THE UNIVERSITY OF MALTA DEPARTMENT OF MATHEMATICS MATHEMATICAL FORMULAE

THE UNIVERSITY OF MALTA DEPARTMENT OF MATHEMATICS MATHEMATICAL FORMULAE UNIVERSITY PRESS, MSIDA, MALTA 208

THE UNIVERSITY OF MALTA DEPARTMENT OF MATHEMATICS MATHEMATICAL FORMULAE UNIVERSITY PRESS, MSIDA, MALTA 208

THIS BOOKLET IS NOT AN ORIGINAL COPY, BUT A RE-TRANSCRIPTION OF THE BOOKLET PROVIDED TO STUDENTS DURING MATSEC EXAMINATION SESSIONS FOR PURE MATHEMATICS (AM27/IM27). EFFORT WAS MADE TO ENSURE THAT THIS RE-TRANSCRIPTION IS AS ACCURATE AS POSSIBLE. TRANSCRIBED BY LUKE COLLINS CREATED WITH L A TEX 2ε AVAILABLE TO DOWNLOAD FOR FREE AT https://mths.com.mt MMXVIII

CONTENTS Pge MENSURATION ALGEBRA 2 HYPERBOLIC FUNCTIONS 3 CIRCULAR FUNCTIONS 4 COORDINATE GEOMETRY 5 CALCULUS I INFINITE SERIES 6 II DERIVATIVES 7 III INTEGRALS 8 IV APPLICATIONS 9 V APROXIMATIONS 2 VECTORS 2 MECHANICS 3 PROBABILITY 4 STATISTICS 6 I FORMULAE 6 II TABLES 8

MENSURATION Circle Are of circle, rdius r is πr 2 Circumference of circle is 2πr Sphere Volume of sphere, rdius r, is 4 3 πr3 Surfce re of sphere is 4πr 2 Right circulr cylinder Volume of cylinder, rdius r nd height h is πr 2 h Curved surfce re is 2πrh Right circulr cone Volume of cone, rdius r, nd height h is 3 πr2 h Curved surfce re is πrl where l is the slnt height of the cone.

ALGEBRA Fctors 3 + b 3 = ( + b)( 2 b + b 2 ) 3 b 3 = ( b)( 2 + b + b 2 ) Permuttions nd Combintions ( ) n n n! C r = = r r!(n r)! n P r = n! (n r)! Finite Series n q=0 ( + qd) = n 2 [2 + (n )d]; n q=0 r q = ( rn ) r n = r=r n 2 n(n + ); r=r 2 = n 6 n(n + )(2n + ); r 3 = r= 4 n2 (n + ) 2 ( + x) n = + nx + n(n ).2 x 2 + + ( ) n x r + + x n r (n +ve int.) de Moivre s Theorem If n is n integer, (cosθ + isinθ) n = cosnθ + isinnθ. If n is rtionl number, cosnθ + isinnθ is one of the vlues of (cosθ + isinθ) n. 2

HYPERBOLIC FUNCTIONS sinhx = ex e x 2 coshx = ex + e x 2 sinh x = ln[x + (x 2 + )] Principl vlue of cosh x = ln[x + (x 2 )] (x ) tnh x = / 2 ln + x x ( x < ) 3

CIRCULAR FUNCTIONS sin 2 A + cos 2 A = sec 2 A = + tn 2 A cosec 2 A = + cot 2 A If sinθ = sinα, If cosθ = cosα, If tnθ = tnα then θ = nπ + ( ) n α then θ = 2nπ ± α then θ = nπ + α where n = 0,±,±2,... sin(a ± B) = sinacosb ± cosasinb cos(a ± B) = cosacosb sinasinb tna ± tnb tn(a ± B) = tnatnb sina + sinb = 2sin / 2 (A + B)cos / 2 (A B) sina sinb = 2cos / 2 (A + B)sin / 2 (A B) cosa + cosb = 2cos / 2 (A + B)cos / 2 (A B) cosa cosb = 2sin / 2 (A + B)sin / 2 (A B) 2sinAcosB = sin(a + B) + sin(a B) 2cosAsinB = sin(a + B) sin(a B) 2cosAcosB = cos(a + B) + cos(a B) 2sinAsinB = cos(a B) cos(a + B) sin2a = 2sinAcosA cos2a = cos 2 A sin 2 A = 2sin 2 A = 2cos 2 A tn2a = 2tnA tn 2 A If tn A 2 = t, then sina = 2t t2 ; cosa = + t2 + t 2 4

