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1 2012
2 i
3 ii RDD RDD RDD
4 iii R Excel
5 iv p Yates Fisher R Excel CO GDP
6 v t t t t t t t
7 U U 1 1 U = {1, 2, 3, 4, 5, 6} A A = {2, 4, 6} 1 A A P (A) A P (A) P (A) = n(a) n(u) n(u), n(a) U A 1 1 A P (A) = 3 6 = 1 2 ( ) 1. A 0 P (A) 1 U P (U) = 1 P ( ) = 0 2. A, B A, B A B A, B 1
8 2 1 A B P (A B) = P (A) + P (B) P (A B) A, B A, B P (A B) = P (A) + P (B) 3. A A A P (A) = 1 P (A) ( ) 1. A 0 P (A) 1 2. A 1, A 2, A 3, P (A 1 A 2 A 3 ) = P (A 1 ) + P (A 2 ) + P (A 3 ) (100 ) 1 48 (40 )
9 A B A B A B P A (B) 1 P B (A) A, B A A B n(a B) n(a B) n(b) B n(a B) n(a B) n(b) n(a) n(a) n(u) P A (B) = n(a B) n(a) = n(a B) n(u) n(a) n(u) = P (A B) P (A) P A (B) = P (A B) P (A) 2 60% 42% ( ) 1
10 4 1 ( ) A B P (A B) P (A) = P A (B) = 2 4 P (A B) = P (A) P A (B) = = (1) (2) 4 ( ) 2 3 a,b,c a b c 1 ( ) = 6 A B P A (B) A a 2 c 1 A B c 1 P (A) = = 1 6 2, P (A B) = 1 6 P A(B) = P (A B)P (A) = = ( ) 1.3 A B P A (B) = P (B) P B (A) = P (A) B A 2 A, B 2
11 P (A B) = P (A) P (B) A 2 B (1) 1 2 ) (2) 1 2 ) (1) P (A B) = = = P (A)P (B) 10 A, B (2) P (A B) = = , P (A B) = = P (B) = P (A B) + P (A B) = = 7 10 P (A B) = 42 ( ) = P (A)P (B) 10 A, B ( ) k A k P (A 1 ), P (A 2 ), P (A 3 ) P (A 1 ) = 3/7
12 6 1 P (A 2 ) A 1 A 2, A 1 A 2 P (A 1 A 2 ) = P (A 1 ) P A1 (A 2 ) = = 1 7 P (A 1 A 2 ) = P (A 1 ) P A1 (A 2 ) = = 2 7 P (A 2 ) = P (A 3 ) A 1 A 2 A 3, A 1 A 2 A 3, A 1 A 2 A 3, A 1 A 2 A 3 P (A 3 ) = P (A 1 A 2 A 3 ) + P (A 1 A 2 A 3 ) + P (A 1 A 2 A 3 ) + P (A 1 A 2 A 3 ) P (A 1 A 2 A 3 ) = P (A 1 A 2 ) P A1 A 2 (A 3 ) = P (A 1 ) P A1 (A 2 ) P A1 A 2 (A 3 ) = = 1 35 P (A 1 A 2 A 3 ) = P (A 1 ) P A1 (A 2 ) P A1 A 2 (A 3 ) = = 4 35 P (A 1 A 2 A 3 ) = P (A 1 ) P A1 (A 2 ) P A1 A 2 (A 3 ) = = 4 35 P (A 1 A 2 A 3 ) = P (A 1 ) P A1 (A 2 ) P A1 A 2 (A 3 ) = = 6 35 P (A 3 ) = = A 3 2 B 2 3 A 3 B B 1
13 X P (X = 0), P (X = 1), P (X = 2), P (X = 3) bwr 3! = 6 rbw, rwb, brw, bwr, wrb, wbr X 3, 1, 1, 0, 0, X X X k x 1, x 2,, x k i = 1, 2,, k X = x i p i x i p i 1 X x 1 x 2 x k p 1 p 2 p k X p 1 + p p k = 1
14 X X = = = = = = 99 09, 19, 29,, = X X p
15 p X X = = = = = X X X 1 E(X) = x 1 p 1 + x 2 p x k p k
