.9756 3.80 0.9999 0.48 0.6844 0.98 0.8365 1.48 0.9306 1.98 0.9761 3.90 1.0000 0.49 0.6879 0.99 0.8389 1.49 0.9319 1.99 0.9767 4.00 1.0000 0.50 0.6915 1.00 0.8413 1.50 0.9332 2.00 0.9772 Edexcel AS/A level Mathematics Formulae List: Statistics S1 – Issue 1 – September 2009 19 PERCENTAGE POINTS OF THE NORMAL DISTRIBUTION The values z in the table are those which a random variable Z ∼ N(0, 1) exceeds with probability p; that is, P(Z > z) = 1 − Φ(z) = p. p z p z 0.5000 0.0000 0.0500 1.6449 0.4000 0.2533 0.0250 1.9600 0.3000 0.5244 0.0100 2.3263 0.2000 0.8416 0.0050 2.5758 0.1500 1.0364 0.0010 3.0902 0.1000 1.2816 0.0005 3.2905 20 Edexcel AS/A level Mathematics Formulae List: Statistics S2 – Issue 1 – September 2009 Statistics S2 Candidates sitting S2 may also require those formulae listed under Statistics S1, and also those listed under Core Mathematics C1 and C2. Discrete distributions Standard discrete distributions: Distribution of X )P( xX = Mean Variance Binomial ),B( pn xnx pp x n −−⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ )1( np )1( pnp − Poisson )Po(λ ! e x xλλ− λ λ Continuous distributions For a continuous random variable X having probability density function f Expectation (mean): ∫== xxxX d)f()E( μ Variance: ∫ ∫ −=−== 2222 d)f(d)f()()Var( μμσ xxxxxxX For a function )g(X : ∫= xxxX d)f()g())E(g( Cumulative distribution function: ⎮⌡ ⌠=≤= ∞− 0 00 d)(f)P()F( x ttxXx Standard continuous distribution: Distribution of X P.D.F. Mean Variance Uniform (Rectangular) on [a, b] ab − 1 )(2 1 ba + 212 1 )( ab − Edexcel AS/A level Mathematics Formulae List: Statistics S2 – Issue 1 – September 2009 23 p = 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 n = 25, x = 0 0.2774 0.0718 0.0172 0.0038 0.0008 0.0001 0.0000 0.0000 0.0000 0.0000 1 0.6424 0.2712 0.0931 0.0274 0.0070 0.0016 0.0003 0.0001 0.0000 0.0000 2 0.8729 0.5371 0.2537 0.0982 0.0321 0.0090 0.0021 0.0004 0.0001 0.0000 3 0.9659 0.7636 0.4711 0.2340 0.0962 0.0332 0.0097 0.0024 0.0005 0.0001 4 0.9928 0.9020 0.6821 0.4207 0.2137 0.0905 0.0320 0.0095 0.0023 0.0005 5 0.9988 0.9666 0.8385 0.6167 0.3783 0.1935 0.0826 0.0294 0.0086 0.0020 6 0.9998 0.9905 0.9305 0.7800 0.5611 0.3407 0.1734 0.0736 0.0258 0.0073 7 1.0000 0.9977 0.9745 0.8909 0.7265 0.5118 0.3061 0.1536 0.0639 0.0216 8 1.0000 0.9995 0.9920 0.9532 0.8506 0.6769 0.4668 0.2735 0.1340 0.0539 9 1.0000 0.9999 0.9979 0.9827 0.9287 0.8106 0.6303 0.4246 0.2424 0.1148 10 1.0000 1.0000 0.9995 0.9944 0.9703 0.9022 0.7712 0.5858 0.3843 0.2122 11 1.0000 1.0000 0.9999 0.9985 0.9893 0.9558 0.8746 0.7323 0.5426 0.3450 12 1.0000 1.0000 1.0000 0.9996 0.9966 0.9825 0.9396 0.8462 0.6937 0.5000 13 1.0000 1.0000 1.0000 0.9999 0.9991 0.9940 0.9745 0.9222 0.8173 0.6550 14 1.0000 1.0000 1.0000 1.0000 0.9998 0.9982 0.9907 0.9656 0.9040 0.7878 15 1.0000 1.0000 1.0000 1.0000 1.0000 0.9995 