COORDINATE GEOMETRY Perpendiculr distnce from (h,k) to x+by+c = 0 is h + bk + c ( 2 + b 2 ) The cute ngle between two lines with grdients m, m 2 is tn m m 2 + m m 2 Are of Tringle is x x 2 x 3 / 2 [x (y 2 y 3 ) + x 2 (y 3 y ) + x 3 (y y 2 )] = / 2 y y 2 y 3 Circle The eqution x 2 + y 2 + 2gx + 2 f y + c = 0 represents circle with centre t ( g, f ) nd rdius (g 2 + f 2 c). The prmetric equtions of circle with centre t (,b) nd rdius r re x = + r cost, y = b + r sint. Point dividing P P 2 in the rtio k : hs coordintes ( x + kx 2 + k, y + ky 2 + k, z ) + kz 2 + k Angle φ between two lines with direction cosines l, m, n : l, m, n ±(ll + mm + nn ) is given by cosφ = (l 2 + m 2 + n 2 ) (l 2 + m 2 + n 2 ) Distnce from P (x,y,z ) to plne Ax + By +Cz + D = 0 is Ax + By +Cz + D (A 2 + B 2 +C 2 ) Plne distnce p from origin, direction cosines of norml l, m, n, lx + my + nz = p. Line through (x,y,z ), direction cosines l, m, n. x x l = y y m = z z = t. n 5

CALCULUS I. INFINITE SERIES Tylor s Theorem f (+x) = f ()+x f ()+ x2 2! f ()+ + xr (r )! f (r ) ()+, with reminder term xr r! f (r) ( + θx), where 0 < θ <. Mclurin s Theorem f (x) = f (0) + x f (0) + x2 2! f (0) + + xr (r )! f (r ) (0) +, with reminder term xr r! f (r) (θx), where 0 < θ <. expx e x = + x + x2 xr + + 2! r! + * log e ( + x) ln( + x) = x x2 2 + x3 3 vlid for < x. + ( )r xr r + sinx = x x3 3! + x5 5! + ( )r x 2r+ (2r + )! + * cosx = x2 2! + x4 4! + ( )r x 2r (2r)! + * sinhx = / 2 (e x e x ) = x + x3 3! + x5 x2r+ + + 5! (2r + )! + * coshx = / 2 (e x + e x ) = + x2 2! + x4 x2r + + 4! (2r)! + * * These series re vlid for ll finite x. 6

II DERIVATIVES f (x) x n sinx cosx tnx cotx secx cosecx e x x ( > o) log e x lnx sinhx coshx uv u v f (x) nx n cosx sinx sec 2 x cosec 2 x secxtnx cosecxcotx e x x ln x coshx sinhx uv + u v (vu uv )/v 2 7

III INTEGRALS (Constnts of integrtion re omitted; ln log e ) f (x) ( 2 x 2 ) ( 2 + x 2 ) ( 2 + x 2 ) x ( 2 + x 2 ) ( 2 x 2 ) sinx cosx f (x)dx ( sin x ) ( x ) tn ln{x + (x 2 + 2 ( )} or sinh x ) ( 2 + x 2 ) ln{x + (x 2 2 ( )} or cosh x ) cosx sinx tn x ln(sec x) cot x ln(sin x) secx ln(secx + tnx) { ( x or ln tn 2 + π )} 4 cosecx lntn x 2 coshx sinhx u dv dx sinhx coshx uv v du dx dx 8

IV APPLICATIONS For curve y = f (x), x b. Are between curve nd x-xis = y dx Men vlue = y dx b Volume of revolution bout x-xis = π y 2 dx Centroid of re between curve nd x-xis hs coordintes x = xy dx y dx ; y = / 2 y 2 dx y dx Centroid of solid of revolution bout x-xis: x = xy 2 dx y 2 dx 9

For the re shown in Figure y First moment bout x-xis = 2 y 2 dx y = f (x) First moment bout y-xis = xy dx Second moment bout x-xis = 3 y3 dx 0 Fig. b x Second moment bout y-xis = x 2 y dx For the solid of revolution shown in Figure 2 First moment bout xy-plne = 0 y y = f (x) First moment bout yz-plne = π xy 2 dx 0 b x Second moment bout x-xis = π 2 y4 dx Second moment bout y-xis = π y 2 (x 2 + y2 4 ) dx Fig. 2 0

Length of rc = (dy) t 2 2} { + dx = (ẋ 2 + ẏ 2 )dt dx Are of surfce of revolution = 2π y { + ( dy) 2} dx dx t t 2 = 2π y (ẋ 2 + ẏ 2 )dt t { + ( dy) 2} 3/2 Rdius of curvture ρ = dx d 2 y dx 2 = (ẋ2 + ẏ 2 ) 3/2 ẋÿ ẍẏ Polr Coordintes θ 2 Are enclosed by curve = / 2 r 2 dθ θ Length of rc = θ 2 θ {r 2 + ( dr dθ ) r 2 2} dθ = r { + r 2 ( dθ ) 2} dr dr / d p Rdius of curvture ρ = r dr