16 10 2 X expectation 1 1 X E(X) = = X E(X) = ( ) ( ) ( ) ( ) p = = = = 46% X 2 1 X
17 X 1 m = E(X) X m V (X) = E((X m) 2 ) = k (x i m) 2 p i i=1 X variance m X m 2 (X m) 2 X m E(X m) = k (x i m)p i = i=1 k x i p i m i=1 k p i = m m 1 = 0 i=1 V (X) (X m) 2 X cm cm 2 m = E(X) cm σ(x) = V (X) standard deviation 1 1 X V (X) = (0 1) (1 1) (2 1) (3 1)2 1 6 = 1 2 A B 100
18 12 2 A B A 1 X B 1 Y E(X) = E(Y ) = 50 V (X) = ( ) 2 V (Y ) = (200 50) 2 = (150 50) ( ) = (50 50)2 σ(x) : σ(y ) = : 30 5 = (0 50)2 A B X X 2 2 X X X X 2.4 m = E(X) V (X) = E(X 2 ) m 2
19 V (X) = = k k (x i m) 2 p i = (x 2 i 2mx i + m 2 )p i i=1 k x 2 i p i 2m i=1 k x i p i + m 2 k i=1 i=1 i=1 = E(X 2 ) 2m m + m 2 1 = E(X 2 ) m 2 p i a, b E(aX + b) = ae(x) + b k k E(aX + b) = (ax i + b)p i = a x i p i + b i=1 i=1 = a E(X) + b 1 = ae(x) + b k i=1 p i a V (ax) = a 2 V (X) m = E(X) E(aX) = ae(x) = am V (ax) = E((aX) 2 ) (am) 2 = a 2 E(X 2 ) a 2 m 2 = a 2 (E(X 2 ) m 2 ) = a 2 V (X) X E(X 2 ) = ( ) (108 ) (105 ) (107 ) ( ) (10 6 ) (104 ) ( ) ( ) p = = 138 5
20 BIG J1 J , 2, , 2, % 14 78% 1 10% 2 4% 3 4% 4 4% BIG 1 X
21 X x 1, x 2,, x k P (X = x i ) E(X) = x 1 P (X = x 1 ) + x 2 P (X = x 2 ) + + x k P (X = x k ) A X = x i P A (X = x i ) A E A (X) E A (X) = x 1 P A (X = x 1 ) + x 2 P A (X = x 2 ) + + x k P A (X = x k ) A 1, A 2, A 3, A 1 A 2 A 3 P (B) = P (A 1 )P A1 (B) + P (A 2 )P A2 (B) + P (A 3 )P A3 (B) + A 1, A 2, A 3, A 1 A 2 A 3 E(X) = P (A 1 )E A1 (X) + P (A 2 )E A2 (X) + P (A 3 )E A3 (X) + E(X) = x i P (X = x i ) = i i x i P (A j )P Aj (X = x i ) = P (A j ) x i P Aj (X = x i ) = P (A j )E Aj (X) j i j j
22 p q p + q = 1 X X P (X = k) = q k 1 p k = 1, 2, 3, E(X) = kq k 1 p k=1 III 1 1 A A, A A A A X = 1 E A (X) = 1 A E A (X) A P A (X = k) Y P A (X = k) = P (Y = k) E A (X) = 1 + E(Y ) E(Y ) = E(X) E A (X) = 1 + E(X) E(X) = P (A) 1 + P (A) (1 + E(X)) E(X) = p + q(1 + E(X)) = 1 + qe(x) E(X) E(X) = 1 q 1
23 r w X E(X) a w 1 A A X = 1 E A (X) = 1 A 2 1 Y E A (X) = 1 + E(Y ) 1 r w 1 E(Y ) = a w 1 E A (X) = 1 + a w 1 a w = r w + r 1 + w w + r (1 + a w 1) a w = 1 + w w + r a w 1 1 a w 3.4 X a A E A (X) e a a < b a+ e a < b+ e b a = 0, b = 1 1 A P (A) = p 0 p 0 1 E(X) = P (A) E A (X) + P (A) E A (X)
