(2015 ) 1 NHK 2012 5 28 2013 7 3 2014 9 17 2015 4 8!? New York Times 2009 8 5 For Today s Graduate, Just Oe Word: Statistics Google Hal Varia I keep sayig that the sexy job i the ext 10 years will be statisticias. 2011 1903 SF H.G. Wells Statistical thikig will oe day be as ecessary for efficiet citizeship as the ability to read ad write. Advaced Theory of Statistics M. 2 a) b) c) d) 17 e) f ) g) h) i) j ) χ 2 1
populatio sample II, B II? http://www.med.oita-u.ac.jp/is/eshimahome.htm [1] I. [2] 1 [3] David J. Had [4] 2 [5] 3 2
2 variable descriptive statistics A, B, C A, B, C 1 0 100 1 100 * =1: 0: 1 72 76 1 26 72 94 1 51 53 83 0 76 59 58 0 2 40 59 1 27 66 73 1 52 49 61 0 77 33 44 0 3 87 94 1 28 31 44 1 53 71 69 0 78 67 73 0 4 66 60 1 29 43 69 1 54 64 94 0 79 47 62 0 5 75 66 1 30 37 46 1 55 80 100 0 80 58 77 0 6 61 87 1 31 56 72 1 56 61 55 0 81 42 78 0 7 30 40 1 32 62 61 1 57 31 58 0 82 57 69 0 8 84 90 1 33 57 65 1 58 63 64 0 83 50 82 0 9 65 83 1 34 70 86 1 59 67 86 0 84 49 45 0 10 76 80 1 35 64 89 1 60 65 76 0 85 75 74 0 11 73 84 1 36 41 67 1 61 32 53 0 86 59 69 0 12 58 72 1 37 37 45 1 62 76 93 0 87 69 81 0 13 53 86 1 38 58 85 1 63 57 52 0 88 80 76 0 14 55 78 1 39 52 68 1 64 66 83 0 89 66 72 0 15 48 49 1 40 87 72 1 65 61 71 0 90 60 53 0 16 47 56 1 41 29 70 1 66 48 48 0 91 71 82 0 17 39 68 1 42 70 94 1 67 78 75 0 92 48 44 0 18 44 59 1 43 58 59 1 68 58 67 0 93 71 75 0 19 65 71 1 44 72 66 1 69 60 61 0 94 47 65 0 20 67 77 1 45 83 69 1 70 63 71 0 95 70 60 0 21 60 54 1 46 85 74 1 71 64 81 0 96 84 70 0 22 55 47 1 47 52 48 1 72 54 50 0 97 44 73 0 23 31 64 1 48 67 84 1 73 74 90 0 98 68 77 0 24 57 78 1 49 52 54 1 74 70 88 0 99 73 42 0 25 40 37 1 50 68 53 1 75 87 76 0 100 82 73 0 2.1 1 50 2 class frequecy 2 4 relative frequecy 5 6 cumulative frequecy cumulative relative frequecy 3
2 50 0 10 5 0 0.00 0 0.00 10 20 15 0 0.00 0 0.00 20 30 25 1 0.02 1 0.02 30 40 35 6 0.12 7 0.14 40 50 45 7 0.14 14 0.28 50 60 55 12 0.24 26 0.52 60 70 65 11 0.22 37 0.74 70 80 75 8 0.16 45 0.90 80 90 85 5 0.10 50 1.00 90 100 95 0 0.00 50 1.00 histogram 1 Frequecy 0 2 4 6 8 10 12 0 20 40 60 80 100 Test score 1 50 2.1. 50 2.2. 2.2 averages mea 4