V APPROXIMATIONS Trpezoidl Rule: y dx / 2 h{(y o + y n ) + 2(y + y 2 + + y n )} Simpson s rule (n even) y dx / 3 h{(y o + y n ) + 4(y + y 3 + + y n ) + 2(y 2 + y 4 + + y n 2 ) Newton s pproximtion to root of f (x) = 0: x n+ = x n f (x n) f (x n ) VECTORS Line through point, position vector, prllel to b r = +tb Position vector of point dividing the line joining P, Q with position λp + µq vectors p, q in the rtio λ : µ is λ + µ Plne through point, position vector, perpendiculr to n (r ). n = 0 Sclr product =. 2 = 2 cosθ = x x 2 + y y 2 + z z 2 i j k Vector product = 2 = 2 = 2 sinθ ˆn = x y z x 2 y 2 z 2 2

MECHANICS Centre of mss Arc, rdius r, ngle 2θ r sinθ/θ from centre Sector of circle, rdius r, ngle 2θ 2 3 r sinθ/θ from centre Hemisphere, rdius r 3 8 r from centre Hemisphericl shell, rdius r 2 r from centre Solid cone, height h 2 from vertex Conicl shell, height h 2 3 h from vertex Moments of inerti Rod, length 2l, bout perpendiculr xis through centre 3 ml 2 Disc, rdius r, bout perpendiculr xis through centre 2 mr 2 Hoop, rdius r, bout dimeter 2 mr 2 Solid sphere, rdius r, bout dimeter 2 5 mr 2 Sphericl shell, rdius r, bout dimeter 2 3 mr 2 Prllel xes theorem I A = I G + M(GA) 2 Perpendiculr xes theorem for lmin I oz = I ox + I oy Simple hrmonic motion d 2 y dt 2 = ω2 x, ( dx) 2 = ω 2 ( 2 x 2 ), x = sin(ωt + ε) dt Compound pendulum Period = 2π (k 2 + h 2 )/gh Components of ccelertion r r θ 2 long rdius vector 2ṙ θ + r θ perpendiculr to rdius vector 3

PROBABILITY Probbility lws P(A B) = P(A) + P(B) P(A B) P(A B) = P(A) P(B A) Discrete vrible X with probbility function P(X = x) Continuous vrible X with probbility density function f (X) Distribution function F(X) F(x o ) = P(X x o ) = x x o P(x) Expecttion of X E(X) = ΣxP(X = x) Expecttion of g(x) E[g(x)] = Σg(x)P(X = x) F(x) = P(X < x) x o = f (x) dx E(X) = x f (x) dx E[g(X)] = g(x) f (x)dx Vrince σ 2 Vr(X) = E[{X E(X) 2 }] Covrince Cov(X,X 2 ) = E[{X E(X )}{X 2 E(X 2 )}] Correltion coefficient ρ 2 (X,X 2 ) ρ 2 = Cov(X,X 2 ) {Vr(X )Vr(X 2 )} Liner regression coefficient, β 2, for X on X 2 β 2 = Cov(X,X 2 ) Vr(X 2 ) 4

Probbility generting function G(z) G(z) = P(0) + P()z + P(2)z 2 + + P(r)z r +, where P(r) = P(X = r) Binomil distribution (X, p,n) P(X = k) = ( N k E(X) = N p ) p k ( p) N k Vr(X) = N p( p) G(z) = [pz + ( p)] N Poisson distribution (X,m) P(X = k) = e m m k k! E(X) = m Vr(X) = m G(z) = e m e mz Norml distribution If X is distributed N(µ,σ 2 ) then X µ σ is distributed N(0,) where σ is the stndrd devition nd σ 2 is the vrince. 5

STATISTICS µ, σ 2 popultion men nd vrince X i ith rndom selection in smple size n Smple men X = n ΣX i Smple vrince S 2 = n Σ(X i X) 2 E(S 2 ) = σ 2 E(X) = µ Vr(X) = Vr(X) n = σ 2 n One smple t-test t n = X µ o S/ n Two smple t-test t n +n 2 2 = X X 2 S ( + ) n n 2 where S 2 = Σ(x x ) 2 + Σ(x 2 x 2 ) 2 n + n 2 2 nd n, n 2 re the sizes of the two smples 6

Pired smple t-test t n = s 2 = Vr(y) Y S ( ) where Y j = X j X 2 j ( j =,2,3,...,n) nd n Spermn s rnk correltion coefficient ρ ρ = 6Σ d2 n(n 2 ) Kendll s rnk correltion coefficient r ( ) ( ) Number of greed Number of different pir rnkings pir rnkings r = = Number of pirs S 2n(n ) Pired smple Wilcoxon (n > 8) T = (sum of the rnks with the less frequent sign) Z = T T s distributed N(0,); T = n(n + ) ; s 2 n(n + )(2n + ) = 4 24 7

