List MF9 List of fomulae ad statistical tables Cambidge Iteatioal AS & A Level Mathematics (9709) ad Futhe Mathematics (93) Fo use fom 00 i all papes fo the above syllabuses. CST39 *50870970*
PURE MATHEMATICS Mesuatio Volume of sphee = 4 3 π 3 Suface aea of sphee = 4π Volume of coe o pyamid = base aea height 3 Aea of cuved suface of coe = π slat height Ac legth of cicle = θ (θ i adias) Aea of secto of cicle = (θ i adias) θ Algeba Fo the quadatic equatio Fo a aithmetic seies: Fo a geometic seies: a + b + c = 0 : b± b 4ac = a u = a+ ( ) d, S = ( a+ l) = { a+ ( ) d} u = a a( ), S = ( ) Biomial seies: ( a+ b) = a + a b+ a b + a b + K + b 3 3 3! ad =!( )! a, S = ( < ), whee is a positive itege ( ) ( )( ) 3 ( + ) = + + + +K, whee is atioal ad <! 3!
Tigoomety cos θ + si θ, siθ taθ cosθ + ta θ sec θ, si( A± B) si Acos B± cos Asi B cos( A± B) cos Acos Bm si Asi B cot θ + cosec θ ta A± ta B ta( A± B) m taatab si A si Acos A cos A cos A si A cos A si A taa ta A ta A Picipal values: π si π, 0 cos π, ta π < < π Diffeetiatio f( ) f( ) l e si cos e cos si ta sec sec sec ta cosec cosec cot cot ta uv If = f( t) ad y = g( t) the dy = dy d d dt dt u v cosec + du dv v + u d d du dv v u d d v 3
Itegatio (Abitay costats ae omitted; a deotes a positive costat.) f( ) e si cos sec a dv du u d= uv v d d d f( ) d = l f( ) f( ) + a a f( ) d + + l e cos si ta ta a a a l a + a a+ l a a ( ) ( > a) ( < a) Vectos a= ai+ a j+ a k ad b= bi+ bj+ b3k the If 3 ab. = ab + ab + ab = a b cosθ 3 3 4
FURTHER PURE MATHEMATICS Algeba Summatios: = ( + ), = = ( + )(+ ), 6 = = 3 = ( + ) 4 Maclaui s seies: ( ) f( ) = f(0) + f (0) + f (0) + K+ f (0) + K!! e = ep( ) = + + + K+ + K (all )!! 3 + l( + ) = + K+ ( ) + K ( < ) 3 3 5 + si = + K+ ( ) + K (all ) 3! 5! ( + )! 4 cos = + K+ ( ) + K (all )! 4! ( )! 3 5 + ta = + K+ ( ) + K ( ) 3 5 + 3 5 + sih = + + + K+ + K (all ) 3! 5! ( + )! 4 cosh = + + + K+ + K (all )! 4! ( )! tah 3 5 + = + + + K+ + K ( < < ) 3 5 + Tigoomety If t = ta the: t si = ad + t t cos = + t Hypebolic fuctios cosh sih, sih sih cosh, ( ) sih = l + + ( ) cosh = l + ( ) + tah = l ( ) < cosh cosh + sih 5
Diffeetiatio f( ) f( ) si cos sih cosh tah sih cosh tah cosh sih sech + Itegatio (Abitay costats ae omitted; a deotes a positive costat.) f( ) f( ) d sec l sec ta l ta( ) + = + π ( < π ) 4 cosec l cosec + cot = l ta ( ) (0 < <π ) sih cosh cosh sih sech a a a + si cosh sih tah a a a ( < a) ( > a) 6
MECHANICS Uifomly acceleated motio v= u+ at, s = ( u+ v) t, s = ut + at, v = u + as Motio of a pojectile Equatio of tajectoy is: FURTHER MECHANICS g y = taθ V cos θ Elastic stigs ad spigs λ T =, l λ E = l Motio i a cicle Fo uifom cicula motio, the acceleatio is diected towads the cete ad has magitude ω o v Cetes of mass of uifom bodies Tiagula lamia: alog media fom vete 3 Solid hemisphee of adius : 3 fom cete 8 Hemispheical shell of adius : fom cete Cicula ac of adius ad agle α: siα fom cete α Cicula secto of adius ad agle α: si α fom cete 3α Solid coe o pyamid of height h: 3 h fom vete 4 7