arithmetic mea *1 x 1, x 2,, x x = x 1 + x 2 + + x = 1 i=1 x i 50 58.34 2 x = 5 0 + 15 0 + 25 1 + 35 6 + 45 7 + 55 12 + 65 11 + 75 8 + 85 5 + 95 0 50 = 59.00 (1, 1, 1, 1, 2, 3, 4, 5, 16, 20) 5.4 10 8 2 media x (1), x (2),, x () = 2m + 1 m + 1 x (m+1) = 2m m m + 1 (x (m) + x (m+1) )/2 = 10 5 x (5) = 2 6 x (6) = 3 2.5 50 58 mode 2 50 60 12 55 2 1 2.3. 50 2.1 2.4. 2.2 2.3 2 rage 50 29 87 58 *1 x G x H x G = x 1 x 2 x, ( ) 1 = 1 1 + + 1 x H x 1 x 5
Iter Quartile Rage (IQR) 50% 100p%(0 p 1) 100p quartile 4 3 1 Q 1 25% 2 Q 2 50% 3 Q 3 75% Q 3 Q 1 50 Q 1 = 47.25, Q 3 = 69.5 IQR= 22.25 variace stadard deviatio x 1, x 2,, x x i x (i = 1, 2,, ) 2 S 2 S 2 = 1 { (x1 x) 2 + (x 2 x) 2 + + (x x) 2} = 1 S 2 = 1 * 2 (x i x) 2 = 1 i=1 x 2 i x 2 i=1 (x i x) 2 2 S S = S 2 = 1 (x i x) 2 50 240.7 15.5 2.5. 50 2.6. i=1 i=1 2.4 box-ad-whisker plot 2 100 Q 1 Q 3 1.5 2.7. 2.8. 2.5 x i (i = 1, 2,, ) x S z i = x i x S *2 1/ 1/( 1) 6
0 20 40 60 80 100 2 100 z i (i = 1,, 2,, ) 0 1 z z 50 10 50 + 10z i 100 59.98 14.46 1 72 59.98 50 + 10 = 58.3 14.46 2.6 2 cotigecy table cross table 100 60 60 3 2 2 3 24 26 50 30 20 50 54 46 100 i(= 1,, ) 1 (x i, y i ) x y x y scatter plot 2 50 3 2 2 7
3 50 correlatio coefficiet (x 1, y 1 ), (x 2, y 2 ),, (x, y ) x y covariace S xy = 1 (x i x)(y i ȳ) i=1 x y ( x, ȳ) (x i x)(y i ȳ) x i, y i 1 3 x i, y i 2 4 (x i, y i ) 1 3 2 4 4 x S x y S y r r = S xy S x S y = 1 1 i=1 (x i x)(y i ȳ) i=1 (x i x) 2 1 i=1 (y i ȳ) 2 50 15.51 15.02 141.05 0.605 x y z i = (x i x)/s x, w i = (y i ȳ)/s y r xy = 1 z i w i 1 r xy 1 1 i=1 8
1. 2 2 2. 3. 2 x y 3 z 2.9. 50 2.10. 3 3.1 2 (1, 1) (1, 2) (1, 3) (6, 5) (6, 6) 36 1 1 36 Ω ω Ω = {(1, 1), (1, 2), (1, 3),, (6, 5), (6, 6)} evet A, B, 2 2 A A = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)} 1 φ 1 2 1 1 {1} {1, 3, 5} 5 2 A, B A B A B A, B A B A B A B A B = φ. 1 A B 3 A B = {1, 2, 3, 5} A B = {1, 3} C A C = φ A C 9
