KURSSIN TILASTOMATEMATIIKKA KAAVOJA

KURSSIN TILASTOMATEMATIIKKA KAAVOJA X = S = s = Otossuureita X i tai x = x i (otoskeskiarvo) (X i X) = (x i x) = Xi x i E(X) =µ, var(x) = σ X x tai, E(S )=σ (otosvariassi) Normaalijakautuee populaatio otossuureet V = Z = X µ σ/ N(0, ) T = X µ S/ t vapausasteella ( )S σ = (X i X) Z = (X X ) (µ µ ) σ / + σ / N(0, ) σ χ vapausasteella T = (X X ) (µ µ ) S p / +/ t vapausastei +, missä S p = ( )S +( )S + olettae, että σ = σ W = (X X ) (µ µ ) (a + a ) t vapausastei S / + S/ a /( ) + a /( ), missä a = s ja a = s (Welch Satterthwaite-approksimaatio) F = S /σ F vapausastei S/σ ja i Piste-estimoiti Parametri θ Estimaatti ˆθ Estimaattori ˆΘ µ ˆµ = x X σ σ = s S m ˆm = q(0.5) Q(0.5) Odotusarvo väliestimoiti Z = X µ σ/ : x ± z σ α/ T = X µ S/ : x ± t s α/ (vapausastei ) Z = (X X ) (µ µ ) σ : (x x ) ± z α/ + σ σ / + σ/ T = (X X ) (µ µ ) : (x x ) ± t α/ s p + S p / +/ (vapausastei + ) W = (X X ) (µ µ ) s : (x x ) ± t α/ + s (Welch Satterthwaite) S / + S/ (vapausastei = V = (a + a ) a /( ) + a /( ), missä a = s ja a = s ) Suhdeluvu estimoiti biomijakaumalle ( ) P(X = x) = p x ( p) x x E(X) = p, var(x) = p( p) i=x F = S /σ : S/σ ( ) ˆp i i L( ˆp L ) i = α ( )S σ : ˆp = x x, i=0 Variassi väliestimoiti s s f,α/ ( )s h,α/ ja s s ja ( ) ˆp i i U( ˆp U ) i = α ( )s h,α/ (vapausastei ) f,α/ (vapausastei ja ) ii

Odotusarvoje testaus z = x µ 0 σ/ ja H 0 : µ = µ 0 : µ>µ 0 z z α Φ(z) µ<µ 0 z z α Φ(z) µ µ 0 z z α/ mi ( Φ(z), Φ(z) ) Φ o stadardiormaali jakauma kertymäfuktio. t = x µ 0 s/ ja H 0 : µ = µ 0 : µ>µ 0 t t α F (t) µ<µ 0 t t α F (t) µ µ 0 t t α/ mi ( F (t), F (t) ) F o t-jakauma kertymäfuktio vapausasteella. v = Variassie testaus ( )s σ 0 ja H 0 : σ = σ 0 : σ > σ0 v h,α F (v) σ < σ0 v h,α F (v) σ σ0 v h,α/ tai v h,α/ mi ( F (v), F (v) ) F o χ -jakauma kertymäfuktio vapausasteella. f = k s s ja H 0 : σ = kσ : σ >kσ f f,α G(f) σ <kσ f f,α G(f) σ kσ f f,α/ tai f f,α/ mi ( G(f), G(f) ) G o F-jakauma kertymäfuktio vapausastei ja. z = x x d 0 ja H 0 : µ µ = d 0 : σ / + σ/ µ µ >d 0 z z α Φ(z) µ µ <d 0 z z α Φ(z) µ µ d 0 z z α/ mi ( Φ(z), Φ(z) ) Φ o stadardiormaali jakauma kertymäfuktio. t = x x d 0, missä s p = ( )s +( )s, ja H 0 : µ µ = d 0 : s p / +/ + µ µ >d 0 t t α F (t) µ µ <d 0 t t α F (t) µ µ d 0 t t α/ mi ( F (t), F (t) ) F o t-jakauma kertymäfuktio vapausastei +. t = x x d 0 ja H 0 : µ µ = d 0 (Welch Satterthwaite) : s / + s / µ µ >d 0 t t α F (t) µ µ <d 0 t t α F (t) µ µ d 0 t t α/ mi ( F (t), F (t) ) F o approksimatiivisesti t-jakauma kertymäfuktio vapausastei (a + a ) a /( ) + a /( ), missä a = s ja a = s. iii H = H = Jakauma sopivuustesti H o : P(T )=p,..., P(T k )=p k k (F i p i ) p i χ k vapausasteella Riippumattomuustesti. Kotigessitaulut P(T )=p,..., P(T k )=p k ja P(S )=q,..., P(S l )=q l k j= P(T i S j )=p i,j (i =,..., k ja j =,..., l) H 0 : p i,j = p i q j (i =,..., k ja j =,..., l) l (F i,j F i G j /) χ F i G j / Kotigessitaulu: (k )(l ) vapausasteella S S S l Σ T f, f, f,l f T f, f, f,l f........ T k f k, f k, f k,l f k Σ g g g l iv