24 18 3 e 0 = p (1 p 0) (1+ e 1 ) ( e 0 = ( e 1 +1) p 0 e ) < e e a < e a+1 +1 < e a+2 +2 < < e b +(b a)
25
26
27 total fertility rate b a X b,a Y b = X b,15 + X b, X b,49 5. Y b E(Y b ) = E(X b,15 ) + E(X b,16 ) + + E(X b,49 ) 1000 crude birth rate
28 a = E(X b,15 ) + E(X b,16 ) + E(X b,17 ) + E(X b,18 ) + E(X b,19 ) = = a = E(X b,20 ) + E(X b,21 ) + E(X b,22 ) + E(X b,23 ) + E(X b,24 ) = = E(X b,25 ) + E(X b,26 ) + E(X b,27 ) + E(X b,28 ) + E(X b,29 ) = E(X b,30 ) + E(X b,31 ) + E(X b,32 ) + E(X b,33 ) + E(X b,34 ) = E(X b,35 ) + E(X b,36 ) + E(X b,37 ) + E(X b,38 ) + E(X b,39 ) = = = = E(X b,40 ) + E(X b,41 ) + E(X b,42 ) + E(X b,43 ) + E(X b,44 ) = = E(X b,45 ) + E(X b,46 ) + E(X b,47 ) + E(X b,48 ) + E(X b,49 ) = = E(Y b ) = = b = 1961/7/1 1965/7/1
29 a a = 15, 16,, a a E(X 2010 a,a ) 3. E(X ,15 ) + E(X ,16 ) + + E(X ,49 ) = =
30
31 life expectancy x q x l x d x e x 0 q 0 l 0 d 0 e0 1 q 1 l 1 d 1 e1 2 q 2 l 2 d 2 e q x x x x + 1 l x d x x x l x x + 1 e x x l x Y x Y x E(Y x ) x
32 = = = e / = / = / = / = 0.09 X E(Y 0 ) = = 1.49
33 x e x = E(Y x ) Y x d x l x d x+1 l x d x+2 l x E(Y x ) = 1 2 dx + 3 l x 2 dx l x 2 dx+2 + l x = 1 [ 1 l x 2 d x d x ] 2 d x+2 + = 1 [ 1 l x 2 (l x l x+1 ) (l x+1 l x+2 ) + 5 ] 2 (l x+2 l x+3 ) + = 1 [ ] 1 l x 2 l x + l x+1 + l x+2 + l x+3 + (4.1) (4.2) (4.1) (4.2) = =
34
35 29 5 population sample random sampling , 1, 2, 3, 4, 5, 6, 7, 8, 9 20
36 , 01, 02,,
37 A,B,C,D,E 100, 200, 300, 400, ( ) A,B,C,D,E , 2 15, 3 15, 4 15, 5 15
38 A 01,02 B 03,04,05 C 06,07,08,09 D 10,11,12,13,14 E B D ( ) A ( ) 51 A 75 B , , 523
39 5.2. RDD RDD RDD Random Digit Dialing NTT RDD 1 RDD RDD RDD RDD RDD [ ]-[ ]-[ ]
40 RDD
41 5.2. RDD : : 30
42 36 5 =3000 =2000 =1000 = = 67% = = 50% n t n/t n = 1 t = 1 n/t = 1 n = 3 t = 2 n/t = 3/2 = % Graham Walden 40% 1, RDD RDD 3 (George Horace Gallup)
43 % /02/06 5
44 , % BS CS CATV
45 = RDD ( % ( PM ,20 FNN 5 19, ,27
46 % FNN FNN (2012 )
47
48 [?] 1 10 RDD [ ] IP DTMF 0 [?] ISBN
49 quality % discrete quantity
50 , , , , , , , continuous quantity 0, 1, 2, , 750 7, 559 7, , 020 7, 662 7, , 369 7, 884 7, , 635 8, 369 9, , 936 8, 265 8,
51
52 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 492 3, , , ,