3 A, B, C (A B) C = (A C) (B C) (A B) C = (A C) (B C). 2 A 2 B 2 C 2 5 A = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)} B = {(1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3, 5), (5, 1), (5, 3), (5, 5)} C = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (4, 1)} A B = {(1, 1), (1, 3), (1, 5), (2, 2), (3, 1), (3, 3), (3, 5), (4, 4), (5, 1), (5, 3), (5, 5), (6, 6)} (A B) C = {(1, 1), (1, 3), (2, 2), (3, 1)} A C = {(1, 1), (2, 2)}, B C = {(1, 1), (1, 3), (3, 1)} (A C) (B C) = {(1, 1), (1, 3), (2, 2), (3, 1)} 1 2 A A A c φ c = Ω, Ω c = φ A A c = Ω, A A c = φ Ω A = A, φ A = A (A c ) c = A A, B (A B) c = A c B c, (A B) c = A c B c. 1 A B 3 A B = {1, 2, 3, 5} (A B) c = {4, 6} A c = {2, 4, 6}, B c = {4, 5, 6} A c B c = {4, 6} 1 2 3.2 (a), (b), (c) (a) A 0 P (A) 1 (b) P (Ω) = 1 (c) A 1, A 2, A 3, P (A 1 A 2 A 3 ) = P (A 1 ) + P (A 2 ) + P (A 3 ) + 10
1749-1827 N A R A P (A) P (A) = R N 1 1 1 1 6 A A A α P (A) = α 100 (a), (b), (c) 3 3.3 A B 3 A B c, A B, A c B A B = (A B c ) (A B) (A c B) (c) P (A B) = P (A B c ) + P (A B) + P (A c B) A A B c, A B A = (A B c ) (A B) (c) P (A) = P (A B c ) + P (A B) P (B) = P (A c B) + P (A B) P (A B) = [P (A) P (A B)] + P (A B) + [P (B) P (A B)] = P (A) + P (B) P (A B) P (A B) = P (A) + P (B) P (A B) A B A B = φ P (A B) = P (A) + P (B) 11
A A c = φ, Ω = A A c (a), (c) 1 = P (Ω) = P (A) + P (A c ) A = Ω A c = φ P (φ) = 0 1. 1 A B 3 A = {1, 3, 5}, B = {1, 2, 3} P (A) = 3/6 = 1/2, P (B) = 3/6 = 1/2 A B = {1, 3} P (A B) = 2/6 = 1/3 P (A B) = P (A) + P (B) P (A B) = 1 2 + 1 2 1 3 = 2 3 A B = {1, 2, 3, 5} P (A B) = 4/6 = 2/3 2. 2 A B 5 A = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}, B = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (4, 1)} P (A) = 6/36 = 1/6, P (B) = 10/36 = 5/18 A B = {(1, 1), (2, 2)} P (A B) = 2/36 = 1/18 P (A B) = P (A) + P (B) P (A B) = 6 36 + 10 36 2 36 = 14 36 = 7 18 A B = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3), (4, 1), (4, 4), (5, 5), (6, 6)} P (A B) = 14/36 = 7/18 3.1. 4 100 1 A B 4 49 24 73 5 22 27 54 46 100 3.4 1 5 5 1 3 A B P (A) = 1/5 3 1/3 B A P (A B) B B A 1, 3, 5 3 12