0.9971 0.9868 0.9560 0.8852 16 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9992 0.9957 0.9826 0.9461 17 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9988 0.9942 0.9784 18 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9984 0.9927 19 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9996 0.9980 20 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9995 21 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 22 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 n = 30, x = 0 0.2146 0.0424 0.0076 0.0012 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 1 0.5535 0.1837 0.0480 0.0105 0.0020 0.0003 0.0000 0.0000 0.0000 0.0000 2 0.8122 0.4114 0.1514 0.0442 0.0106 0.0021 0.0003 0.0000 0.0000 0.0000 3 0.9392 0.6474 0.3217 0.1227 0.0374 0.0093 0.0019 0.0003 0.0000 0.0000 4 0.9844 0.8245 0.5245 0.2552 0.0979 0.0302 0.0075 0.0015 0.0002 0.0000 5 0.9967 0.9268 0.7106 0.4275 0.2026 0.0766 0.0233 0.0057 0.0011 0.0002 6 0.9994 0.9742 0.8474 0.6070 0.3481 0.1595 0.0586 0.0172 0.0040 0.0007 7 0.9999 0.9922 0.9302 0.7608 0.5143 0.2814 0.1238 0.0435 0.0121 0.0026 8 1.0000 0.9980 0.9722 0.8713 0.6736 0.4315 0.2247 0.0940 0.0312 0.0081 9 1.0000 0.9995 0.9903 0.9389 0.8034 0.5888 0.3575 0.1763 0.0694 0.0214 10 1.0000 0.9999 0.9971 0.9744 0.8943 0.7304 0.5078 0.2915 0.1350 0.0494 11 1.0000 1.0000 0.9992 0.9905 0.9493 0.8407 0.6548 0.4311 0.2327 0.1002 12 1.0000 1.0000 0.9998 0.9969 0.9784 0.9155 0.7802 0.5785 0.3592 0.1808 13 1.0000 1.0000 1.0000 0.9991 0.9918 0.9599 0.8737 0.7145 0.5025 0.2923 14 1.0000 1.0000 1.0000 0.9998 0.9973 0.9831 0.9348 0.8246 0.6448 0.4278 15 1.0000 1.0000 1.0000 0.9999 0.9992 0.9936 0.9699 0.9029 0.7691 0.5722 16 1.0000 1.0000 1.0000 1.0000 0.9998 0.9979 0.9876 0.9519 0.8644 0.7077 17 1.0000 1.0000 1.0000 1.0000 0.9999 0.9994 0.9955 0.9788 0.9286 0.8192 18 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9986 0.9917 0.9666 0.8998 19 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9996 0.9971 0.9862 0.9506 20 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9991 0.9950 0.9786 21 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9984 0.9919 22 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9996 0.9974 23 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9993 24 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 25 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 24 Edexcel AS/A level Mathematics Formulae List: Statistics S2 – Issue 1 – September 2009 p = 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 n = 40, x = 0 0.1285 0.0148 0.0015 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1 0.3991 0.0805 0.0121 0.0015 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 2 0.6767 0.2228 0.0486 0.0079 0.0010 0.0001 0.0000 0.0000 0.0000 0.0000 3 0.8619 0.4231 0.1302 0.0285 0.0047 0.0006 0.0001 0.0000 0.0000 