Tble The stndrdised norml distribution Entry represents re under the stndrdized norml distribution from the men to Z Z.00.0.02.03.04.05.06.07.08.09 0.0.0000.0040.0080.020.060.099.0239.0279.039.0359 0..0398.0438.0478.057.0557.0596.0636.0675.074.0754 0.2.0793.0832.087.090.0948.0987.026.064.03.4 0.3.79.27.255.293.33.368.406.443.480.57 0.4.554.59.628.664.700.736.772.808.844.879 0.5.95.950.985.209.2054.2088.223.257.290.2224 0.6.2258.229.2324.2357.2389.2422.2454.2486.258.2549 0.7.2580.262.2642.2673.2704.2734.2764.2794.2823.2852 0.8.288.290.2939.2967.2996.3023.305.3079.306.333 0.9.359.386.322.3238.3264.3289.335.3340.3365.3389.0.343.3438.346.3485.3508.353.3554.3577.3599.362..3643.3665.3686.3708.3729.3749.3770.3790.380.3830.2.3849.3869.3888.3907.3925.3944.3962.3980.3997.405.3.4032.4049.4066.4082.4099.45.43.447.462.477.4.492.4207.4222.4236.425.4265.4279.4292.4306.439.5.4332.4345.4357.4370.4382.4394.4406.448.4430.444.6.4452.4463.4474.4485.4495.4505.455.4525.4535.4545.7.4554.4564.4573.4582.459.4599.4608.466.4625.4633.8.464.4649.4656.4664.467.4678.4686.4693.4700.4706.9.473.479.4726.4732.4738.4744.4750.4756.4762.4767 2.0.4773.4778.4783.4788.4793.4798.4803.4808.482.487 2..482.4826.4830.4834.4838.4842.4846.4850.4854.4857 2.2.486.4865.4868.487.4875.4878.488.4884.4887.4890 2.3.4893.4896.4898.490.4904.4906.4909.49.493.496 2.4.498.4920.4922.4925.4927.4929.493.4932.4934.4936 2.5.4938.4940.494.4943.4945.4946.4948.4949.495.4952 2.6.4953.4955.4956.4957.4959.4960.496.4962.4963.4964 2.7.4965.4966.4967.4968.4969.4970.497.4972.4973.4974 2.8.4974.4975.4976.4977.4977.4978.4979.4980.4980.498 2.9.498.4982.4983.4983.4984.4984.4985.4985.4986.4986 3.0.49865.49869.49874.49878.49882.49886.49889.49893.49896.49900 3..49903.49906.4990.4993.4996.4998.4992.49924.49926.49929 3.2.4993.49934.49936.49938.49940.49942.49944.49946.49948.49950 3.3.49952.49953.49955.49957.49958.49960.4996.49962.49964.49965 3.4.49966.49968.49969.49970.4997.49972.49973.49974.49975.49976 3.5.49977.49978.49978.49979.49980.4998.4998.49982.49983.49983 3.6.49984.49985.49985.49986.49986.49987.49987.49988.49988.49989 3.7.49989.49990.49990.49990.4999.4999.49992.49992.49992.49992 3.8.49993.49993.49993.49994.49994.49994.49994.49995.49995.49995 3.9.49995.49995.49996.49996.49996.49996.49996.49996.49997.49997 0 Z 8

Tble 2 Percentge points of Student s t-distribution α.0.05.025.0.005.00 v 3.078 6.34 2.706 3.82 63.657 38.309 2.886 2.920 4.303 6.965 9.925 22.327 3.638 2.353 3.82 4.54 5.84 0.25 4.533 2.32 2.776 3.747 4.604 7.73 5.476 2.05 2.57 3.365 4.032 5.893 6.440.943 2.447 3.43 3.707 5.208 7.45.895 2.365 2.998 3.499 4.785 8.397.860 2.306 2.896 3.355 4.50 9.383.833 2.262 2.82 3.250 4.297 0.372.82 2.228 2.764 3.69 4.44.363.796 2.20 2.78 3.06 4.025 2.356.782 2.79 2.68 3.055 3.930 3.350.77 2.60 2.650 3.02 3.852 4.345.76 2.45 2.624 2.977 3.787 5.34.753 2.3 2.602 2.947 3.733 6.337.746 2.20 2.583 2.92 3.686 7.333.740 2.0 2.567 2.898 3.646 8.330.734 2.0 2.552 2.878 3.60 9.328.729 2.093 2.539 2.86 3.579 20.325.725 2.086 2.528 2.845 3.552 2.323.72 2.080 2.58 2.83 3.527 22.32.77 2.074 2.508 2.89 3.505 23.39.74 2.069 2.500 2.807 3.485 24.38.7 2.064 2.492 2.797 3.467 25.36.708 2.060 2.485 2.787 3.450 26.35.706 2.056 2.479 2.779 3.435 27.34.703 2.052 2.473 2.77 3.42 28.33.70 2.048 2.467 2.763 3.408 29.3.699 2.045 2.462 2.756 3.396 30.30.697 2.042 2.457 2.750 3.385 40.303.684 2.02 2.423 2.704 3.307 60.296.67 2.000 2.390 2.660 3.232 20.289.658.980 2.358 2.67 3.60.282.645.960 2.326 2.576 3.090 t α,v α t 9