P (A B) B A P (A B) = P (A B) P (B) P (B) = 3/5, P (A B) = 1/5 P (A B) = 1/5 3/5 = 1 3 P (A B) = P (B)P (A B) A B. 3 10 1 2 1 1 A 2 B 2 A B 1 P (A) = 3/10 1 2 P (B A) = 2/9 2 P (A B) = P (A)P (B A) = 3 2 10 9 = 1 15 A B P (A B) = P (A)P (B) A B P (A) = P (A B) P (B) = P (B A) A B. 2 1 1 1 A 2 1 B 1 A B P (A B) = 1/36 P (A)P (B) = (1/6) (1/6) = 1/36 A B 3.2. (i), (ii) (i) B 1 B 2 = φ P (B 1 B 2 A) = P (B 1 A) + P (B 2 A) (ii) P (B A) + P (B c A) = 1 3.3. 2 (i) A 6 B 4 A B (ii) C 7 B C 3.4. 3.1 P (A B) P (B A) 3.5 A = A Ω = A (B B c ) = (A B) (A B c ) A B A B c (c) P (A) = P (A B) + P (A B c ) 13
P (A) = P (A B)P (B) + P (A B c )P (B c ) P (B A) = P (A B) P (A) = P (A B)P (B) P (A B)P (B) + P (A B c )P (B c ) 1. 2 1 3 1 2 1 2 1 A B 1 B c 2 P (B) = P (B c ) = 1/2 P (A B) = 3/4, P (A B c ) = 1/3 P (B A) = (3/4)(1/2) (3/4)(1/2) + (1/3)(1/2) = 9 13, P (1/3)(1/2) (Bc A) = (1/3)(1/2) + (3/4)(1/2) = 4 13 P (B c A) P (B A) + P (B c A) = 1 2. A B 95 90 P (A B) = 0.95, P (A c B c ) = 0.90 P (B) = 0.005 P (B A) P (B A) = 0.95 0.005 0.95 0.005 + (1 0.90) (1 0.005) = 0.046 3.5. 2012 10 30 3.11 (1) (2) 99% 99% 99% 99% 99.1% 99.9% ) sesitivity specificity (1) 3 1% (2) 2% 1000 1 35 300 1 14
4 4.1 1 6 1/6 radom variable X, Y, X P (X = 1) = 1/6, P (X = 2) = 1/6,, P (X = 6) = 1/6 X x X X. 2 X X 2, 3,, 12 P (X = 2) = 1/36, P (X = 3) = 2/36, P (X = 4) = 3/36, P (X = 5) = 4/36, P (X = 6) = 5/36, P (X = 7) = 6/36, P (X = 8) = 5/36, P (X = 9) = 4/36, P (X = 10) = 3/36, P (X = 11) = 2/36, P (X = 12) = 1/36 X 6 X x 1, x 2,, x K f(x 1 ), f(x 2 ),, f(x K ) X f(x k ) = P (X = x k ) (k = 1,, K) X f f(x k ) 0 K f(x k ) = 1 k=1 X 1 X a b P (a X b) = X f b a f(x)dx f(x) 0 15 f(x)dx = 1
X P (X = a) = 0 0 X x F (x) = P (X x) X cumulative distributio fuctio F (x) = x f(u)du F (x) = f(x) F (X) = u x f(u) 4.2 expectatio E(X) E(X) = xf(x), E(X) = x xf(x)dx x x X g(x) E(g(X)) = g(x)f(x), x E(g(X)) = a, b g(x)f(x)dx E(aX + b) = ae(x) + b a = 0 E(b) = b b = 0 E(aX) = ae(x) X x 1,, x K K K K K E(aX + b) = (ax i + b)f(x i ) = a x i f(x i ) + b f(x i ) = ae(x) + b i=1 1. 1 X i=1 E(X) = 1 1 6 + 2 1 6 + 3 1 6 + 4 1 6 + 5 1 6 + 6 1 6 = 21 6 = 7 2 1 7/2 X 2 E(X 2 ) = 1 2 1 6 + 22 1 6 + 32 1 6 + 42 1 6 + 52 1 6 + 62 1 6 = 91 6 2. 3 X k 3C k (1/2) k (1/2) 3 k = 3 C k (1/2) 3 (k = 0, 1, 2, 3) ( ) 3 ( 1 1 E(X) = 0 3 C 0 + 1 3 C 1 2 2 ) 3 + 2 3 C 2 ( 1 2 i=1 ) 3 ( ) 3 1 + 3 3 C 3 = 3 2 2 1 3 3/2 4.1. 4 3 3 X X 16