PROBABILITY & STATISTICS Summay statistics Fo ugouped data: Fo gouped data: Σ =, stadad deviatio Σ( ) Σ = = Σf =, stadad deviatio Σ f Σ( ) f Σ f = = Σf Σf Discete adom vaiables E( X ) = Σ p, Va( X) = Σ p {E( X)} Fo the biomial distibutio B(, p ) : p = p ( p), µ = p, Fo the geometic distibutio Geo(p): p = p( p), σ = p( p) µ = p Fo the Poisso distibutio Po( λ ) p λ λ = e, µ = λ,! σ = λ Cotiuous adom vaiables E( X ) = f( ) d, Va( X) = f( ) d {E( X)} Samplig ad testig Ubiased estimatos: Σ =, s = = Σ Σ( ) ( Σ ) Cetal Limit Theoem: σ X ~N µ, Appoimate distibutio of sample popotio: p( p) N p, 8
Samplig ad testig Two-sample estimate of a commo vaiace: FURTHER PROBABILITY & STATISTICS s Σ( ) + Σ( ) = + Pobability geeatig fuctios X G X ( t ) = E( t ), E( X ) = G (), X Va( X ) = G () + G () {G ()} X X X 9
If Z has a omal distibutio with mea 0 ad vaiace, the, fo each value of z, the table gives the value of Φ(z), whee Φ(z) = P(Z z). Fo egative values of z, use Φ( z) = Φ(z). THE NORMAL DISTRIBUTION FUNCTION z 0 3 4 5 6 7 8 9 3 4 5 6 7 8 9 ADD 0.0 0.5000 0.5040 0.5080 0.50 0.560 0.599 0.539 0.579 0.539 0.5359 4 8 6 0 4 8 3 36 0. 0.5398 0.5438 0.5478 0.557 0.5557 0.5596 0.5636 0.5675 0.574 0.5753 4 8 6 0 4 8 3 36 0. 0.5793 0.583 0.587 0.590 0.5948 0.5987 0.606 0.6064 0.603 0.64 4 8 5 9 3 7 3 35 0.3 0.679 0.67 0.655 0.693 0.633 0.6368 0.6406 0.6443 0.6480 0.657 4 7 5 9 6 30 34 0.4 0.6554 0.659 0.668 0.6664 0.6700 0.6736 0.677 0.6808 0.6844 0.6879 4 7 4 8 5 9 3 0.5 0.695 0.6950 0.6985 0.709 0.7054 0.7088 0.73 0.757 0.790 0.74 3 7 0 4 7 0 4 7 3 0.6 0.757 0.79 0.734 0.7357 0.7389 0.74 0.7454 0.7486 0.757 0.7549 3 7 0 3 6 9 3 6 9 0.7 0.7580 0.76 0.764 0.7673 0.7704 0.7734 0.7764 0.7794 0.783 0.785 3 6 9 5 8 4 7 0.8 0.788 0.790 0.7939 0.7967 0.7995 0.803 0.805 0.8078 0.806 0.833 3 5 8 4 6 9 5 0.9 0.859 0.886 0.8 0.838 0.864 0.889 0.835 0.8340 0.8365 0.8389 3 5 8 0 3 5 8 0 3.0 0.843 0.8438 0.846 0.8485 0.8508 0.853 0.8554 0.8577 0.8599 0.86 5 7 9 4 6 9. 0.8643 0.8665 0.8686 0.8708 0.879 0.8749 0.8770 0.8790 0.880 0.8830 4 6 8 0 4 6 8. 0.8849 0.8869 0.8888 0.8907 0.895 0.8944 0.896 0.8980 0.8997 0.905 4 6 7 9 3 5 7.3 0.903 0.9049 0.9066 0.908 0.9099 0.95 0.93 0.947 0.96 0.977 3 5 6 8 0 3 4.4 0.99 0.907 0.9 0.936 0.95 0.965 0.979 0.99 0.9306 0.939 3 4 6 7 8 0 3.5 0.933 0.9345 0.9357 0.9370 0.938 0.9394 0.9406 0.948 0.949 0.944 4 5 6 7 8 0.6 0.945 0.9463 0.9474 0.9484 0.9495 0.9505 0.955 0.955 0.9535 0.9545 3 4 5 6 7 8 9.7 0.9554 0.9564 0.9573 0.958 0.959 0.9599 0.9608 0.966 0.965 0.9633 3 4 4 5 6 7 8.8 0.964 0.9649 0.9656 0.9664 0.967 0.9678 0.9686 0.9693 0.9699 0.9706 3 4 4 5 6 6.9 0.973 0.979 0.976 0.973 0.9738 0.9744 0.9750 0.9756 0.976 0.9767 3 4 4 5 5.0 0.977 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.98 0.987 0 3 3 4 4. 0.98 0.986 0.9830 0.9834 0.9838 0.984 0.9846 0.9850 0.9854 0.9857 0 3 3 4. 