医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/009192 このサンプルページの内容は, 第 2 版 1 刷発行時のものです.
i 2 t 1. 2. 3 2 3. 6 4. 7 5. n 2 ν 6. 2 7. 2003
ii 2 2013 10
iii 1987 7
iv 1. 2. 3. 1 1987 7
v 1 2 1 2.1 3 1 3 2 4 3 5 2.2 6 1 6 2 7 3 7 2.3 9 1 9 2 9 3 13 2.4 14 1 14 2 16 18 3 2 3.1 2 19 1 19 2 2 20 3.2 22 3.3 2 26 1 26 2 29 34 4 4.1 35 1 35 2 36 3 37 4 39
vi 5 40 4.2 42 1 42 2 46 4.3 47 1 47 2 49 3 51 4.4 51 1 52 2 χ 2 59 3 t 60 4 F 61 63 5 5.1 64 5.2 65 1 65 2 66 5.3 71 1 71 2 73 5.4 78 1 78 2 79 80 6 6.1 82 1 P p 82 2 83 3 85 6.2 85 1 86 2 87 6.3 89 1 89 2 91 3 92 6.4 95 6.5 97
vii 6.6 100 1 2 100 2 2 102 6.7 108 6.8 110 1 110 2 σ 2 111 6.9 113 1 2 113 2 2 118 6.10 1 119 6.11 124 125 7 7.1 128 1 128 2 130 3 130 7.2 132 7.3 135 136 138 157 170 178 179 A α I ι P ρ B β K κ Σ σ Γ γ Λ λ T τ Δ, δ M μ Υ υ E ε N ν Φ ϕ, φ Z ζ Ξ ξ X χ H η O o Ψ ψ Θ θ Π π Ω ω
3 1 2.1 1 1 1 22 2010 2.1 class frequency 2.1 2.1 histogram
4 2 1 2.1 50 59 [mmhg] 40 0 40 49 0 50 59 6 60 69 54 70 79 150 80 89 131 90 99 62 100 109 15 110 119 1 120 129 0 130 139 0 140 0 419 22 2.1 2.1 2 1 2.2 [mmhg] 20 29 50 59 40 0 0 0 0 40 49 2 0.017 0 0 50 59 16 0.136 6 0.014 60 69 54 0.458 54 0.129 70 79 37 0.314 150 0.358 80 89 8 0.068 131 0.313 90 99 0 0 62 0.148 100 109 1 0.008 15 0.036 110 119 0 0 1 0.002 120 129 0 0 0 0 130 139 0 0 0 0 140 0 0 0 0 118 1 419 1 2.1
2.1 5 2.2 2.2 2.2 20 29 50 59 2.2 50 59 20 29 3 2.2 50 59 2.3 2.3 2.3 [mmhg] 50 59 40 0 0 0 0 40 49 0 0 0 0 50 59 6 6 0.014 0.014 60 69 54 60 0.129 0.143 70 79 150 210 0.358 0.501 80 89 131 341 0.313 0.814 90 99 62 403 0.148 0.962 100 109 15 418 0.036 0.998 110 119 1 419 0.002 1.000 120 129 0 419 0 1.000 130 139 0 419 0 1.000 140 0 419 0 1.000 2.1
6 2 1 2.3 2.3 90 mmhg 89 mmhg 419 341 81.4% variable variate 2 1 2 2.2 1 3 1 n 1 x 1,x 2,...,x n n mean arithmetic mean x
4.4 51 3 4.14 P (X = x) = e λ λ x x! λ 1 k A 1,A 2,...,A k p 1,p 2,...,p k 1 n A 1,A 2,...,A k X 1,X 2,...,X k A 1 X 1 = x 1 A 2 X 2 = x 2 A k X k = x k P (X 1 = x 1,X 2 = x 2,..., X k = x k )= n! x 1!x 2! x k! px1 1 px2 2 px k k p 1 + p 2 + + p k =1, x 1 + x 2 + + x k = n (4.22) multinomial distribution k =2 4.4 χ 2 t F
52 4 1 X f(x) = 1 (x μ) 2 σ 2π e 2σ 2 (4.23) normal distribution 4.15 μ σ 2 X 1 μ σ 2 N(μ, σ 2 ) X N(μ, σ 2 ) μ =0,σ 2 =1 N(0, 1) 4.15 N(μ, σ 2 ) 1 N(μ, σ 2 ) f(x) <x< 4.15 x = μ 2 μ σ 2 (> 0) 4.16 P (μ 1σ X μ +1σ) 0.683 P (μ 2σ X μ +2σ) 0.954 P (μ 3σ X μ +3σ) 0.997