4.3 1 1 7/2 X µ = E(X) variace V (X) = E[(X µ) 2 ] V (X) 0 V (X) X V (X) stadard deviatio σ 2 σ V (X) = x (x µ) 2 f(x), V (X) = (x µ) 2 f(x)dx V (X) = E(X 2 ) 2µE(X) + µ 2 = E(X 2 ) µ 2 = E(X 2 ) (E(X)) 2 a, b V (ax + b) = a 2 V (X) a = 0 V (b) = 0 X x 1,, x K K V (ax + b) = K {(ax i + b) (aµ + b)} 2 f(x i ) = a 2 i=1 1. 1 E(X) = 7/2, E(X 2 ) = 91/6 2. 2 E(X 2 ) = 0 2 3 C 0 ( 1 2 ) 3 + 1 2 3 C 1 ( 1 2 V (X) = 91 6 K i=1 ( ) 2 7 = 35 2 12 ) 3 + 2 2 3 C 2 ( 1 2 (x i µ) 2 f(x i ) = a 2 V (X) ) 3 ( ) 3 1 + 3 2 3 C 3 = 3 + 12 + 9 = 3 2 8 V (X) = 3 ( ) 2 3 = 3 2 4 4.2. 4.1 X 4.3. X { 1 (0 x 1) f(x) = 0 (x < 0, 1 < x) 17
4.4 1 2 2 52 1 X 1 Y X, Y 0, 1 X = x, Y = y f(x, y) = P (X = x, Y = y) f(0, 0) = 39 38 13 39 39 13 13 12, f(1, 0) =, f(0, 1) =, f(1, 1) = 2 X, Y 52 51 52 51 52 51 52 51 X Y f(x, y) X, Y f(x, y) 0 f(x, y) = 1 x y f(0, 0), f(1, 0), f(0, 1), f(1, 1) 0 f(x, y) = f(0, 0) + f(1, 0) + f(0, 1) + f(1, 1) = 1 x=0,1 y=0,1 X, Y f(x, y) f(x, y) 0 f(x, y)dxdy = 1 X = 0 P (X = 0) = P (X = 0, Y = 0) + P (X = 0, Y = 1) = f(0, 0) + f(0, 1) X = 1 Y = 0 Y = 1 g(x) = P (X = x), h(y) = P (Y = y) g(x) = f(x, 0) + f(x, 1), h(y) = f(0, y) + f(1, y) X, Y X, Y g(x) = y f(x, y), h(y) = x f(x, y) X, Y g(x) = X, Y P (X = x Y = y) = f, g, h f(x, y)dy, h(y) = f(x, y)dx P (X = x, Y = y) P (X = x, Y = y), P (Y = y X = x) = P (Y = y) P (X = x) g(x y) = f(x, y) f(x, y), h(y x) = h(y) g(x) h(y) 0, g(x) 0 g(x y) y x h(y x) x y 18
x, y f(x, y) = g(x)h(y) X Y X, Y X, Y g(x y) = g(x)h(y) h(y) = g(x), h(y x) = g(x)h(y) g(x) = h(y) X Y Y X 2 X, Y X, Y 5 2 5 2 X, Y X 1 2 3 4 5 6 h(y) 1 1/36 1/36 1/36 1/36 1/36 1/36 1/6 2 1/36 1/36 1/36 1/36 1/36 1/36 1/6 Y 3 1/36 1/36 1/36 1/36 1/36 1/36 1/6 4 1/36 1/36 1/36 1/36 1/36 1/36 1/6 5 1/36 1/36 1/36 1/36 1/36 1/36 1/6 6 1/36 1/36 1/36 1/36 1/36 1/36 1/6 g(x) 1/6 1/6 1/6 1/6 1/6 1/6 1 5 5.1 (a), (b), (c) (a) 2 (b) p (c) X x F x f(x) = P (X = x) = C x p x (1 p) x (x = 0, 1,, ) biomial distributio B(, p) C x p x (1 p) x = (p + (1 p)) = 1 x=0 1. 10 X 2 0.5 X B(10, 0.5) f(x) 7 f(x) = 10 C x 0.5 x 0.5 10 x (x = 0, 1,, 10) 19