53 A B C D E F , , , , , , =SUM(B1:B23) 1. A (a) A1 2.5 (b) A2 =A1+5 (c) A2 A3:A23 ] ] 2. B C D 3. B25 =SUM(B1:B23) 4. F (a) F1 =B1/B$25 $ 25 (b) F1 F2:F23 ] ] F 5. (a) F1:F23 ] ] (b) ] (c) ] A1:A23.. (1) G H (2) (3)
54 mode mean median (1) 2, 4, 7, 9, 12, 15, 17 (2) 2, 4, 7, 9, 12, 15, 17, 46, 54 (3) 4, 6, 9, 10, 11, 12 (1) (2) (2) (1) (3) 9 10 (9 + 10)/2 = (1) (2)
55 , , , , , , , % 50% = standard deviation
56 50 6 inter-quartile range Q 1 3 Q 3 1 Q 1 Q 1 1 = 25% 4 3 Q 3 Q 3 3 = 75% 4 3 Q 3 1 Q 1 2 = 50% % %
57 = = Q 1 = = Q 3 = = Q 3 Q 1 = = % % , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,
58 x 1, x 2, x 3 x = (x 1 + x 2 + x 3 )/3 x 1 x, x 2 x, x 3 x (x 1 x) + (x 2 x) + (x 3 x) = 0 (x 1 x) 2 + (x 2 x) 2 + (x 3 x) 2 n x 1, x 2, x 3,, x n n n s 2 s 2 = 1 n [ (x1 x) 2 + (x 2 x) 2 + (x 3 x) (x n x) 2] s 2 s standard deviation s 2 s cm s 2 cm 2 s 2 1 A B 5 A B B A A s 2 A = 1 [ (50 60) 2 + (55 60) 2 + (60 60) 2 + (65 60) 2 + (70 60) 2] 5 = 50
59 s A = 5 2 B s 2 B = 1 [ (20 60) 2 + (40 60) 2 + (60 60) 2 + (80 60) 2 + (100 60) 2] 5 = 800 s B = 20 2 B A 4 2 A B A B A B 50 s 2 A = 1 [ ( ) (0 50) 2 99 ] = s 2 B = 1 [ (200 50) (150 50) (50 50) (0 50) 2 55 ] 100 = s A = , s B = 30 5 A B 7.4 A B s s 2 = {}}{{}}{ 100 (10 x) 2 + (10 x) 2 + (17 x) 2 + (17 x) 2 = (10 x) (17 x)
60 54 6 x 1. (a) B10 B2 / $S2 B10 : R15 2. (a) B18 B$1 * B10 B18 : R23 (b) S18 SUM(B18:R18) S18 : S23 3. (a) B26 (B$1 - $S18)^2 * B10 B26 : R31 (b) S26 SUM(B26:R26) S26 : S31 (c) T26 SQRT(S26) T26 : T31
61 se02.xls 6.3.4
62 56 6 n x 1, x 2,, x n x n x k x, k=1 n (x k x) 2, k=1 n (x k x) 4 k=1 f(x) = n (x k x) 2 k=1 x f(x) x = 1 n n x k, x 2 = 1 n n x 2 k. k=1 k=1 ] f(x) = nx 2 2nxx + nx 2 = n [(x x) 2 + x 2 x 2 f(x) x = x g g f(x) = n g(x x k ) k=1 f(x) g
63 f(x) = n x k x k=1 x
64
65 59 7 k x 1, x 2,, x k x 1 + x x k k , 1 5, 2 5, 1 5, 2 5 k x 1, x 2,, x k w 1, w 2,, w k w 1 x 1 + w 2 x 2 + w k x k 1 k p 1, p 2,, p k 7.1 X B(n, p) n E(X) = k nc k p k q n k k=0