0.0000 4 0.9520 0.6290 0.2633 0.0759 0.0160 0.0026 0.0003 0.0000 0.0000 0.0000 5 0.9861 0.7937 0.4325 0.1613 0.0433 0.0086 0.0013 0.0001 0.0000 0.0000 6 0.9966 0.9005 0.6067 0.2859 0.0962 0.0238 0.0044 0.0006 0.0001 0.0000 7 0.9993 0.9581 0.7559 0.4371 0.1820 0.0553 0.0124 0.0021 0.0002 0.0000 8 0.9999 0.9845 0.8646 0.5931 0.2998 0.1110 0.0303 0.0061 0.0009 0.0001 9 1.0000 0.9949 0.9328 0.7318 0.4395 0.1959 0.0644 0.0156 0.0027 0.0003 10 1.0000 0.9985 0.9701 0.8392 0.5839 0.3087 0.1215 0.0352 0.0074 0.0011 11 1.0000 0.9996 0.9880 0.9125 0.7151 0.4406 0.2053 0.0709 0.0179 0.0032 12 1.0000 0.9999 0.9957 0.9568 0.8209 0.5772 0.3143 0.1285 0.0386 0.0083 13 1.0000 1.0000 0.9986 0.9806 0.8968 0.7032 0.4408 0.2112 0.0751 0.0192 14 1.0000 1.0000 0.9996 0.9921 0.9456 0.8074 0.5721 0.3174 0.1326 0.0403 15 1.0000 1.0000 0.9999 0.9971 0.9738 0.8849 0.6946 0.4402 0.2142 0.0769 16 1.0000 1.0000 1.0000 0.9990 0.9884 0.9367 0.7978 0.5681 0.3185 0.1341 17 1.0000 1.0000 1.0000 0.9997 0.9953 0.9680 0.8761 0.6885 0.4391 0.2148 18 1.0000 1.0000 1.0000 0.9999 0.9983 0.9852 0.9301 0.7911 0.5651 0.3179 19 1.0000 1.0000 1.0000 1.0000 0.9994 0.9937 0.9637 0.8702 0.6844 0.4373 20 1.0000 1.0000 1.0000 1.0000 0.9998 0.9976 0.9827 0.9256 0.7870 0.5627 21 1.0000 1.0000 1.0000 1.0000 1.0000 0.9991 0.9925 0.9608 0.8669 0.6821 22 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9970 0.9811 0.9233 0.7852 23 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9989 0.9917 0.9595 0.8659 24 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9996 0.9966 0.9804 0.9231 25 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9988 0.9914 0.9597 26 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9996 0.9966 0.9808 27 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9988 0.9917 28 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9996 0.9968 29 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9989 30 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 31 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 32 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 Edexcel AS/A level Mathematics Formulae List: Statistics S2 – Issue 1 – September 2009 25 p = 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 n = 50, x = 0 0.0769 0.0052 0.0003 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1 0.2794 0.0338 0.0029 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 2 0.5405 0.1117 0.0142 0.0013 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 3 0.7604 0.2503 0.0460 0.0057 0.0005 0.0000 0.0000 0.0000 0.0000 0.0000 4 0.8964 0.4312 0.1121 0.0185 0.0021 0.0002 0.0000 0.0000 0.0000 0.0000 5 0.9622 0.6161 0.2194 0.0480 0.0070 0.0007 0.0001 0.0000 0.0000 0.0000 6 0.9882 0.7702 0.3613 0.1034 0.0194 0.0025 0.0002 0.0000 0.0000 0.0000 7 0.9968 0.8779 0.5188 0.1904 0.0453 0.0073 0.0008 0.0001 0.0000 0.0000 