2. 10 1 X 1 1 2 1/6 1 1 X B(10, 1/6) f(x) 8 f(x) = 10 C x ( 1 6 ) x ( ) 10 x 5 (x = 0, 1,, 10) 6 7 B(10, 0.5) 8 B(10, 1/6) X B(, p) E(X) = p, V (X) = p(1 p) 5.1. B(, p) X E(X) = p, V (X) p(1 p) 5.2 p p x f(x) = C x p x (1 p) x p = λ, p 0 f(x) = e λ λ x x! (x = 0, 1, 2, ) Poisso distributio P o(λ) e x e x = 1 + x + x2 2! + x3 3! + = e λ λ x = e λ x! x=0 X P o(λ) x=0 r=0 x r r! λ x x! = e λ e λ = 1 E(X) = λ, V (X) = λ P o(λ) λ B(, p) E(X) = p, V (X) = p(1 p) 5.2. 4 X λ = 2 P o(2) 20
5.3 ormal distributio { } 1 f(x) = exp (x µ)2 2πσ 2 2σ 2 ( < x < ) σ > 0 E(X) = µ, V (X) = σ 2 µ σ 2 N(µ, σ 2 ) N(µ, σ 2 ) x = µ x = µ x ± f(x) 0 µ x σ 2 9 9 N(20, 2 2 ), N(20, 4 2 ), N(20, 6 2 ) X N(µ, σ 2 ) X Z = X µ N(0, 1) 0 1 σ stadard ormal distributio f(z) Φ(z) f(z) = 1 2π exp ) z ( z2, Φ(z) = 2 Φ( z) = 1 Φ(z) 1 2π exp ) ( x2 dx 2 Φ(z) Excel Φ(1.645) 0.95, Φ(1.96) 0.975 Z = X µ N(0, 1) Z k k (k > 0) σ P ( k Z k) = P (Z k) P (Z < k) = Φ(k) Φ( k) k = 2 P ( 2 Z 2) = Φ(2) Φ( 2) 0.975 (1 0.975) = 0.95 Z 95% 2 2 X X 95% µ 2σ µ + 2σ 1. X N(9, 25) P (3 X 21) Z = X 9 5 N(0, 1) ( 3 9 P (3 X 21) = P X 9 21 9 ) = P ( 1.2 Z 2.4) = Φ(2.4) Φ( 1.2) 5 5 5 z = 2.4 Φ(2.4) 0.5 + 0.4918 = 0.9918. Φ( 1.2) = 1 Φ(1.2) Φ(1.2) 0.5 + 0.3849 = 0.8849 Φ( 1.2) 1 0.8849 = 0.1151. P (3 X 21) = 0.9918 0.1151 = 0.8767 21
2. 17 N(172, 4 2 ) 17 1 180cm 17 % X X N(172, 4 4 ) P (X 180) ( ) X 172 P (X 180) = P 2 = 1 Φ(2) 1 0.975 = 0.025 4 17 180cm 2.5% 5.3. T 50 10 T N(50, 10 2 ) 40 T 60, T 70, T 75, T 55, 50 T 51 5.4. 25 20 257 BMI (Body Mass Idex, kg/ m 2 ) 22.65(kg/m 2 ) 4.01(kg/m 2 ) 20 BMI N(22.65, 4.01 2 ) 20 BMI 25(kg/m 2 ) % B(, p) X N(p, p(1 p)) B(, p) N(p, p(1 p)) 1 X B(, 1/6) /6, 5/36 10, 20, 30, 50, 100 B(, 1/6) 10 B(, 1/6) X B(, p) ( ) a p P (X a) Φ p(1 p) p > 5 (1 p) > 5 p = 1/2 10 p 0 1. 720 1 X X 100 X B(720, 1/6) 120 100 Z = X 120 10 ( ) X 120 P (X 100) = P 2 = P (Z 2) 0.025 10 0.0228 5.5. < > 2 8 450 3 [1] 22
6 6.1 populatio sample X 1, X 2,, X X 1, X 2,, X X 1 X 1 X 2 *3 X 1, X 2,, X f(x) sample size (i) (ii) 2 (i) X 1 X 2,, X µ σ (ii) 6.2 1936 2 237 2 100 1 1000 1000 24% *3 23