Suurimma uskottavuude estimoiti L(θ,..., θ m ; x,..., x )=f(x ; θ,..., θ m ) f(x ; θ,..., θ m ) l(θ,..., θ m ; x,..., x ) = l f(x ; θ,..., θ m )+ + l f(x ; θ,..., θ m ) (ˆθ,..., ˆθ m ) = argmax θ,...,θm (ˆθ,..., ˆθ m ) = argmax θ,...,θm Malli: Data: L(θ,..., θ m ; x,..., x ) l(θ,..., θ m ; x,..., x ) Lieaarie regressio y = β 0 + β x + + β k x k + ɛ x x x k y x, x, x,k y x, x, x,k y.... x, x, x,k y x, x, x,k y ɛ β 0 x, x, x,k X =......., y = y., ɛ = ɛ., β = β. x, x, x,k y ɛ β k Datamalli: y = Xβ + ɛ. ˆβ = b =(X T X) X T y = β +(X T X) X T ɛ C =(c ij ) = (X T X) H = XCX T (hattumatriisi) P = I H E(b i )=β i, var(b i )=c ii σ ja cov(b i,b j )=c ij σ σ = ŷ = Xb e = y ŷ = Py = Pɛ SSE = e = e i = (y i ŷ i ) SSE k = MSE, MSE = RMSE v tai SST = (y i y), SSR = (ŷ i y) SST = SSE + SSR MST = SST, MSR = SSR k ANOVA-taulu: Variaatio lähde Vapausasteet Neliösummat Keskieliöt F Regressio Residuaali Kokoaisvariaatio H 0 : β = = β k = 0 : k k SSR SSE SST MSR σ = MSE (MST) F = MSR MSE F = MSR F vapausastei k ja k MSE H 0 : β i = β 0,i ; T i = b i β 0,i RMSE c ii t vapausastei k R = SSR SSE = SST SST Radj = MSE MST = SSE k SST, y = β 0 + β x + + β k x k + Kategoriset regressorit: z i z i, z i, z i,mi A i, 0 0 A i, 0 0.... A i,mi 0 0 A i,mi 0 0 0 l (β i, z i, + + β i,mi z i,mi )+ɛ Logistie regressio P(y = A) = +e β0 βx βkxk L(β 0,..., β k )=L (β 0,..., β k ) L (β 0,..., β k ), missä p i = +e, jos y β0 βxi, βkxi,k i =A L i (β 0,..., β k )= e β0 βxi, βkxi,k p i = +e, jos y β0 βxi, βkxi,k i =B vi

H = r S = r S = (b 0,b,..., b k ) = argmax L(β 0, β,..., β k ) β0,β...,βk ( + ) ˆp 0 = +e b0 bx0, bkx0,k. Kruskal Wallis-testi k j= H 0 : µ = = µ k W j j 3( + ) χ k vapausasteella Spearmai järjestyskorrelaatiokerroi Otokset: x,,...,x, ja x,,...,x, Järjestysluvut: r,,...,r, ja r,,...,r, (r,i r)(r,i r) (r,i r) 6 ( ) (r,i r), missä r = d i, missä d i = r,i r,i + (Olettae, että otosarvot ovat erilliset!) Stadardiormaalijakauma kvatiileja Stadardiormaalijakauma z 0.00 0.0 0.0 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.5000 0.5040 0.5080 0.50 0.560 0.590 0.539 0.579 0.539 0.5359 0. 0.5398 0.5438 0.5478 0.557 0.5557 0.5596 0.5636 0.5675 0.574 0.5753 0. 