8 0.9992 0.9421 0.6681 0.3073 0.0916 0.0183 0.0025 0.0002 0.0000 0.0000 9 0.9998 0.9755 0.7911 0.4437 0.1637 0.0402 0.0067 0.0008 0.0001 0.0000 10 1.0000 0.9906 0.8801 0.5836 0.2622 0.0789 0.0160 0.0022 0.0002 0.0000 11 1.0000 0.9968 0.9372 0.7107 0.3816 0.1390 0.0342 0.0057 0.0006 0.0000 12 1.0000 0.9990 0.9699 0.8139 0.5110 0.2229 0.0661 0.0133 0.0018 0.0002 13 1.0000 0.9997 0.9868 0.8894 0.6370 0.3279 0.1163 0.0280 0.0045 0.0005 14 1.0000 0.9999 0.9947 0.9393 0.7481 0.4468 0.1878 0.0540 0.0104 0.0013 15 1.0000 1.0000 0.9981 0.9692 0.8369 0.5692 0.2801 0.0955 0.0220 0.0033 16 1.0000 1.0000 0.9993 0.9856 0.9017 0.6839 0.3889 0.1561 0.0427 0.0077 17 1.0000 1.0000 0.9998 0.9937 0.9449 0.7822 0.5060 0.2369 0.0765 0.0164 18 1.0000 1.0000 0.9999 0.9975 0.9713 0.8594 0.6216 0.3356 0.1273 0.0325 19 1.0000 1.0000 1.0000 0.9991 0.9861 0.9152 0.7264 0.4465 0.1974 0.0595 20 1.0000 1.0000 1.0000 0.9997 0.9937 0.9522 0.8139 0.5610 0.2862 0.1013 21 1.0000 1.0000 1.0000 0.9999 0.9974 0.9749 0.8813 0.6701 0.3900 0.1611 22 1.0000 1.0000 1.0000 1.0000 0.9990 0.9877 0.9290 0.7660 0.5019 0.2399 23 1.0000 1.0000 1.0000 1.0000 0.9996 0.9944 0.9604 0.8438 0.6134 0.3359 24 1.0000 1.0000 1.0000 1.0000 0.9999 0.9976 0.9793 0.9022 0.7160 0.4439 25 1.0000 1.0000 1.0000 1.0000 1.0000 0.9991 0.9900 0.9427 0.8034 0.5561 26 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9955 0.9686 0.8721 0.6641 27 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9981 0.9840 0.9220 0.7601 28 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9993 0.9924 0.9556 0.8389 29 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9966 0.9765 0.8987 30 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9986 0.9884 0.9405 31 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9995 0.9947 0.9675 32 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9978 0.9836 33 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9991 0.9923 34 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9967 35 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9987 36 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9995 37 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 38 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 28 Edexcel AS/A level Mathematics Formulae List: Statistics S3 – Issue 1 – September 2009 PERCENTAGE POINTS OF THE χ 2 DISTRIBUTION The values in the table are those which a random variable with the χ 2 distribution on ν degrees of freedom exceeds with the probability shown. ν 0.995 0.990 0.975 0.950 0.900 0.100 0.050 0.025 0.010 0.005 1 0.000 0.000 0.001 0.004 0.016 2.705 3.841 5.024 6.635 7.879 2 0.010 0.020 0.051 0.103 0.211 4.605 5.991 7.378 9.210 10.597 3 0.072 0.115 0.216 0.352 0.584 6.251 7.815 9.348 11.345 12.838 4 0.207 0.297 0.484 0.711 1.064 7.779 9.488 11.143 13.277 14.860 5 0.412 0.554 0.831 