6.3 µ σ 2 µ = xf(x), σ 2 = (x µ) 2 f(x) x x µ = xf(x)dx, σ 2 = (x µ) 2 f(x)dx µ σ 2 X 1, X 2, X sample mea sample variace X 1, X 2,, X f(x) X = X 1 + X 2 + + X X, Y E(X + Y ) = E(X) + E(Y ) E( X) = E(X 1) + E(X 2 ) + + E(X ) = 1 i=1 X i = µ + µ + + µ = µ = µ X, Y V (X +Y ) = V (X)+V (Y ) V ( X) = 1 2 V (X 1 + X 2 + + X ) = σ2 + σ 2 + + σ 2 X 0 N X µ = N µ s 2 = 1 1 {(X 1 X) 2 + (X 2 X) 2 + + (X X) } = 1 1 2.3. S 2 = 1 {(X 1 X) 2 + (X 2 X) 2 + + (X X) } 2 = σ2 (X i X) 2 s 2 1 E(s 2 ) = σ 2 s 2 σ 2 S 2 E(S 2 ) = 1 σ2 s 2 S 2 = 10 1 i=1 6.4 X X/ X/ 0.5 11 10 10000 10 X/ 0.5 24
11 6.5 5.3. X 1 + + X µ, σ 2 X i (i = 1,, ) X 1 + + X N(µ, σ 2 ) X = (X 1 + + X )/ N(µ, σ 2 /) B(, p) i (i = 1,, ) 1 0 X i X i B(1, p) S S = X 1 + + X E(X i ) = p, V (X i ) = p(1 p) S N(p, p(1 p)) 7 7.1 µ σ 2 µ σ 2 θ ˆθ µ ˆµ = X 1 + + X X 1 = x 1, X 2 = x 2,, X = x θ ˆθ X µ s 2 σ 2 25
p x X X B(, p) p P (X = x) = C x p x (1 p) x p L(p) = C x p x (1 p) x p p L(p) dl(p)/dp = 0 p ˆp = x/ L(p) log L(p) d log L(p) = 1 dl(p) dl(p) = 0 dp L(p) dp dp d log L(p) = 0 dp 1 N(µ, σ 2 ) µ 1. ˆθ θ ˆθ θ E(ˆθ) = θ X = (X 1 + + X )/ µ s 2 σ 2 1/ S 2 2. θ 7.2 θ 1 α P (L θ U) 1 α L, U L, U 1 α [L, U] 100(1 α)% cofidece iterval α 0.01, 0.05, 0.1 99% 95% 90% 100 100 100 95% 95% 100 95% 95 θ 90% 100 90% 90 θ µ p 95% 26
7.2.1 µ N(µ, σ 2 ) X 1, X 2,, X µ i) X N(µ, σ 2 ) ax + b N(aµ + b, a 2 σ 2 ) ii) X, Y N(µ 1, σ 2 1), N(µ 2, σ 2 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 1, X 2,, X N(µ, σ 2 ) ii) X 1 + X 2 + + X N(µ, σ 2 ) X = (X 1 + X 2 + + X )/ i) N(µ, σ 2 /) X σ/ (a) X µ (b) 1/ X µ 2 σ/ 1/2 4 10 σ/ 1/10 100 X N(µ, σ 2 /) X Z Z = X µ σ/ Z N(0, 1) σ 2 X N(0, 1) N(0, 1) Φ(z) = 1 α z Z α Φ(z) = 0.975 z Z 0.025 1.96 Φ(z) = 0.95 z Z 0.05 1.65 P ( Z α/2 X ) µ σ/ Z α/2 = 1 α µ ( ) σ P X Z α/2 µ X σ + Z α/2 = 1 α σ 2 µ 1 α 100(1 α)% [ ] σ σ X Z α/2, X + Zα/2 95% [ X 1.96 σ, X + 1.96 σ ] * σ 2 s 2 = 1 i=1 1 (X i X) 2 ( X µ)/s 1 t t σ 2 1. 2. - - 1-50 240.7 1 27