0.5793 0.583 0.587 0.590 0.5948 0.5987 0.606 0.6064 0.603 0.64 0.3 0.679 0.67 0.655 0.693 0.633 0.6368 0.6406 0.6443 0.6480 0.657 0.4 0.6554 0.659 0.668 0.6664 0.6700 0.6736 0.677 0.6808 0.6844 0.6879 0.5 0.695 0.6950 0.6985 0.709 0.7054 0.7088 0.73 0.757 0.790 0.74 0.6 0.757 0.79 0.734 0.7357 0.7389 0.74 0.7454 0.7486 0.757 0.7549 0.7 0.7580 0.76 0.764 0.7673 0.7704 0.7734 0.7764 0.7794 0.783 0.785 0.8 0.788 0.790 0.7939 0.7969 0.7995 0.803 0.805 0.8078 0.806 0.833 0.9 0.859 0.886 0.8 0.838 0.864 0.889 0.835 0.8340 0.8365 0.8389.0 0.843 0.8438 0.846 0.8485 0.8508 0.853 0.8554 0.8577 0.859 0.86. 0.8643 0.8665 0.8686 0.8708 0.879 0.8749 0.8770 0.8790 0.880 0.8830. 0.8849 0.8869 0.8888 0.8907 0.895 0.8944 0.896 0.8980 0.8997 0.905.3 0.903 0.9049 0.9066 0.908 0.9099 0.95 0.93 0.947 0.96 0.977.4 0.99 0.907 0.9 0.936 0.95 0.965 0.979 0.99 0.9306 0.939.5 0.933 0.9345 0.9357 0.9370 0.938 0.9394 0.9406 0.948 0.949 0.944.6 0.945 0.9463 0.9474 0.9484 0.9495 0.9505 0.955 0.955 0.9535 0.9545.7 0.9554 0.9564 0.9573 0.958 0.959 0.9599 0.9608 0.966 0.965 0.9633.8 0.964 0.9649 0.9656 0.9664 0.967 0.9678 0.9686 0.9693 0.9699 0.9706.9 0.973 0.979 0.976 0.973 0.9738 0.9744 0.9750 0.9756 0.976 0.9767.0 0.977 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.98 0.987. 0.98 0.986 0.9830 0.9834 0.9838 0.984 0.9846 0.9850 0.9854 0.9857. 0.986 0.9864 0.9868 0.987 0.9875 0.9878 0.988 0.9884 0.9887 0.9890.3 0.9893 0.9896 0.9898 0.990 0.9904 0.9906 0.9909 0.99 0.993 0.996.4 0.998 0.990 0.99 0.995 0.997 0.999 0.993 0.993 0.9934 0.9936.5 0.9938 0.9940 0.994 0.9943 0.9945 0.9946 0.9948 0.9949 0.995 0.995.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.996 0.996 0.9963 0.9964.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.997 0.997 0.9973 0.9974.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.998.9 0.998 0.998 0.998 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986 3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990 3. 0.9990 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.9993 0.9993 3. 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995 3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997 3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998 vii viii

t-jakauma khi-toisee-jakauma alkuhätäkvatiileja χ -jakauma v.a. Kvatiili (alkuhätä) 0.005 0.00 0.05 0.05 0.0 0.5 0.50 0.75 0.90 0.95 0.975 0.99 0.995 0.39E-4 0.0006 0.00098 0.0039 0.058 0.0 0.455.3.7 3.84 5.0 6.63 7.88 0.000 0.00 0.0506 0.03 0. 0.575.39.77 4.6 5.99 7.38 9. 0.6 3 0.077 0.5 0.6 0.35 0.584..37 4. 6.5 7.8 9.35.3.8 4 0.07 0.97 0.484 0.7.06.9 3.36 5.39 7.78 9.49. 3.3 4.9 5 0.4 0.554 0.83.5.6.67 4.35 6.63 9.4..8 5. 6.7 6 0.676 0.87.4.64.0 3.45 5.35 7.84 0.6.6 4.4 6.8 8.5 7 0.989.4.69.7.83 4.5 6.35 9.04.0 4. 6.0 8.5 0.3 8.34.65.8.73 3.49 5.07 7.34 0. 3.4 5.5 7.5 0..0 9.73.09.70 3.33 4.7 5.9 8.34.4 4.7 6.9 9.0.7 3.6 0.6.56 3.5 3.94 4.87 6.74 9.34.5 6.0 8.3 0.5 3. 