1.145 1.610 9.236 11.070 12.832 15.086 16.750 6 0.676 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.812 18.548 7 0.989 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.475 20.278 8 1.344 1.646 2.180 2.733 3.490 13.362 15.507 17.535 20.090 21.955 9 1.735 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666 23.589 10 2.156 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 25.188 11 2.603 3.053 3.816 4.575 5.580 17.275 19.675 21.920 24.725 26.757 12 3.074 3.571 4.404 5.226 6.304 18.549 21.026 23.337 26.217 28.300 13 3.565 4.107 5.009 5.892 7.042 19.812 22.362 24.736 27.688 29.819 14 4.075 4.660 5.629 6.571 7.790 21.064 23.685 26.119 29.141 31.319 15 4.601 5.229 6.262 7.261 8.547 22.307 24.996 27.488 30.578 32.801 16 5.142 5.812 6.908 7.962 9.312 23.542 26.296 28.845 32.000 34.267 17 5.697 6.408 7.564 8.672 10.085 24.769 27.587 30.191 33.409 35.718 18 6.265 7.015 8.231 9.390 10.865 25.989 28.869 31.526 34.805 37.156 19 6.844 7.633 8.907 10.117 11.651 27.204 30.144 32.852 36.191 38.582 20 7.434 8.260 9.591 10.851 12.443 28.412 31.410 34.170 37.566 39.997 21 8.034 8.897 10.283 11.591 13.240 29.615 32.671 35.479 38.932 41.401 22 8.643 9.542 10.982 12.338 14.042 30.813 33.924 36.781 40.289 42.796 23 9.260 10.196 11.689 13.091 14.848 32.007 35.172 38.076 41.638 44.181 24 9.886 10.856 12.401 13.848 15.659 33.196 36.415 39.364 42.980 45.558 25 10.520 11.524 13.120 14.611 16.473 34.382 37.652 40.646 44.314 46.928 26 11.160 12.198 13.844 15.379 17.292 35.563 38.885 41.923 45.642 48.290 27 11.808 12.879 14.573 16.151 18.114 36.741 40.113 43.194 46.963 49.645 28 12.461 13.565 15.308 16.928 18.939 37.916 41.337 44.461 48.278 50.993 29 13.121 14.256 16.047 17.708 19.768 39.088 42.557 45.722 49.588 52.336 30 13.787 14.953 16.791 18.493 20.599 40.256 43.773 46.979 50.892 53.672 Edexcel AS/A level Mathematics Formulae List: Statistics S3 – Issue 1 – September 2009 29 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 0.10 0.05 Level 0.025 0.01 0.005 Sample Level 0.05 Level 0.025 0.01 0.8000 0.9000 0.9500 0.9800 0.9900 4 1.0000 - - 0.6870 0.8054 0.8783 0.9343 0.9587 5 0.9000 1.0000 1.0000 0.6084 0.7293 0.8114 0.8822 0.9172 6 0.8286 0.8857 0.9429 0.5509 0.6694 0.7545 0.8329 0.8745 7 0.7143 0.7857 0.8929 0.5067 0.6215 0.7067 0.7887 0.8343 8 0.6429 0.7381 0.8333 0.4716 0.5822 0.6664 0.7498 0.7977 9 0.6000 0.7000 0.7833 0.4428 0.5494 0.6319 0.7155 0.7646 10 0.5636 0.6485 0.7455 0.4187 0.5214 0.6021 0.6851 0.7348 11 0.5364 0.6182 0.7091 0.3981 0.4973 0.5760 0.6581 0.7079 12 0.5035 0.5874 0.6783 0.3802 0.4762 0.5529 0.6339 0.6835 13 0.4835 0.5604 0.6484 0.3646 0.4575 0.5324 0.6120 0.6614 14 0.4637 0.5385 0.6264 0.3507 0.4409 0.5140 0.5923 0.6411 15 0.4464 0.5214 0.6036 0.3383 0.4259 0.4973 0.5742 0.6226 16 0.4294 0.5029 0.5824 0.3271 0.4124 0.4821 0.5577 0.6055 17 0.4142 0.4877 0.5662 