5 X = (72 + 40 + 87 + 66 + 75)/5 = 68 95% [ X 1.96 σ ] [ σ 240.7, X + 1.96 = 68 1.96, 68 + 1.96 5 ] 240.7 = [54.4, 81.6] 5 2. 2. - - 1-50 = 50 50 58.34 245.6 95% [ X 1.96 σ ] [ σ 245.6, X + 1.96 = 58.34 1.96, 58.34 + 1.96 50 ( ) t 49 t [53.9, 62.8] ] 245.6 = [54.0, 62.7] 50 7.1. 25 20 257 BMI ( kg/ m 2 ) 22.65(kg/m 2 ) 4.01(kg/m 2 ) 20 BMI 95% 7.2. σ 2 = 9 µ 95% 1 7.2.2 p B(1, p) X 1, X 2,, X p p ˆp = X = i X i/ i X i B(, p) p i X i N(p, p(1 p)) ( i X i p)/ p(1 p) P ( Z α/2 i X i p p(1 p) Z α/2 ) 1 α ˆp = i X i/ p ( p(1 p) P ˆp Z α/2 ) p(1 p) p ˆp + Z α/2 1 α p ˆp p P ( ˆp Z α/2 ˆp(1 ˆp) ) ˆp(1 ˆp) p ˆp + Z α/2 1 α p 100(1 α)% [ ] ˆp(1 ˆp) ˆp(1 ˆp) ˆp Z α/2, ˆp + Z α/2 7.3. B(1, p) = 100 54 p 95% 7.4. 100 40 A 100 A p 95% 95% 40% ±5% 28
8 8.1 0.5 10 0.5 1 0.5 0.5 X B(10, 0.5) 1 P (X = 1) = 10 C 1 0.5 1 0.5 9 = 0.0098 1 P (X 1) = P (X = 0) + P (X = 1) = 0.0107 0.5 1 1.07% 1.07% 0.5 hypothesis testig 8.2 1) H 0 H 1 α 2) p 3) p < α H 0 H 1 sigificat p α 10 p 0.5 H 0 : p = 0.5 ( p 0.5) H 1 : p < 0.5 H 0 1.07% α 5% 10% p 10 p P (X 1 H 0 ) = 0.0107 29
p α p α 10 α = 0.05 0.0107 < 0.05 0.5 α = 0.01 0.0107 > 0.01 0.5 1 µ µ 0 α σ 2 H 0 : µ = µ 0 H 1 : µ µ 0 H 1 : µ > µ 0 H 1 : µ < µ 0 (X 1, X 2,, X ) X N(µ 0, σ 2 /) Z = X µ 0 σ/ N(0, 1) Z Z 0 0 α Z > Z α/2 Z Z α/2 Z z p P ( Z z H 0 ) α X * σ 2 s 2 µ 0 s/ 1 t t t σ 2 1. 2. - - 1-50 240.7 µ = 58.34 5 µ 50 5% H 0 : µ = 50 H 1 : µ 50 1 5 X = (72 + 40 + 87 + 66 + 75)/5 = 68 z = 68 50 240.7/5 2.594 5% Z α/2 = Z 0.05/2 = 1.96 z = 2.594 > 1.96 p P ( Z 2.594) = P (Z 2.594, Z 2.594) = P (Z 2.594) + P (Z 2.594) = Φ( 2.594) + 1 Φ(2.594) = 2[1 Φ(2.594)] 2(1 0.9952) = 0.0096 µ 50 2. 25 20 257 BMI (Body Mass Idex, kg/ m 2 ) 22.65(kg/m 2 ) 4.01(kg/m 2 ) 20 BMI 23kg/m 2 5% = 257 z = 22.65 23 4.01/ 257 1.399 30
z = 1.399 < 1.96 p P ( Z 1.399) = P (Z 1.399, Z 1.399) = P (Z 1.399) + P (Z 1.399) = Φ( 1.399) + 1 Φ(1.399) = 2[1 Φ(1.399)] 2(1 0.91924) = 0.16152 5% BMI 23kg/m 2 * 2 2 BMI 23kg/m 2 23kg/m 2 8.1. 2. - - 100 = 100 60 5% 69.01 211.95 1 10 p p p 0 α H 0 : p = p 0 H 1 : p p 0 (X 1, X 2,, X ) X i (i = 1,, ) 0 1 B(1, p 0 ) S = X 1 + + X B(, p 0 ) N(p 0, p 0 (1 p 0 )) Z = S p 0 p0 (1 p 0 ) = ˆp p 0 p0 (1 p 0 )/ ( ˆp = S ) N(0, 1). 50% 100 5% H 0 : p = 0.5 H 1 : p 0.5 60 p Z = 0.6 0.5 0.5 0.5/100 = 2.0 P ( Z 2.0) = 2[1 Φ(2.0)] 2(1 0.97725) = 0.0455 60/100 = 0.6 8.2. 2. - - 100 60 54 = 100 0.5 5% 31
8.3 2 α 2 i) 1 (α ): ii) 2 (β ): 2 2 α 1 α 5% 1 5% p α 1 α 2 2 2 2 2 1 β power 32