5..60 3.05 3.8 4.57 5.58 7.58 0.3 3.7 7.3 9.7.9 4.7 6.8 3.07 3.57 4.40 5.3 6.30 8.44.3 4.8 8.5.0 3.3 6. 8.3 3 3.57 4. 5.0 5.89 7.04 9.3.3 6.0 9.8.4 4.7 7.7 9.8 4 4.07 4.66 5.63 6.57 7.79 0. 3.3 7.. 3.7 6. 9. 3.3 5 4.60 5.3 6.6 7.6 8.55.0 4.3 8..3 5.0 7.5 30.6 3.8 6 5.4 5.8 6.9 7.96 9.3.9 5.3 9.4 3.5 6.3 8.8 3.0 34.3 7 5.70 6.4 7.56 8.67 0..8 6.3 0.5 4.8 7.6 30. 33.4 35.7 8 6.6 7.0 8.3 9.39 0.9 3.7 7.3.6 6.0 8.9 3.5 34.8 37. 9 6.84 7.63 8.9 0..7 4.6 8.3.7 7. 30. 3.9 36. 38.6 0 7.43 8.6 9.59 0.9.4 5.5 9.3 3.8 8.4 3.4 34. 37.6 40.0 8.03 8.90 0.3.6 3. 6.3 0.3 4.9 9.6 3.7 35.5 38.9 4.4 8.64 9.54.0.3 4.0 7..3 6.0 30.8 33.9 36.8 40.3 4.8 3 9.6 0..7 3. 4.8 8..3 7. 3.0 35. 38. 4.6 44. 4 9.89 0.9.4 3.8 5.7 9.0 3.3 8. 33. 36.4 39.4 43.0 45.6 5 0.5.5 3. 4.6 6.5 9.9 4.3 9.3 34.4 37.7 40.6 44.3 46.9 6.. 3.8 5.4 7.3 0.8 5.3 30.4 35.6 38.9 4.9 45.6 48.3 7.8.9 4.6 6. 8..7 6.3 3.5 36.7 40. 43. 47.0 49.6 8.5 3.6 5.3 6.9 8.9.7 7.3 3.6 37.9 4.3 44.5 48.3 5.0 9 3. 4.3 6.0 7.7 9.8 3.6 8.3 33.7 39. 4.6 45.7 49.6 5.3 30 3.8 5.0 6.8 8.5 0.6 4.5 9.3 34.8 40.3 43.8 47.0 50.9 53.7 3 4.5 5.7 7.5 9.3.4 5.4 30.3 35.9 4.4 45.0 48. 5. 55.0 3 5. 6.4 8.3 0..3 6.3 3.3 37.0 4.6 46. 49.5 53.5 56.3 33 5.8 7. 9.0 0.9 3. 7. 3.3 38. 43.7 47.4 50.7 54.8 57.6 34 6.5 7.8 9.8.7 4.0 8. 33.3 39. 44.9 48.6 5.0 56. 59.0 35 7. 8.5 0.6.5 4.8 9. 34.3 40. 46. 49.8 53. 57.3 60.3 36 7.9 9..3 3.3 5.6 30.0 35.3 4.3 47. 5.0 54.4 58.6 6.6 37 8.6 0.0. 4. 6.5 30.9 36.3 4.4 48.4 5. 55.7 59.9 6.9 38 9.3 0.7.9 4.9 7.3 3.8 37.3 43.5 49.5 53.4 56.9 6. 64. 39 0.0.4 3.7 5.7 8. 3.7 38.3 44.5 50.7 54.6 58. 6.4 65.5 40 0.7. 4.4 6.5 9. 33.7 39.3 45.6 5.8 55.8 59.3 63.7 66.8 4.4.9 5. 7.3 9.9 34.6 40.3 46.7 5.9 56.9 60.6 65.0 68. 4. 3.7 6.0 8. 30.8 35.5 4.3 47.8 54. 58. 6.8 66. 69.3 43.9 4.4 6.8 9.0 3.6 36.4 4.3 48.8 55. 59.3 63.0 67.5 70.6 44 3.6 5. 7.6 9.8 3.5 37.4 43.3 49.9 56.4 60.5 64. 68.7 7.9 45 4.3 5.9 8.4 30.6 33.4 38.3 44.3 5.0 57.5 6.7 65.4 70.0 73. 0.005 0.00 0.05 0.05 0.0 0.5 0.50 0.75 0.90 0.95 0.975 0.99 0.995 This table was produced usig APL programs writte by William Kight. t-jakauma kvatiileja 0.0 0.05 0.05 0.0 0.005 0.00 0.0005 (loppuhätä) 0.0 0.0 0.05 0.0 0.0 0.00 0.00 (kaksi hätää) -------+---------------------------------------------------------+----- V 3.078 6.34.7 3.8 63.66 38.3 637 a.886.90 4.303 6.965 9.95.330 3.6 p 3.638.353 3.8 4.54 5.84 0.0.9 3 a 4.533.3.776 3.747 4.604 7.73 8.60 4 u 5.476.05.57 3.365 4.03 5.893 6.869 5 s 6.440.943.447 3.43 3.707 5.08 5.959 6 a 7.45.895.365.998 3.499 4.785 5.408 7 s 8.397.860.306.896 3.355 4.50 5.04 8 t 9.383.833.6.8 3.50 4.97 4.78 9 e 0.37.8.8.764 3.69 4.44 4.587 0 e.363.796.0.78 3.06 4.05 4.437 t.356.78.79.68 3.055 3.930 4.38 3.350.77.60.650 3.0 3.85 4. 