66 60 7 k 1 E(X) = k nc k = n n 1 C k 1 (7.1) n k nc k p k q n k = k=1 = np j = k 1 n n n 1 C k 1 p k q n k k=1 n n 1C k 1 p k 1 q n k k=1 E(X) = np n 1 j=0 n 1 j=0 n 1C j p j q n 1 j n 1C j p j q n 1 j B(n 1, p) 1 E(X) = np m = E(X) V (X) = E(X 2 ) m 2 = E(X(X 1) + X) m 2 = E(X(X 1)) + m m 2 V (X) E(X(X 1)) (??) k 2 E(X(X 1)) = k(k 1) nc k = n(n 1) n 2 C k 2 (7.2) n k(k 1) nc k p k q n k = k=2 = n(n 1)p 2 j = k 2 n n(n 1) n 2 C k 2 p k q n k k=2 n n 2C k 2 p k 2 q n k k=2 E(X(X 1)) = n(n 1)p 2 n 2 j=0 n 2C j p j q n 2 j = n(n 1)p 2 1 = n(n 1)p 2
67 V (X) = n(n 1)p 2 + m m 2 = npq B(n, p) = np, = npq X HG(n, r, w)
68
69 63 8 p q = 1 p n X X n k n k n C k p k q n k P (X = k) = n C k p k q n k binomial distribution B(n, p) binomial expansion (a + b) n = n nc k a k b n k k=0 1 5 (1) 1 3 (2) (1) 3 2 (2) a b 1 p q = 1 p
70 64 8 P Q P : Q P, Q Fermat n = a+b 1 n a n a 1 P = n nc k p k q n k. Q = k=a a 1 k=0 nc k p k q n k Monmort a k k < b a 1 + k a 1 a 1+kC a 1 p a 1 q k p = a 1+k C a 1 p a q k n 2n Excel b k = n C k p k q n k n = 20, p = 0.1 Excel COMBIN(n,k) b k (k = 0, 1, 2,, 20) (combinatorial number) n C k
71 65 1. A k = 0, 1, 2,..., B b k B1 =COMBIN(20,A1)*0.1^A1*0.9^(20-A1) B2:B21 nc k nc k 1 = n! (k 1)!(n k + 1)! k!(n k)! n! = n k + 1 k b k = b k 1 (n k + 1)p kq (8.1) 1. A k = 0, 1, 2,..., B1 b 0 =0.9^(20) 3. B2:B21 B2 =B1*(21-A2)/A2/9 (1) 1 (2)
72
73 67 9 r w n ( n ) X X = k r + w n r+w C n r k r C k w n k w C n k P (X = k) = r C k wc n k r+wc n k n r hyper-geometric distribution 1 HG(n, r, w) n X B(n, p) p = r r + w 7 3 (1) (2) (3) X X 1
74
75 r w X w f w (t) = E(t X w ) 1 A X w = { 1 A 1 + X w 1 A f w (t) = X 0 = 1 f 0 (t) = t (10.1) f 1 (t) = f 2 (t) = f 3 (t) = r w + r t + w w + r tf w 1(t) (10.1) r [rt + t2 ], 1 (1 + r)(2 + r) [r(1 + r)t + 2rt2 + 2t 3 ], 1 (1 + r)(2 + r)(3 + r) [r(1 + r)(2 + r)t + 3r(1 + r)t2 + 6rt 3 + 6t 4 ] f w (t) f w (t) t w k µ (k) w = E(X w (X w 1)(X w 2) (X w k + 1))
76 70 10 k = 1 (10.1) f w(t) = t = 1 r w + r + w w + r (f w 1(t) + f w 1(t)) µ (1) w = 1 + w w + r µ(1) w 1 µ (1) w = w 1 + r + 1 w k 2 (10.1) f w (k) (t) = w [ ] tf (k) w + r w 1(t) + k C 1 f (k 1) w 1 (t) µ (k) w = w w + r ( ) µ (k) w 1 + kµ (k 1) w 1 (10.2) (10.2) 1 x [x] n = x(x 1)(x 2) (x n + 1), [x] n = x(x + 1)(x + 2) (x + n 1) µ (k) w = k! [1 + r] k (w r)[w] k 1 (w + r)µ (k) w = wµ (k) w 1 k! [1 + r] k {(w + r) (w r)[w] k 1 w (w + r)[w 1] k 1 } 1