0.986 0.9864 0.9868 0.987 0.9875 0.9878 0.988 0.9884 0.9887 0.9890 0 3 3.3 0.9893 0.9896 0.9898 0.990 0.9904 0.9906 0.9909 0.99 0.993 0.996 0.4 0.998 0.990 0.99 0.995 0.997 0.999 0.993 0.993 0.9934 0.9936 0 0.5 0.9938 0.9940 0.994 0.9943 0.9945 0.9946 0.9948 0.9949 0.995 0.995 0 0 0.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.996 0.996 0.9963 0.9964 0 0 0 0.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.997 0.997 0.9973 0.9974 0 0 0 0 0.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.998 0 0 0 0 0 0 0.9 0.998 0.998 0.998 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986 0 0 0 0 0 0 0 0 0 If Z has a omal distibutio with mea 0 ad vaiace, the, fo each value of p, the table gives the value of z such that P(Z z) = p. Citical values fo the omal distibutio p 0.75 0.90 0.95 0.975 0.99 0.995 0.9975 0.999 0.9995 z 0.674.8.645.960.36.576.807 3.090 3.9 0
CRITICAL VALUES FOR THE t-distribution If T has a t-distibutio with ν degees of feedom, the, fo each pai of values of p ad ν, the table gives the value of t such that: P(T t) = p. p 0.75 0.90 0.95 0.975 0.99 0.995 0.9975 0.999 0.9995 ν =.000 3.078 6.34.7 3.8 63.66 7.3 38.3 636.6 0.86.886.90 4.303 6.965 9.95 4.09.33 3.60 3 0.765.638.353 3.8 4.54 5.84 7.453 0..9 4 0.74.533.3.776 3.747 4.604 5.598 7.73 8.60 5 0.77.476.05.57 3.365 4.03 4.773 5.894 6.869 6 0.78.440.943.447 3.43 3.707 4.37 5.08 5.959 7 0.7.45.895.365.998 3.499 4.09 4.785 5.408 8 0.706.397.860.306.896 3.355 3.833 4.50 5.04 9 0.703.383.833.6.8 3.50 3.690 4.97 4.78 0 0.700.37.8.8.764 3.69 3.58 4.44 4.587 0.697.363.796.0.78 3.06 3.497 4.05 4.437 0.695.356.78.79.68 3.055 3.48 3.930 4.38 3 0.694.350.77.60.650 3.0 3.37 3.85 4. 4 0.69.345.76.45.64.977 3.36 3.787 4.40 5 0.69.34.753.3.60.947 3.86 3.733 4.073 6 0.690.337.746.0.583.9 3.5 3.686 4.05 7 0.689.333.740.0.567.898 3. 3.646 3.965 8 0.688.330.734.0.55.878 3.97 3.60 3.9 9 0.688.38.79.093.539.86 3.74 3.579 3.883 0 0.687.35.75.086.58.845 3.53 3.55 3.850 0.686.33.7.080.58.83 3.35 3.57 3.89 0.686.3.77.074.508.89 3.9 3.505 3.79 3 0.685.39.74.069.500.807 3.04 3.485 3.768 4 0.685.38.7.064.49.797 3.09 3.467 3.745 5 0.684.36.708.060.485.787 3.078 3.450 3.75 6 0.684.35.706.056.479.779 3.067 3.435 3.707 7 0.684.34.703.05.473.77 3.057 3.4 3.689 8 0.683.33.70.048.467.763 3.047 3.408 3.674 9 0.683.3.699.045.46.756 3.038 3.396 3.660 30 0.683.30.697.04.457.750 3.030 3.385 3.646 40 0.68.303.684.0.43.704.97 3.307 3.55 60 0.679.96.67.000.390.660.95 3.3 3.460 0 0.677.89.658.980.358.67.860 3.60 3.373 0.674.8.645.960.36.576.807 3.090 3.9
If X has a χ -distibutio with ν degees of feedom the, fo each pai of values of p ad ν, the table gives the value of such that P(X ) = p. CRITICAL VALUES FOR THE χ -DISTRIBUTION p 0.0 0.05 0.05 0.9 0.95 0.975 0.99 0.995 0.999 ν = 0.0 3 57 0.0 3 98 0.0 393.706 3.84 5.04 6.635 7.879 0.83 0.000 0.05064 0.06 4.605 5.99 7.378 9.0 0.60 3.8 3 0.48 0.58 0.358 6.5 7.85 9.348.34.84 6.7 4 0.97 0.4844 0.707 7.779 9.488.4 3.8 4.86 8.47 5 0.5543 0.83.45 9.36.07.83 5.09 6.75 0.5 6 0.87.37.635 0.64.59 4.45 6.8 8.55.46 7.39.690.67.0 4.07 6.0 8.48 0.8 4.3 8.647.80.733 3.36 5.5 7.53 0.09.95 6. 