Tble 3 Percentge points of the χ 2 distribution α.995.99.975.95.50.20.0.05.025.0.005 v 0.000 0.0002 0.000 0.0039 0.45.64 2.7 3.84 5.02 6.63 7.88 2 0.00 0.020 0.05 0.03.39 3.22 4.6 5.99 7.38 9.2 0.60 3 0.072 0.5 0.26 0.352 2.37 4.64 6.25 7.8 9.35.34 2.84 4 0.207 0.30 0.484 0.7 3.36 5.99 7.78 9.49.4 3.28 4.86 5 0.42 0.55 0.83.5 4.35 7.29 9.24.07 2.83 5.09 6.75 6 0.676 0.87.24.64 5.35 8.56 0.64 2.59 4.45 6.8 8.55 7 0.989.24.69 2.7 6.35 9.80 2.02 4.07 6.0 8.48 20.28 8.34.65 2.8 2.73 7.34.03 3.36 5.5 7.53 20.09 2.95 9.73 2.09 2.70 3.33 8.34 2.24 4.68 6.92 9.02 2.67 23.59 0 2.6 2.56 3.25 3.94 9.34 3.44 5.99 8.3 20.48 23.2 25.9 2.60 3.05 3.82 4.57 0.34 4.63 7.28 9.68 2.92 24.72 26.76 2 3.07 3.57 4.40 5.23.34 5.8 8.55 2.03 23.34 26.22 28.30 3 3.57 4. 5.0 5.89 2.34 6.99 9.8 22.36 24.74 27.69 29.82 4 4.07 4.66 5.63 6.57 3.34 8.5 2.06 23.68 26.2 29.4 3.32 5 4.60 5.23 6.26 7.26 4.34 9.3 22.3 25.00 27.49 30.58 32.80 6 5.4 5.8 6.9 7.96 5.34 20.47 23.54 26.30 28.85 32.00 34.27 7 5.70 6.4 7.56 8.67 6.34 2.62 24.77 27.59 30.9 33.4 35.72 8 6.26 7.0 8.23 9.39 7.34 22.76 25.99 28.87 3.53 34.8 37.6 9 6.84 7.63 8.9 0.2 8.34 23.90 27.20 30.4 32.85 36.9 38.58 20 7.43 8.26 9.59 0.85 9.34 25.04 28.4 3.4 34.7 37.57 40.00 2 8.03 8.90 0.28.59 20.34 26.7 29.62 32.67 35.48 38.93 4.40 22 8.64 9.54 0.98 2.34 2.34 27.30 30.8 33.92 36.78 40.29 42.80 23 9.26 0.20.69 3.09 22.34 28.43 32.0 35.7 38.08 4.64 44.8 24 9.89 0.86 2.40 3.85 23.34 29.55 33.20 36.42 39.36 42.98 45.56 25 0.52.52 3.2 4.6 24.34 30.68 34.38 37.65 40.65 44.3 46.93 26.6 2.20 3.84 5.38 25.34 3.80 35.56 38.89 4.92 45.64 48.29 27.8 2.88 4.57 6.5 26.34 32.9 36.74 40. 43.9 46.96 49.64 28 2.46 3.56 5.3 6.93 27.34 34.03 37.92 4.34 44.46 48.28 50.99 29 3.2 4.26 6.05 7.7 28.34 35.4 39.09 42.56 45.72 49.59 52.34 30 3.79 4.95 6.79 8.49 29.34 36.25 40.26 43.77 46.98 50.89 53.67 40 20.7 22.6 24.43 26.5 39.34 47.27 5.8 55.76 59.34 63.69 66.77 50 27.99 29.7 32.36 34.76 49.33 58.6 63.7 67.5 7.4 76.5 79.49 60 35.53 37.48 40.48 43.9 59.33 68.97 74.40 79.08 83.30 88.38 9.95 70 43.28 45.44 48.76 5.74 69.33 79.7 85.53 90.53 95.02 00.43 04.2 80 5.7 53.34 57.5 60.39 79.33 90.4 96.58 0.88 06.63 2.33 6.3 90 59.20 6.75 65.85 69.3 89.33 0.05 07.57 3.5 8.4 24.2 28.3 00 67.33 70.06 74.22 77.93 99.33.67 8.50 24.34 29.56 35.8 40.2 α χ 2 α,v 20