3 4.345.76.45.64.977 3.787 4.40 4 5.34.753.3.60.947 3.733 4.073 5 6.337.746.0.583.9 3.686 4.05 6 7.333.740.0.567.898 3.646 3.965 7 8.330.734.0.55.878 3.60 3.9 8 9.38.79.093.539.86 3.579 3.883 9 0.35.75.086.58.845 3.55 3.850 0.33.7.080.58.83 3.57 3.89.3.77.074.508.89 3.505 3.79 3.39.74.069.500.807 3.485 3.768 3 4.38.7.064.49.797 3.467 3.745 4 5.36.708.060.485.787 3.450 3.75 5 6.35.706.056.479.779 3.435 3.707 6 7.34.703.05.473.77 3.4 3.690 7 8.33.70.048.467.763 3.408 3.674 8 9.3.699.045.46.756 3.396 3.659 9 30.30.697.04.457.750 3.385 3.646 30 3.309.694.037.449.738 3.365 3.6 3 34.307.69.03.44.78 3.348 3.60 34 36.306.688.08.434.79 3.333 3.58 36 38.304.686.04.49.7 3.39 3.566 38 40.303.684.0.43.704 3.307 3.55 40 4.30.68.08.48.698 3.96 3.538 4 44.30.680.05.44.69 3.86 3.56 44 46.300.679.03.40.687 3.77 3.55 46 48.99.677.0.407.68 3.69 3.505 48 50.99.676.009.403.678 3.6 3.496 50 55.97.673.004.396.668 3.45 3.476 55 60.96.67.000.390.660 3.3 3.460 60 65.95.669.997.385.654 3.0 3.447 65 70.94.667.994.38.648 3. 3.435 70 80.9.664.990.374.639 3.95 3.46 80 00.90.660.984.364.66 3.74 3.390 00 50.87.655.976.35.609 3.45 3.357 50 00.86.653.97.345.60 3.3 3.340 00 -------+---------------------------------------------------------+----- 0.0 0.0 0.05 0.0 0.0 0.00 0.00 (kaksi hätää) 0.0 0.05 0.05 0.0 0.005 0.00 0.0005 (loppuhätä) This table was calculated by APL programs writte by William Kight. ix x

F-jakauma Merkity järjestykse testi F-jakauma loppuhätäkvatiilit f,α todeäköisyyksille α = 0.05, 0.05, 0.0 vapausasteille v ja v. Iverssit ovat F-jakauma alkuhätäkvatiilit f,α samoille todeäköisyyksille vapausasteille v ja v. Merkity järjestykse testi kriittisiä arvoja riskitasoille α = 0.05 ja α = 0.0 Kaksipuolie testi Toispuolie testi v v : 3 4 5 6 7 8 9 0 5 0 5 50 00 v 6 99 6 5 30 34 37 39 4 4 44 46 48 49 5 53 648 799 864 900 9 937 948 957 963 969 977 985 993 998 008 03 405 5000 5403 565 5764 5859 598 598 60 6056 606 657 609 640 6303 6334 8.5 9.00 9.6 9.5 9.30 9.33 9.35 9.37 9.38 9.40 9.4 9.43 9.45 9.46 9.48 9.49 38.5 39.00 39.7 39.5 39.30 39.33 39.36 39.37 39.39 39.40 39.4 39.43 39.45 39.46 39.48 39.49 98.50 99.00 99.7 99.5 99.30 99.33 99.36 99.37 99.39 99.40 99.4 99.43 99.45 99.46 99.48 99.49 0.3 9.55 9.8 9. 9.0 8.94 8.89 8.85 8.8 8.79 8.74 8.70 8.66 8.63 8.58 8.55 3 7.44 6.04 5.44 5.0 4.88 4.73 4.6 4.54 4.47 4.4 4.34 4.5 4.7 4. 4.0 3.96 3 34. 30.8 9.46 8.7 8.4 7.9 7.67 7.49 7.35 7.3 7.05 6.87 6.69 6.58 6.35 6.4 7.7 6.94 6.59 6.39 6.6 6.6 6.09 6.04 6.00 5.96 5.9 5.86 5.80 5.77 5.70 5.66 4. 0.65 9.98 9.60 9.36 9.0 9.07 8.98 8.90 8.84 8.75 8.66 8.56 8.50 8.38 8.3 4.0 8.00 6.69 5.98 5.5 5. 