4.4 53 4.16 N(μ, σ 2 ) 3 X, Y N(μ 1, σ1), 2 N(μ 2, σ2) 2 X +Y N(μ 1 + μ 2,σ 2 1 + σ 2 2) X Y N(μ 1 μ 2,σ 2 1 + σ 2 2) X Y = log X Y X a Z N(0, 1) f(z) f(z) = 1 2π e z2 2 (4.24) 1 I 4.17 Z = z 0 0 Z z 0 Z =1.24 0 Z 1.24 P (0 Z 1.24) I 4.18 1 1 1.2 2 2 0.04 3 1 2 = 0.3925 P (0 Z 1.24) I (0 Z) z =0
54 4 4.17 N(0, 1) 4.18 I 0 Z 1.24 P (0 Z 1.24) 1 0.5 Z 1.2 P ( 0.5 Z 1.2) 4.19 0.5 Z 0 0 Z 0.5 P ( 0.5 Z 1.2) = P ( 0.5 Z 0) + P (0 Z 1.2) = P (0 Z 0.5) + P (0 Z 1.2) =0.1915 + 0.3849 = 0.5764
4.4 55 4.19 0.5 Z 1.2 P ( 0.5 Z 1.2) 2 0.8 Z 2.0 P (0.8 Z 2.0) 4.20 2 P (0.8 Z 2.0) = P (0 Z 2.0) P (0 Z 0.8) =0.4772 0.2881 = 0.1891 4.20 0.8 Z 2.0 P (0.8 Z 2.0) b X N(μ, σ 2 ) a X b P (a X b) I X N(μ, σ 2 ) Z = X μ σ Z N(0, 1) (4.25) 4.21 X a X b P (a X b) z a = a μ σ, z b = b μ σ Z z a Z z b P (z a Z z b ) (4.26)
56 4 4.21 N(μ, σ 2 ) σ 4.21 A X = a μ (a μ)/σ A Z = z a =1 = 0 (z a 0)/1 =z a (4.26) A A Z N(0, 1) z a, z b (4.26) 4.7 [cm] = 156 =5 153 cm 160 cm % X X N(156, 5 2 ) a = 153, b = 160 P (a X b) =P (153 X 160) (4.26) z a = 153 156 5 = 0.6, z b = 160 156 5 =0.8 Z N(0, 1) P (153 X 160) = P ( 0.6 Z 0.8) P ( 0.6 Z 0.8) = P (0 Z 0.6) + P (0 Z 0.8) =0.2257 + 0.2881 = 0.5138 (51.38%)
6.10 1 119 5 W μ w 0 σ 2 H 0 : μ w =0 H 1 : μ w \=0 1 W 42 s 2 s 2 = 694 5 1 = 173.5 2 (6.16) t 0 5(42 0) t 0 = 7.130 173.5 3 t 0 > 2.7764 4 t 5% H 0 1 n 2 I (X 1 = x 1,...,X i = x i,...,x n = x n ) II (Y 1 = y 1,..., Y i = y i,..., Y n = y n ) I II 2 II I (Y 1 X 1 = y 1 x 1,...,Y n X n = y n x n ) n H 0 : μ =0 H 1 : μ 0 H 0 : μ =0 H 1 : μ>0 μ<0 σ 2 6.10 1 1 k 2 1 1 38 40 42 44 k 4 k k 6.18 σ 2 k
120 6 k 1 k =2 2 6.18 1 H 0 : μ 1 = μ 2 = = μ k H 1 : 6.5 x H 0 6.5 I x 11,x 12,...,x 1n1 n 1 x 1 II x 21,x 22,...,x 2n2 n 2 x 2.... K x k1,x k2,,x knk n k x k I+ + K n = n 1 + + n k x x ij x 2 S t S t = = k n i (x ij x) 2 (6.21) i=1 j=1 k n i x 2 ij 1 n i=1 j=1 k n i x ij i=1 j=1 2 (6.22)
6.10 1 121 S b x i x 2 k S b = n 1 (x 1 x) 2 + n 2 (x 2 x) 2 + + n k (x k x) 2 = = k n i (x i x) 2 (6.23) i=1 k 1 n i n i i=1 x ij j=1 2 1 k n i n x ij i=1 j=1 2 (6.24) S w x ij x i 2 k n 1 n 2 n k S w = (x 1j x 1 ) 2 + (x 2j x 2 ) 2 + + (x kj x k ) 2 = = j=1 j=1 j=1 k n i (x ij x i ) 2 (6.25) i=1 j=1 k n i x 2 ij i=1 j=1 k 1 n i n i i=1 j=1 x ij 2 (6.26) S t S b S w 6.15 S t = S b + S w (6.27) 6.19 k =3 1 6.19 k =3