Tble 4 Upper percentge points of the F-distribution () α = 0.0 v 2 3 4 5 6 7 8 9 0 2 5 20 24 30 v 2 4052.2 4999.5 5403.4 5624.6 5763.7 5859.0 5928.4 598. 6022.5 6055.8 606.3 657.3 6208.7 6234.6 6260.6 2 98.50 99.00 99.7 99.25 99.30 99.33 99.36 99.37 99.39 99.40 99.42 99.43 99.45 99.46 99.47 3 34.2 30.82 29.46 28.7 28.24 27.9 27.67 27.49 27.35 27.23 27.05 26.87 26.69 26.60 26.5 4 2.20 8.00 6.69 5.98 5.52 5.2 4.98 4.80 4.66 4.55 4.37 4.20 4.02 3.93 3.84 5 6.26 3.27 2.06.39 0.97 0.67 0.46 0.29 0.6 0.05 9.89 9.72 9.55 9.47 9.38 6 3.75 0.93 9.78 9.5 8.75 8.47 8.26 8.0 7.98 7.87 7.72 7.56 7.40 7.3 7.23 7 2.25 9.55 8.45 7.85 7.46 7.9 6.99 6.84 6.72 6.62 6.47 6.3 6.6 6.07 5.99 8.26 8.65 7.59 7.0 6.63 6.37 6.8 6.03 5.9 5.8 5.67 5.52 5.36 5.28 5.20 9 0.56 8.02 6.99 6.42 6.06 5.80 5.6 5.47 5.35 5.26 5. 4.96 4.8 4.73 4.65 0 0.04 7.56 6.55 5.99 5.64 5.39 5.20 5.06 4.94 4.85 4.7 4.56 4.4 4.33 4.25 9.65 7.2 6.22 5.67 5.32 5.07 4.89 4.74 4.63 4.54 4.40 4.25 4.0 4.02 3.94 2 9.33 6.93 5.95 5.4 5.06 4.82 4.64 4.50 4.39 4.30 4.6 4.0 3.86 3.78 3.70 3 9.07 6.70 5.74 5.2 4.86 4.62 4.44 4.30 4.9 4.0 3.96 3.82 3.67 3.59 3.5 4 8.86 6.52 5.56 5.04 4.70 4.46 4.28 4.4 4.03 3.94 3.80 3.66 3.5 3.43 3.35 5 8.68 6.36 5.42 4.89 4.56 4.32 4.4 4.00 3.90 3.8 3.67 3.52 3.37 3.29 3.2 6 8.53 6.23 5.29 4.77 4.44 4.20 4.03 3.89 3.78 3.69 3.55 3.4 3.26 3.8 3.0 7 8.40 6. 5.9 4.67 4.34 4.0 3.93 3.79 3.68 3.59 3.46 3.3 3.6 3.08 3.00 8 8.29 6.0 5.09 4.58 4.25 4.02 3.84 3.7 3.60 3.5 3.37 3.23 3.08 3.00 2.92 9 8.9 5.93 5.0 4.50 4.7 3.94 3.77 3.63 3.52 3.43 3.30 3.5 3.00 2.93 2.84 20 8.0 5.85 4.94 4.43 4.0 3.87 3.70 3.56 3.46 3.37 3.23 3.09 2.94 2.86 2.78 2 8.02 5.78 4.87 4.37 4.04 3.8 3.64 3.5 3.40 3.3 3.7 3.03 2.88 2.80 2.72 22 7.95 5.72 4.82 4.3 3.99 3.76 3.59 3.45 3.35 3.26 3.2 2.98 2.83 2.75 2.67 23 7.88 5.66 4.77 4.26 3.94 3.7 3.54 3.4 3.30 3.2 3.07 2.93 2.78 2.70 2.62 24 7.82 5.6 4.72 4.22 3.90 3.67 3.50 3.36 3.26 3.7 3.03 2.89 2.74 2.66 2.58 25 7.77 5.57 4.68 4.8 3.86 3.63 3.46 3.32 3.22 3.3 2.99 2.85 2.70 2.62 2.54 26 7.72 5.53 4.64 4.4 3.82 3.59 3.42 3.29 3.8 3.09 2.96 2.82 2.66 2.59 2.50 27 7.68 5.49 4.60 4. 3.79 3.56 3.39 3.26 3.5 3.06 2.93 2.78 2.63 2.55 2.47 28 7.64 5.45 4.57 4.07 3.75 3.53 3.36 3.23 3.2 3.03 2.90 2.75 2.60 2.52 2.44 29 7.60 5.42 4.54 4.05 3.73 3.50 3.33 3.20 3.09 3.0 2.87 2.73 2.57 2.50 2.4 30 7.56 5.39 4.5 4.02 3.70 3.47 3.30 3.7 3.07 2.98 2.84 2.70 2.55 2.47 2.39 40 7.3 5.8 4.3 3.83 3.5 3.29 3.2 2.99 2.89 2.80 2.67 2.52 2.37 2.29 2.20 60 7.08 4.98 4.3 3.65 3.34 3.2 2.95 2.82 2.72 2.63 2.50 2.35 2.20 2.2 2.03 20 6.85 4.79 3.95 3.48 3.7 2.96 2.79 2.66 2.56 2.47 2.34 2.9 2.04.95.86 6.64 4.6 3.78 3.32 3.02 2.80 2.64 2.5 2.4 2.32 2.9 2.04.88.79.70 α F α,v,v 2 2