4.98 4.80 4.66 4.55 4.37 4.0 4.0 3.9 3.69 3.58 6.6 5.79 5.4 5.9 5.05 4.95 4.88 4.8 4.77 4.74 4.68 4.6 4.56 4.5 4.44 4.4 5 0.0 8.43 7.76 7.39 7.5 6.98 6.85 6.76 6.68 6.6 6.5 6.43 6.33 6.7 6.4 6.08 5 6.6 3.7.06.39 0.97 0.67 0.46 0.9 0.6 0.05 9.89 9.7 9.55 9.45 9.4 9.3 5.99 5.4 4.76 4.53 4.39 4.8 4. 4.5 4.0 4.06 4.00 3.94 3.87 3.83 3.75 3.7 6 8.8 7.6 6.60 6.3 5.99 5.8 5.70 5.60 5.5 5.46 5.37 5.7 5.7 5. 4.98 4.9 6 3.75 0.9 9.78 9.5 8.75 8.47 8.6 8.0 7.98 7.87 7.7 7.56 7.40 7.30 7.09 6.99 5.59 4.74 4.35 4. 3.97 3.87 3.79 3.73 3.68 3.64 3.57 3.5 3.44 3.40 3.3 3.7 7 8.07 6.54 5.89 5.5 5.9 5. 4.99 4.90 4.8 4.76 4.67 4.57 4.47 4.40 4.8 4. 7.5 9.55 8.45 7.85 7.46 7.9 6.99 6.84 6.7 6.6 6.47 6.3 6.6 6.06 5.86 5.75 5.3 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3.35 3.8 3. 3.5 3. 3.0.97 8 7.57 6.06 5.4 5.05 4.8 4.65 4.53 4.43 4.36 4.30 4.0 4.0 4.00 3.94 3.8 3.74 8.6 8.65 7.59 7.0 6.63 6.37 6.8 6.03 5.9 5.8 5.67 5.5 5.36 5.6 5.07 4.96 5. 4.6 3.86 3.63 3.48 3.37 3.9 3.3 3.8 3.4 3.07 3.0.94.89.80.76 9 7. 5.7 5.08 4.7 4.48 4.3 4.0 4.0 4.03 3.96 3.87 3.77 3.67 3.60 3.47 3.40 9 0.56 8.0 6.99 6.4 6.06 5.80 5.6 5.47 5.35 5.6 5. 4.96 4.8 4.7 4.5 4.4 4.96 4.0 3.7 3.48 3.33 3. 3.4 3.07 3.0.98.9.85.77.73.64.59 0 6.94 5.46 4.83 4.47 4.4 4.07 3.95 3.85 3.78 3.7 3.6 3.5 3.4 3.35 3. 3.5 0 0.04 7.56 6.55 5.99 5.64 5.39 5.0 5.06 4.94 4.85 4.7 4.56 4.4 4.3 4. 4.0 4.75 3.89 3.49 3.6 3. 3.00.9.85.80.75.69.6.54.50.40.35 6.55 5.0 4.47 4. 3.89 3.73 3.6 3.5 3.44 3.37 3.8 3.8 3.07 3.0.87.80 9.33 6.93 5.95 5.4 5.06 4.8 4.64 4.50 4.39 4.30 4.6 4.0 3.86 3.76 3.57 3.47 4.54 3.68 3.9 3.06.90.79.7.64.59.54.48.40.33.8.8. 5 6.0 4.77 4.5 3.80 3.58 3.4 3.9 3.0 3. 3.06.96.86.76.69.55.47 5 8.68 6.36 5.4 4.89 4.56 4.3 4.4 4.00 3.89 3.80 3.67 3.5 3.37 3.8 3.08.98 4.35 3.49 3.0.87.7.60.5.45.39.35.8.0..07.97.9 0 5.87 4.46 3.86 3.5 3.9 3.3 3.0.9.84.77.68.57.46.40.5.7 0 8.0 5.85 4.94 4.43 4.0 3.87 3.70 3.56 3.46 3.37 3.3 3.09.94.84.64.54 4.4 3.39.99.76.60.49.40.34.8.4.6.09.0.96.84.78 5 5.69 4.9 3.69 3.35 3.3.97.85.75.68.6.5.4.30.3.08.00 5 7.77 5.57 4.68 4.8 3.85 3.63 3.46 3.3 3. 3.3.99.85.70.60.40.9 4.03 3.8.79.56.40.9.0.3.07.03.95.87.78.73.60.5 50 5.34 3.97 3.39 3.05.83.67.55.46.38.3...99.9.75.66 50 7.7 5.06 4.0 3.7 3.4 3.9 3.0.89.78.70.56.4.7.7.95.8 3.94 3.09.70.46.3.9.0.03.97.93.85.77.68.6.48.39 00 5.8 3.83 3.5.9.70.54.4.3.4.8.08.97.85.77.59.48 00 6.90 4.8 3.98 3.5 3..99.8.69.59.50.37..07.97.74.60 This table is computed by Victor L. Bissoette. xi xii