122 6 H 0 S b /σ 2, S w /σ 2 k 1, n k χ 2 F F = S b k 1 S w n k (6.28) (ν 1,ν 2 )=(k 1,n k) F 6.16 1 S b S w (6.24), (6.26) 2 2 ν 1 = k 1 ν 2 = n k 3 S b S w = = S b k 1 = = S w n k 4 F 0 F 0 = = S b k 1 S w n k = S b(n k) S w (k 1) 5 α F 0 (k 1,n k) F α H 0 F 0 < (k 1,n k) F α H 0 6.6 6.6 F S b k 1 S w n k S t n 1 S b k 1 S w n k F = S b k 1 S w n k
6.10 1 123 6.14 I II III 5% I 0, 1, 3, 4 4 II 2, 5, 8 3 III 6, 7, 9, 10, 11 5 I N(μ 1,σ 2 ) II N(μ 2,σ 2 ) III N(μ 3,σ 2 ) H 0, H 1 H 0 : μ 1 = μ 2 = μ 3 H 1 : 1 (6.24) S b (6.26) S w { ( k ni ) 2 } 1 x ij = (0+1+3+4)2 + (2+5+8)2 4 3 i=1 1 n n i j=1 { k } n 2 i x ij = i=1 j=1 + (6+7+9+10+11)2 5 = 460.8 {(0+1+3+4) + (2+5+8) + (6+7+9+10+11)}2 12 = 363 n k i x 2 ij =(0 2 +1 2 +3 2 +4 2 )+(2 2 +5 2 +8 2 ) i=1 j=1 S b, S w +(6 2 +7 2 +9 2 +10 2 +11 2 ) = 506 S b = 460.8 363 = 97.8, S w = 506 460.8 =45.2 2 4 F 97.8 2(= 3 1) 48.9 45.2 9(= 12 3) 5.022 143 11 9.737 5 F 0 9.737 > 4.2565 (2, 9) F 5% H 0 k =2 (6.17) (6.28) T 2 = F 1 2
124 6 σ1 2 = σ2 2 6.13 6.11 X, Y 2 X Y 2 (x i,y i )(i =1, 2,...,n) r r r ρ 0 6.17 H 0 : ρ =0 H 1 : ρ \= 0 H 0 (ρ =0) T T = r n 2 1 r 2 H 1 : ρ>0 ρ<0 (6.29) ν = n 2 t 1 (3.1) (3.2) r t 0 t 0 = r n 2 (6.30) 1 r 2 2 H 1 α t 0 (n 2) t α H 0 t 0 < (n 2) t α H 0 H 1 α H 1 (ρ>0) : t 0 (n 2) t α H 1 (ρ<0) : t 0 (n 2) t α H 0 H 0
125 6.15 19 X Y r r =0.501 5% X Y ρ H 0 : ρ =0 H 1 : ρ \= 0 1 (6.30) n = 19, r =0.501 t 0 t 0 = 0.501 19 2 1 0.501 2 2.387 2 t 0 > 2.1098 17 t 5% H 0 6.1 A 600 1 124 2 76 3 80 4 102 5 108 6 110 A 5% 6.2 22 2010 20 1 2 50 59 5% 312 289 601 422 261 683 734 550 1 284 6.3 A, B 30 A B (1) (3) 5% (1) χ 2 (2) χ 2 (3)
126 6 B B A 2 13 15 A 8 7 15 10 20 30 6.4 50 2 A A 5% A 16 34 50 A 8 42 50 24 76 100 6.5 A, B 2 A 200 120 B 300 150 A, B 2 5% 6.6 22 2010 1 071 304 550 742 520 562 1 5% 6.7 50 150 1:1 150 20 20 (1) (2) 5% 14 36 50 20 80 100 34 116 150 6.8 30 5% 6 2 8 10 12 22 16 14 30 6.9 A 2 [cm] 169.2 A 30 = 172.0
1953 2003 1982 2002 2005 2012 2 C 2013 1987 9 11 1 1 2012 6 27 1 24 2013 10 25 2 1 1 4 11 102 0071 03 3265 8341 FAX 03 3264 8709 http://www.morikita.co.jp/ Printed in Japan ISBN978 4 627 09192 4