Tble 4 (continued) () α = 0.025 v 2 3 4 5 6 7 8 9 0 2 5 20 24 30 v 2 647.79 799.50 864.6 899.58 92.85 937. 948.22 956.66 963.28 968.63 976.7 984.87 993.0 997.25 00.4 2 38.5 39.00 39.7 39.25 39.30 39.33 39.36 39.37 39.39 39.40 39.4 39.43 39.45 39.46 39.47 3 7.44 6.04 5.44 5.0 4.88 4.73 4.62 4.54 4.47 4.42 4.34 4.25 4.7 4.2 4.08 4 2.22 0.65 9.98 9.60 9.36 9.20 9.07 8.98 8.90 8.84 8.75 8.66 8.56 8.5 8.46 5 0.0 8.43 7.76 7.39 7.5 6.98 6.85 6.76 6.68 6.62 6.52 6.43 6.33 6.28 6.23 6 8.8 7.26 6.60 6.23 5.99 5.82 5.70 5.60 5.52 5.46 5.37 5.27 5.7 5.2 5.07 7 8.07 6.54 5.89 5.52 5.29 5.2 4.99 4.90 4.82 4.76 4.67 4.57 4.47 4.42 4.36 8 7.57 6.06 5.42 5.05 4.82 4.65 4.53 4.43 4.36 4.30 4.20 4.0 4.00 3.95 3.89 9 7.2 5.7 5.08 4.72 4.48 4.32 4.20 4.0 4.03 3.96 3.87 3.77 3.67 3.6 3.56 0 6.94 5.46 4.83 4.47 4.24 4.07 3.95 3.85 3.78 3.72 3.62 3.52 3.42 3.37 3.3 6.72 5.26 4.63 4.28 4.04 3.88 3.76 3.66 3.59 3.53 3.43 3.33 3.23 3.7 3.2 2 6.55 5.0 4.47 4.2 3.89 3.73 3.6 3.5 3.44 3.37 3.28 3.8 3.07 3.02 2.96 3 6.4 4.97 4.35 4.00 3.77 3.60 3.48 3.39 3.3 3.25 3.5 3.05 2.95 2.89 2.84 4 6.30 4.86 4.24 3.89 3.66 3.50 3.38 3.29 3.2 3.5 3.05 2.95 2.84 2.79 2.73 5 6.20 4.77 4.5 3.80 3.58 3.4 3.29 3.20 3.2 3.06 2.96 2.86 2.76 2.70 2.64 6 6.2 4.69 4.08 3.73 3.50 3.34 3.22 3.2 3.05 2.99 2.89 2.79 2.68 2.63 2.57 7 6.04 4.62 4.0 3.66 3.44 3.28 3.6 3.06 2.98 2.92 2.82 2.72 2.62 2.56 2.50 8 5.98 4.56 3.95 3.6 3.38 3.22 3.0 3.0 2.93 2.87 2.77 2.67 2.56 2.50 2.45 9 5.92 4.5 3.90 3.56 3.33 3.7 3.05 2.96 2.88 2.82 2.72 2.62 2.5 2.45 2.39 20 5.87 4.46 3.86 3.5 3.29 3.3 3.0 2.9 2.84 2.77 2.68 2.57 2.46 2.4 2.35 2 5.83 4.42 3.82 3.48 3.25 3.09 2.97 2.87 2.80 2.73 2.64 2.53 2.42 2.37 2.3 22 5.79 4.38 3.78 3.44 3.22 3.05 2.93 2.84 2.76 2.70 2.60 2.50 2.39 2.33 2.27 23 5.75 4.35 3.75 3.4 3.8 3.02 2.90 2.8 2.73 2.67 2.57 2.47 2.36 2.30 2.24 24 5.72 4.32 3.72 3.38 3.5 2.99 2.87 2.78 2.70 2.64 2.54 2.44 2.33 2.27 2.2 25 5.69 4.29 3.69 3.35 3.3 2.97 2.85 2.75 2.68 2.6 2.5 2.4 2.30 2.24 2.8 26 5.66 4.27 3.67 3.33 3.0 2.94 2.82 2.73 2.65 2.59 2.49 2.39 2.28 2.22 2.6 27 5.63 4.24 3.65 3.3 3.08 2.92 2.80 2.7 2.63 2.57 2.47 2.36 2.25 2.9 2.3 28 5.6 4.22 3.63 3.29 3.06 2.90 2.78 2.69 2.6 2.55 2.45 2.34 2.23 2.7 2. 29 5.59 4.20 3.6 3.27 3.04 2.88 2.76 2.67 2.59 2.53 2.43 2.32 2.2 2.5 2.09 30 5.57 4.8 3.59 3.25 3.03 2.87 2.75 2.65 2.57 2.5 2.4 2.3 2.20 2.4 2.07 40 5.42 4.05 3.46 3.3 2.90 2.74 2.62 2.53 2.45 2.39 2.29 2.8 2.07 2.0.94 60 5.29 3.93 3.34 3.0 2.79 2.63 2.5 2.4 2.33 2.27 2.7 2.06.94.88.82 20 5.5 3.80 3.23 2.89 2.67 2.52 2.39 2.30 2.22 2.6 2.05.95.82.76.69 5.02 3.69 3.2 2.79 2.57 2.4 2.29 2.9 2. 2.05.94.83.7.64.57 22