0.3170 0.4000 0.4683 0.5425 0.5897 18 0.4014 0.4716 0.5501 0.3077 0.3887 0.4555 0.5285 0.5751 19 0.3912 0.4596 0.5351 0.2992 0.3783 0.4438 0.5155 0.5614 20 0.3805 0.4466 0.5218 0.2914 0.3687 0.4329 0.5034 0.5487 21 0.3701 0.4364 0.5091 0.2841 0.3598 0.4227 0.4921 0.5368 22 0.3608 0.4252 0.4975 0.2774 0.3515 0.4133 0.4815 0.5256 23 0.3528 0.4160 0.4862 0.2711 0.3438 0.4044 0.4716 0.5151 24 0.3443 0.4070 0.4757 0.2653 0.3365 0.3961 0.4622 0.5052 25 0.3369 0.3977 0.4662 0.2598 0.3297 0.3882 0.4534 0.4958 26 0.3306 0.3901 0.4571 0.2546 0.3233 0.3809 0.4451 0.4869 27 0.3242 0.3828 0.4487 0.2497 0.3172 0.3739 0.4372 0.4785 28 0.3180 0.3755 0.4401 0.2451 0.3115 0.3673 0.4297 0.4705 29 0.3118 0.3685 0.4325 0.2407 0.3061 0.3610 0.4226 0.4629 30 0.3063 0.3624 0.4251 0.2070 0.2638 0.3120 0.3665 0.4026 40 0.2640 0.3128 0.3681 0.1843 0.2353 0.2787 0.3281 0.3610 50 0.2353 0.2791 0.3293 0.1678 0.2144 0.2542 0.2997 0.3301 60 0.2144 0.2545 0.3005 0.1550 0.1982 0.2352 0.2776 0.3060 70 0.1982 0.2354 0.2782 0.1448 0.1852 0.2199 0.2597 0.2864 80 0.1852 0.2201 0.2602 0.1364 0.1745 0.2072 0.2449 0.2702 90 0.1745 0.2074 0.2453 0.1292 0.1654 0.1966 0.2324 0.2565 100 0.1654 0.1967 0.2327 30 Edexcel AS/A level Mathematics Formulae List: Statistics S3 – Issue 1 – September 2009 RANDOM NUMBERS 86 13 84 10 07 30 39 05 97 96 88 07 37 26 04 89 13 48 19 20 60 78 48 12 99 47 09 46 91 33 17 21 03 94 79 00 08 50 40 16 78 48 06 37 82 26 01 06 64 65 94 41 17 26 74 66 61 93 24 97 80 56 90 79 66 94 18 40 97 79 93 20 41 51 25 04 20 71 76 04 99 09 39 25 66 31 70 56 30 15 52 17 87 55 31 11 10 68 98 23 56 32 32 72 91 65 97 36 56 61 12 79 95 17 57 16 53 58 96 36 66 02 49 93 97 44 99 15 56 86 80 57 11 78 40 23 58 40 86 14 31 77 53 94 05 93 56 14 71 23 60 46 05 33 23 72 93 10 81 23 98 79 72 43 14 76 54 77 66 29 84 09 88 56 75 86 41 67 04 42 50 97 92 15 10 01 57 01 87 33 73 17 70 18 40 21 24 20 66 62 90 51 94 50 12 48 88 95 09 34 09 30 22 27 25 56 40 76 01 59 31 99 52 24 13 43 27 88 11 39 41 65 00 84 13 06 31 79 74 97 22 96 23 34 46 12 67 11 48 06 99 24 14 83 78 37 65 73 39 47 06 84 55 41 27 06 74 59 14 29 20 14 45 75 31 16 05 41 22 96 08 64 89 30 25 25 71 35 33 31 04 56 12 67 03 74 07 16 49 32 86 87 62 43 15 11 76 49 79 13 78 80 93 89 09 57 07 14 40 74 94 44 97 13 77 04 35 02 12 76 60 91 93 40 81 06 85 85 72 84 63 25 55 14 66 47 99 90 02 90 83 43 16 01 19 69 11 78 87 16 11 22 83 98 15 21 18 57 53 42 91 91 26 52 89 13 86 00 47 61 01 70 10 83 94 71 13 67 11 12 36 54 53 32 90 43 79 01 95 15 Edexcel AS/A level Mathematics Formulae List: Statistics S4 – Issue 1 – September 2009 33 PERCENTAGE POINTS OF THE F DISTRIBUTION The values in the table are those which a random variable with the F distribution on ν1 and ν2 degrees of freedom exceeds with probability 0.05 or 0.01. Probability ν2/ν1 1 2 3 4 5 6 8 10 12 24 ∞ 1 161.4 199.5 215.7 