9.088.700 3.35 4.68 6.9 9.0.67 3.59 7.88 0.558 3.47 3.940 5.99 8.3 0.48 3. 5.9 9.59 3.053 3.86 4.575 7.8 9.68.9 4.73 6.76 3.6 3.57 4.404 5.6 8.55.03 3.34 6. 8.30 3.9 3 4.07 5.009 5.89 9.8.36 4.74 7.69 9.8 34.53 4 4.660 5.69 6.57.06 3.68 6. 9.4 3.3 36. 5 5.9 6.6 7.6.3 5.00 7.49 30.58 3.80 37.70 6 5.8 6.908 7.96 3.54 6.30 8.85 3.00 34.7 39.5 7 6.408 7.564 8.67 4.77 7.59 30.9 33.4 35.7 40.79 8 7.05 8.3 9.390 5.99 8.87 3.53 34.8 37.6 4.3 9 7.633 8.907 0. 7.0 30.4 3.85 36.9 38.58 43.8 0 8.60 9.59 0.85 8.4 3.4 34.7 37.57 40.00 45.3 8.897 0.8.59 9.6 3.67 35.48 38.93 4.40 46.80 9.54 0.98.34 30.8 33.9 36.78 40.9 4.80 48.7 3 0.0.69 3.09 3.0 35.7 38.08 4.64 44.8 49.73 4 0.86.40 3.85 33.0 36.4 39.36 4.98 45.56 5.8 5.5 3. 4.6 34.38 37.65 40.65 44.3 46.93 5.6 30 4.95 6.79 8.49 40.6 43.77 46.98 50.89 53.67 59.70 40.6 4.43 6.5 5.8 55.76 59.34 63.69 66.77 73.40 50 9.7 3.36 34.76 63.7 67.50 7.4 76.5 79.49 86.66 60 37.48 40.48 43.9 74.40 79.08 83.30 88.38 9.95 99.6 70 45.44 48.76 5.74 85.53 90.53 95.0 00.4 04..3 80 53.54 57.5 60.39 96.58 0.9 06.6.3 6.3 4.8 90 6.75 65.65 69.3 07.6 3. 8. 4. 8.3 37. 00 70.06 74. 77.93 8.5 4.3 9.6 35.8 40. 49.4
WILCOXON SIGNED-RANK TEST The sample has size. P is the sum of the aks coespodig to the positive diffeeces. Q is the sum of the aks coespodig to the egative diffeeces. T is the smalle of P ad Q. Fo each value of the table gives the lagest value of T which will lead to ejectio of the ull hypothesis at the level of sigificace idicated. Citical values of T Level of sigificace Oe-tailed 0.05 0.05 0.0 0.005 Two-tailed 0. 0.05 0.0 0.0 = 6 0 7 3 0 8 5 3 0 9 8 5 3 0 0 8 5 3 3 0 7 5 7 3 9 7 3 7 9 4 5 5 5 30 5 9 5 6 35 9 3 9 7 4 34 7 3 8 47 40 3 7 9 53 46 37 3 0 60 5 43 37 Fo lage values of, each of P ad Q ca be appoimated by the omal distibutio with mea + ( ) 4 ad vaiace ( + )(+ ). 4 3
The two samples have sizes m ad, whee m. WILCOXON RANK-SUM TEST R m is the sum of the aks of the items i the sample of size m. W is the smalle of R m ad m( + m + ) R m. Fo each pai of values of m ad, the table gives the lagest value of W which will lead to ejectio of the ull hypothesis at the level of sigificace idicated. Citical values of W Level of sigificace Oe-tailed 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-tailed 0. 0.05 0.0 0. 0.05 0.0 0. 0.05 0.0 0. 0.05 0.0 m = 3 m = 4 m = 5 m = 6 3 6 4 6 0 5 7 6 0 9 7 6 6 8 7 3 0 8 7 8 6 4 7 8 7 6 4 3 0 8 9 7 5 8 9 8 6 5 4 3 9 3 9 7 9 0 8 7 6 4 3 4 0 33 3 8 0 0 9 7 7 5 3 6 3 35 3 9 Level of sigificace Oe-tailed 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-tailed 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 39 36 34 8 4 38 35 5 49 45 9 43 40 37 54 5 47 66 6 59 0 45 4 39 56 53 49 69 65 6 8 78 74 Fo lage values of m ad, the omal distibutio with mea mm ( + + ) ad vaiace m( m + + ) should be used as a appoimatio to the distibutio of R m. 4
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