Tble 4 (continued) () α = 0.05 v 2 3 4 5 6 7 8 9 0 2 5 20 24 30 v 2 6.45 99.50 25.7 224.58 230.6 233.99 236.77 238.88 240.54 24.88 243.9 245.95 248.0 249.05 250.0 2 8.5 9.00 9.6 9.25 9.30 9.33 9.35 9.37 9.38 9.40 9.4 9.43 9.45 9.45 9.46 3 0.3 9.55 9.28 9.2 9.0 8.94 8.89 8.85 8.8 8.79 8.74 8.70 8.66 8.64 8.62 4 7.7 6.94 6.59 6.39 6.26 6.6 6.09 6.04 6.00 5.96 5.9 5.86 5.80 5.77 5.75 5 6.6 5.79 5.4 5.9 5.05 4.95 4.88 4.82 4.77 4.74 4.68 4.62 4.56 4.53 4.50 6 5.99 5.4 4.76 4.53 4.39 4.28 4.2 4.5 4.0 4.06 4.00 3.94 3.87 3.84 3.8 7 5.59 4.74 4.35 4.2 3.97 3.87 3.79 3.73 3.68 3.64 3.57 3.5 3.44 3.4 3.38 8 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3.35 3.28 3.22 3.5 3.2 3.08 9 5.2 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.8 3.4 3.07 3.0 2.94 2.90 2.86 0 4.96 4.0 3.7 3.48 3.33 3.22 3.4 3.07 3.02 2.98 2.9 2.85 2.77 2.74 2.70 4.84 3.98 3.59 3.36 3.20 3.09 3.0 2.95 2.90 2.85 2.79 2.72 2.65 2.6 2.57 2 4.75 3.89 3.49 3.26 3. 3.00 2.9 2.85 2.80 2.75 2.69 2.62 2.54 2.5 2.47 3 4.67 3.8 3.4 3.8 3.03 2.92 2.83 2.77 2.7 2.67 2.60 2.53 2.46 2.42 2.38 4 4.60 3.74 3.34 3. 2.96 2.85 2.76 2.70 2.65 2.60 2.53 2.46 2.39 2.35 2.3 5 4.54 3.68 3.29 3.06 2.90 2.79 2.7 2.64 2.59 2.54 2.48 2.40 2.33 2.29 2.25 6 4.49 3.63 3.24 3.0 2.85 2.74 2.66 2.59 2.54 2.49 2.42 2.35 2.28 2.24 2.9 7 4.45 3.59 3.20 2.96 2.8 2.70 2.6 2.55 2.49 2.45 2.38 2.3 2.23 2.9 2.5 8 4.4 3.55 3.6 2.93 2.77 2.66 2.58 2.5 2.46 2.4 2.34 2.27 2.9 2.5 2. 9 4.38 3.52 3.3 2.90 2.74 2.63 2.54 2.48 2.42 2.38 2.3 2.23 2.6 2. 2.07 20 4.35 3.49 3.0 2.87 2.7 2.60 2.5 2.45 2.39 2.35 2.28 2.20 2.2 2.08 2.04 2 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37 2.32 2.25 2.8 2.0 2.05 2.0 22 4.30 3.44 3.05 2.82 2.66 2.55 2.46 2.40 2.34 2.30 2.23 2.5 2.07 2.03.98 23 4.28 3.42 3.03 2.80 2.64 2.53 2.44 2.37 2.32 2.27 2.20 2.3 2.05 2.0.96 24 4.26 3.40 3.0 2.78 2.62 2.5 2.42 2.36 2.30 2.25 2.8 2. 2.03.98.94 25 4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.28 2.24 2.6 2.09 2.0.96.92 26 4.23 3.37 2.98 2.74 2.59 2.47 2.39 2.32 2.27 2.22 2.5 2.07.99.95.90 27 4.2 3.35 2.96 2.73 2.57 2.46 2.37 2.3 2.25 2.20 2.3 2.06.97.93.88 28 4.20 3.34 2.95 2.7 2.56 2.45 2.36 2.29 2.24 2.9 2.2 2.04.96.9.87 29 4.8 3.33 2.93 2.70 2.55 2.43 2.35 2.28 2.22 2.8 2.0 2.03.94.90.85 30 4.7 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.2 2.6 2.09 2.0.93.89.84 40 4.08 3.23 2.84 2.6 2.45 2.34 2.25 2.8 2.2 2.08 2.00.92.84.79.74 60 4.00 3.5 2.76 2.53 2.37 2.25 2.7 2.0 2.04.99.92.84.75.70.65 20 3.92 3.07 2.68 2.45 2.29 2.8 2.09 2.02.96.9.83.75.66.6.55 3.84 3.00 2.60 2.37 2.2 2.0 2.0.94.88.83.75.67.57.52.46 23