224.6 230.2 234.0 238.9 241.9 243.9 249.1 254.3 2 18.51 19.00 19.16 19.25 19.30 19.33 19.37 19.40 19.41 19.46 19.50 3 10.13 9.55 9.28 9.12 9.01 8.94 8.85 8.79 8.74 8.64 8.53 4 7.71 6.94 6.59 6.39 6.26 6.16 6.04 5.96 5.91 5.77 5.63 5 6.61 5.79 5.41 5.19 5.05 4.95 4.82 4.74 4.68 4.53 4.37 6 5.99 5.14 4.76 4.53 4.39 4.28 4.15 4.06 4.00 3.84 3.67 7 5.59 4.74 4.35 4.12 3.97 3.87 3.73 3.64 3.57 3.41 3.23 8 5.32 4.46 4.07 3.84 3.69 3.58 3.44 3.35 3.28 3.12 2.93 9 5.12 4.26 3.86 3.63 3.48 3.37 3.23 3.14 3.07 2.90 2.71 10 4.96 4.10 3.71 3.48 3.33 3.22 3.07 2.98 2.91 2.74 2.54 11 4.84 3.98 3.59 3.36 3.20 3.09 2.95 2.85 2.79 2.61 2.40 12 4.75 3.89 3.49 3.26 3.11 3.00 2.85 2.75 2.69 2.51 2.30 14 4.60 3.74 3.34 3.11 2.96 2.85 2.70 2.60 2.53 2.35 2.13 16 4.49 3.63 3.24 3.01 2.85 2.74 2.59 2.49 2.42 2.24 2.01 18 4.41 3.55 3.16 2.93 2.77 2.66 2.51 2.41 2.34 2.15 1.92 20 4.35 3.49 3.10 2.87 2.71 2.60 2.45 2.35 2.28 2.08 1.84 25 4.24 3.39 2.99 2.76 2.60 2.49 2.34 2.24 2.16 1.96 1.71 30 4.17 3.32 2.92 2.69 2.53 2.42 2.27 2.16 2.09 1.89 1.62 40 4.08 3.23 2.84 2.61 2.45 2.34 2.18 2.08 2.00 1.79 1.51 60 4.00 3.15 2.76 2.53 2.37 2.25 2.10 1.99 1.92 1.70 1.39 120 3.92 3.07 2.68 2.45 2.29 2.18 2.02 1.91 1.83 1.61 1.25 0.05 ∞ 3.84 3.00 2.60 2.37 2.21 2.10 1.94 1.83 1.75 1.52 1.00 1 4052. 5000. 5403. 5625. 5764. 5859. 5982. 6056. 6106. 6235. 6366. 2 98.50 99.00 99.17 99.25 99.30 99.33 99.37 99.40 99.42 99.46 99.50 3 34.12 30.82 29.46 28.71 28.24 27.91 27.49 27.23 27.05 26.60 26.13 4 21.20 18.00 16.69 15.98 15.52 15.21 14.80 14.55 14.37 13.93 13.45 5 16.26 13.27 12.06 11.39 10.97 10.67 10.29 10.05 9.89 9.47 9.02 6 13.70 10.90 9.78 9.15 8.75 8.47 8.10 7.87 7.72 7.31 6.88 7 12.20 9.55 8.45 7.85 7.46 7.19 6.84 6.62 6.47 6.07 5.65 8 11.30 8.65 7.59 7.01 6.63 6.37 6.03 5.81 5.67 5.28 4.86 9 10.60 8.02 6.99 6.42 6.06 5.80 5.47 5.26 5.11 4.73 4.31 10 10.00 7.56 6.55 5.99 5.64 5.39 5.06 4.85 4.17 4.33 3.91 11 9.65 7.21 6.22 5.67 5.32 5.07 4.74 4.54 4.40 4.02 3.60 12 9.33 6.93 5.95 5.41 5.06 4.82 4.50 4.30 4.16 3.78 3.36 14 8.86 6.51 5.56 5.04 4.70 4.46 4.14 3.94 3.80 3.43 3.00 16 8.53 6.23 5.29 4.77 4.44 4.20 3.89 3.69 3.55 3.18 2.75 18 8.29 6.01 5.09 4.58 4.25 4.01 3.71 3.51 3.37 3.00 2.57 20 8.10 5.85 4.94 4.43 4.10 3.87 3.56 3.37 3.23 2.86 2.42 25 7.77 5.57 4.68 4.18 3.86 3.63 3.32 3.13 2.99 2.62 2.17 30 7.56 5.39 4.51 4.02 3.70 3.47 3.17 2.98 2.84 2.47 2.01 40 7.31 5.18 4.31 3.83 3.51 3.29 2.99 2.80 2.66 2.29 1.80 60 7.08 4.98 4.13 3.65 3.34 3.12 2.82 2.63 2.50 2.12 1.60 120 6.85 4.79 3.95 3.48 3.17 2.96 2.66 2.47 2.34 1.95 1.38 0.01 ∞ 6.63 4.61 3.78 3.32 3.02 2.80 2.51 2.32 2.18 1.79 1.00 If an upper percentage point of the F distribution on ν1 and ν2 degrees of freedom is f , then the corresponding lower percentage point of the F distribution on ν2 and ν1 degrees of freedom is 1/ f . 34 BLANK PAGE 35 BLANK PAGE