77 k! = [1 + r] (w + r) {(w + 1)[w] k k 1 + r[w] k 1 w[w 1] k 1 } k! = [1 + r] (w + r) {[w + 1] k k [w] k + r[w] k 1 } k! = [1 + r] (w + r) {k[w] k k 1 + r[w] k 1 } k! = [1 + r] (w + r)(k + r)[w] k k 1 (k 1)! = k [1 + r] (w + r)[w] k 1 k 1 (k 1)! = kw [1 + r] (w + r)[w 1] k 1 k 2 = kwµ (k 1) w 1 V ar(x w ) = r w(w r) (1 + r) 2 (2 + r)
78
79 n = A1 =INT(6*RAND())+1 2. A2:A20 RAND() 0 1 (0 1 ) *RAND() 0 6 INT(x) x INT(3.14) = 3 =INT(6*RAND()) 0 6 0, 1, 2, 3, 4, 5 p = 2 n = A1 =IF(RAND() < 1/3,0,1) 2. A2:A
80 ( ) 11.3 population sample random sampling
81 A ( ) 51 A 75 B , , A,B,C,...,I,J 100, 200,,
82 ( ) 1. A,B,...,I,J 1 55, 2 55, 3 55,, 9 55, (11.1) 2. C F (11.1) X X x 1 x 2 x k p 1 p 2 p k Excel X VLOOKUP 1. F1:Gk VLOOKUP T (a) E1:Ek p 1, p 2,, p k (b) F1:Fk q 1 = p 1, q 2 = p 1 + p 2, q 3 = p 1 + p 2 + p 3,, q k = p 1 + p p k F1:Fk 0, q 1, q 2,, q k 1 (c) G1:Gk x 1, x 2,, x k 2. A RAND() 0 1
83 B1 =VLOOKUP(A1, F$1:G$k, 2) B B2 VLOOKUP(z, T, 2) z ( ) T F1:Gk 0 x 1 q 1 x 2 q 2 x 3.. q k 1 x k 2 T 2 z < q 1 = x 1 z < q 2 = x 2 z < q 3 = x 3 5 X 10
84
85 p n ˆp p ˆp 1. ˆp ˆp(1 ˆp) 2. ŝ = n n 3. 95% ±1.96 ŝ 99% ±2.545 ŝ 95% p ˆp 1.96 ŝ ˆp ŝ 95% ( ) 15 1
86 80 12 (%) 1990/10/28( ) 18 : : /9/16( ) 18 : : Dr. 1981/12/16( ) 19 : : /2/23( ) 18 : : /1/10/( ) 19 : : 30 TBS /12/8( ) 18 : : /12/22( ) 19 : : /3/13( ) 18 : : /2/11( ) 19 : : /3/22( ) 18 : : /10/28 40% 95% 1. n = 600 ˆp = (1 0.4) 2. ŝ = = % ±1.96 ŝ = ± /3/22 ) 30% 95% %
87 % , , , , % % /20 % 6, 176 3, /20 n ˆp = 0.45 ˆp(1 ˆp) n 95% n = /20 = = = = p = 0.05 p = 0.50
88 82 12
89 , 606 1, 589 1, 619 1, 660 1, 717 1, 824 1, 361 1, 936 1, 872 1, , 934 2, 001 2, 039 2, 092 2, 030 1, 901 1, 833 1, 755 1, 709 1, n = 1, 872, 000 ˆp = 107.1/( ) = % ( ) 1.96 = = , 872, /( ) = % 1980 (1) (2) , , , 882
90 X X B(n, p) ˆp = X n E(ˆp) = 1 n E(X) = 1 n np = p V (ˆp) = 1 n V (X) = 1 pq npq = 2 n2 n ˆp p pq n n - P (p zs ˆp p + zs) = 1 α(z) s = α(z) pq n α(1.645) = 0.10, α(1.96) = 0.05, α(2.575) = 0.01 p(1 p) p(1 p) p z ˆp p + z n n p 2 (p ˆp) 2 z 2 p(1 p) n