Ma Whitey-testi Tolerassivälitaulukko kaksipuoliselle välille Ma Whitey-testi kriittisiä arvoja riskitasoille α = 0.05 ja α = 0.0 (kaksipuolie testi) o otoskoko siiä otoksessa, joka järjestyssumma o pieempi. This table is computed by Victor L. Bissoette. Taulukko ataa kertoime k arvo kaksipuoliselle tolerassivälille. k: γ =0. γ =0.05 γ =0.0 α =0. α =0.05 α =0.0 α =0. α =0.05 α =0.0 α =0. α =0.05 α =0.0 5 3.4993 4.44 5.3868 4.906 5.0767 6.5977 6.6563 7.87 0. 6 3.407 3.75 4.8498 3.735 4.43 5.758 5.3833 6.3656 8.90 7.99 3.4558 4.5087 3.3895 4.096 5.409 4.6570 5.598 7.907 8.754 3.699 4.707 3.560 3.7454 4.889 4.883 4.9694 6.48 9.6367 3.3 4.0945.9864 3.5459 4.638 3.8596 4.580 5.9803 0.5459 3.057 3.9579.8563 3.3935 4.4370 3.66 4.95 5.606.4734.9407 3.8488.7536 3.77 4.88 3.486 4.075 5.343.439.8706 3.759.670 3.748 4.555 3.793 3.8954 5.0956 3.3643.8 3.684.60 3.093 4.0505 3.557 3.7509 4.909 4.39.764 3.600.544 3.04 3.966 3.0537 3.630 4.753 5.855.796 3.5648.493.9648 3.885.9669 3.585 4.6 6.536.68 3.566.4485.935 3.889.896 3.4406 4.5078 7.57.649 3.4740.40.8685 3.7605.877 3.3637 4.4084 8.007.697 3.436.376.883 3.7088.77 3.966 4.33 9.784.5934 3.40.3460.795 3.667.70 3.36 4.433 0.583.5697 3.375.388.7603 3.60.6758 3.838 4.747.40.548 3.3437.94.73 3.583.6346 3.360 4.5.34.585 3.383.78.7047 3.5490.5979 3.094 4.056 3.083.505 3.95.53.6805 3.576.564 3.058 4.0044 4.0943.4940 3.735.35.658 3.4888.534 3.069 3.9580 5.083.4786 3.538.5.6378 3.46.5060.9836 3.947 6.0693.4644 3.354.990.687 3.4375.4797.9533 3.875 7.058.45 3.8.84.60 3.445.4560.947 3.8385 8.0477.4389 3.03.703.5846 3.3933.4340.8983 3.8048 9.0380.474 3.873.573.5693 3.3733.433.8737 3.77 30.089.466 3.73.450.5548 3.3546.3940.8509 3.746 3.003.4065 3.60.337.544 3.3369.3758.899 3.748 3.0.3969 3.477.30.585 3.305.3590.8095 3.6885 33.0045.3878 3.360.8.567 3.3048.3430.7900 3.6638 34.9973.3793 3.48.033.5053 3.90.379.777 3.6405 35.9905.37 3.43.094.4945 3.76.339.7557 3.685 36.9840.3635 3.043.0857.4844 3.68.3003.7396 3.5976 37.9779.356 3.0948.0775.4748 3.503.875.746 3.578 38.970.349 3.0857.0697.4655 3.38.753.705 3.5593 39.9664.345 3.077.063.4568 3.68.638.6966 3.544 40.96.336 3.0688.055.4484 3.58.57.6839 3.544 4.9560.330 3.0609.0485.4404 3.055.44.67 3.5085 4.95.344 3.0533.04.437 3.955.34.6593 3.497 43.9464.388 3.046.0359.454 3.860.8.648 3.4780 44.949.334 3.039.0300.483 3.768.37.637 3.4638 45.9376.3083 3.034.043.47 3.679.049.668 3.450 46.9334.3034 3.060.088.405 3.595.964.667 3.4370 47.994.987 3.099.036.3989 3.55.884.607 3.445 48.956.94 3.039.0086.399 3.435.806.5979 3.45 49.98.897 3.008.0037.387 3.360.734.5890 3.4008 50.983.855 3.006.9990.386 3.87.660.5805 3.3899 55.90.663.9776.9779.3564 3.0960.338.54 3.3395 60.8885.500.9563.9599.335 3.0680.063.5094 3.968 65.8766.359.9378.9444.366 3.0439.087.483 3.604 70.866.35.97.9308.3005 3.08.063.457 3.8 75.8570.6.9074.988.86 3.004.044.4355 3.00 80.8488.09.8947.908.735.9875.08.465 3.753 85.845.94.883.8986.6.976.039.3994 3.59 90.8348.86.878.8899.59.959.0008.3839 3.37 95.887.790.8634.880.45.9468.989.3700 3.43 00.83.73.8548.8748.338.9356.9784.357 3.0977 xiii xiv

Tolerassivälitaulukko toispuoliselle välille Taulukko ataa kertoime k arvo toispuoliselle tolerassivälille. k: γ =0. γ =0.05 γ =0.0 α =0. α =0.05 α =0.0 α =0. α =0.05 α =0.0 α =0. α =0.05 α =0.0 5.743 3.3998 4.6660 3.4066 4.07 5.74 5.367 6.5783 8.9390 6.4937 3.099 4.45 3.0063 3.7077 5.060 4.4 5.4055 7.3346 7.337.8938 3.970.7554 3.3994 4.647 3.859 4.779 6.40 8.86.7543 3.786.589 3.873 4.3539 3.497 4.85 5.88 9.39.6499 3.644.4538 3.03 4.430 3.404 3.973 5.3889 0.0656.5684 3.536.3546.90 3.98 3.0479 3.7383 5.0737.03.506 3.4434.753.850 3.853.8977 3.556 4.890.966.4483 3.3707.0.7364 3.747.7767 3.4099 4.6330 3.98.404 3.3095.554.6705 3.659.6770 3.896 4.470 4.8954.363 3.57.088.644 3.5845.593 3.886 4.337 5.8669.389 3.8.0684.5660 3.50.55 3.04 4.4 6.848.990 3.70.0330.537 3.4640.4594 3.079 4.33 7.895.74 3.369.007.486 3.444.405.967 4.0367 8.7995.486 3.054.9738.4530 3.3703.3570.905 3.9604 9.785.7 3.077.9487.43 3.3308.34.8539 3.894 0.765.078 3.055.960.3960 3.95.757.8079 3.836.7503.90 3.08.9053.374 3.68.408.7663 3.7766.7366.739 3.0069.8864.3490 3.33.09.785 3.768 3.740.589.9873.8690.383 3.06.80.6940 3.68 4.74.45.969.8530.3093 3.8.535.663 3.6395 5.705.33.954.838.97 3.579.90.633 3.60 6.694.04.9367.84.753 3.365.063.606 3.5656 7.680.09.9.84.600 3.65.085.58 3.536 8.673.0988.9085.7993.458 3.0978.0655.5577 3.509 9.6649.0890.8958.7880.34 3.0804.047.5359 3.4733 30.657.0798.8837.7773.98 3.0639.098.555 3.4465 3.6497.07.874.7673.080 3.0484.036.4963 3.44 3.647.069.867.7578.968 3.0338.9984.478 3.3977 33.636.055.855.7489.86 3.000.9840.46 3.3754 34.699.0478.849.7403.76 3.0070.9703.445 3.3543 35.639.0407.838.733.667.9946.9574.498 3.3343 36.68.034.84.746.577.988.945.454 3.355 37.68.077.858.773.49.976.9335.406 3.975 38.6076.06.8080.70.408.9609.94.3885 3.804 39.606.058.8004.7036.330.9507.98.3760 3.64 40.5979.003.793.697.55.9409.907.364 3.486 4.5934.0050.7863.69.83.936.89.358 3.337 4.5890.9998.7796.685.4.96.888.348 3.95 43.5848.9949.7733.6795.048.94.8739.334 3.059 44.5808.990.767.674.0985.9059.8654.34 3.99 45.5769.9857.763.6689.094.8979.8573.38 3.804 46.573.983.7556.6639.0865.8903.8495.305 3.684 47.5695.977.750.659.0808.8830.849.937 3.568 48.566.9730.7449.6544.0753.8759.8346.85 3.457 49.567.969.7398.6499.070.8690.875.768 3.349 50.5595.9653.7349.6455.0650.865.808.689 3.46 55.5447.948.76.658.049.836.790.330 3.0780 60.530.9333.6935.6089.0.8070.764.04 3.038 65.50.904.6769.594.0050.7849.744.759 3.0039 70.5.9090.663.58.9898.7654.76.56.9739 75.505.8990.6493.5697.9765.748.7040.3.9474 80.4947.8899.6377.5594.9644.736.6883.37.937 85.4877.887.67.550.9536.787.674.0973.904 90.483.8743.676.546.9438.706.663.084.883 95.4754.8675.6089.5338.9348.6945.6497.0688.8657 00.470.86.6009.568.965.6839.6390.0563.8496 xv