91 ) ) (1 + z2 p 2 2 (ˆp + z2 n 2n + ˆp 2 0 D = ) 2 ) (ˆp + z2 (1 + z2 ˆp 2 2n n = z 2 ˆp(1 ˆp) + z4 n 4n 2 n 2 1 D = z 2 ˆp(1 ˆp) n ŝ = ˆp(1 ˆp) n (12.1) D = zŝ 2 ( ) ˆp + z2 ( ) D ˆp + z2 + D 2n 2n p 1 + z2 1 + z2 n n ˆp zŝ p ˆp + zŝ (12.2) 2 1 α(z) (12.2) 1 α(z) p ˆp zŝ ˆp zŝ ˆp(1 ˆp) ŝ = n A 95% 2%
92 86 12 [ˆp zs, ˆp + zs] 2zs w% 2zs w ˆp(1 ˆp) n s w 2z ( w ) 2 2z I 2 f(x) = x(1 x) ( w ) 2 ( z ) 2 4n, n (12.3) 2z w 95% z = % w = 0.02 (12.3) n % 1%
93 ( ) A B
94 88 13 A, B A B 6 k X 1, X 2,, X k 1 k n 1 k = n k n n = X 1 + X X k X 1 n k, X 2 n k,, X k n k 3 3 χ χ
95 A Excel A B A B D1 =CHITEST(A1:A6,B1:B6) p p = p p B A B A B D1 =CHITEST(A1:A6,B1:B6) p = p
96 p p p p A p B p 0.05 p p p
97 k X 1, X 2,, X k 1 k χ 2 = (X 1 e) 2 e + (X 2 e) 2 e + + (X k e) 2 e (13.1) e = n/k χ k chi2 n X 1 +X 2 + +X k = n 2 (13.1) chi2 <- function(n,k) { y <- as.integer(runif(n,1,k+1)) x <- table(y) e <- n/k w <- sum((x-e)^2/e) return(w) } y <- as.integer(runif(n,1,k+1)) 1, 2,, k n x <- table(y) y e <- n/k
98 92 13 w <- sum((x-e)^2/e) 2 (13.1) n = w <- c(1:1000) for (k in 1:1000) { w[k] <- chi2(60,6) } hist(w) n = t <- c(1:300)/10 f <- dchisq(t, 11) plot(t,f,type="l") P {a χ 2 b} = 2 t t = a t = b 1 t P {χ 2 b} = 0.95 b t t = b 0.95 b qchisq(0.95,11) χ
99 R A 2 x <- c(12,7,8,10,12,11) chisq.test(x) p Chi-squared test for given probabilities data: x X-squared = 2.2, df = 11, p-value = B Chi-squared test for given probabilities data: x X-squared = 24.6, df = 11, p-value = π
100
101
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105 Excel 1. A B C A5 =A$3*$C1/$C$3 A B E1 p =CHITEST(A1:B2,A5:B6) A1:B6 2. A B C
106 A9 =A$7*$C1/$C$7 A B E1 p =CHITEST(A1:B6,A9:C14)
107 A B C 1 x 11 x 12 x 13 2 x 21 x 22 x 23 3 x 31 x 32 x 33 4 x 41 x 42 x 43 A B C 6 e 11 e 12 e 13 7 e 21 e 22 e 23 8 e 31 e 32 e 33 9 e 41 e 42 e 43 χ 2 = (x 11 e 11 ) 2 e 11 + (x 12 e 12 ) 2 e 12 + (x 13 e 13 ) 2 e 13 + (x 21 e 21 ) 2 e 21 + (x 22 e 22 ) 2 e 22 + (x 23 e 23 ) 2 e 23 + (x 31 e 31 ) 2 e 31 + (x 32 e 32 ) 2 e 32 + (x 33 e 33 ) 2 e 33 + (x 41 e 41 ) 2 e 41 + (x 42 e 42 ) 2 e 42 + (x 43 e 43 ) 2 Excel 4 3 CHITEST =CHITEST(A1:C4,A6:C9) F1 p p e 43
108 p p p 0.05 p
109 Yates Excel p p Yates A B C p χ 2 χ 2 5 5
110 Fisher E F E E F a b a + b F c d c + d a + c b + d N N = a + b + c + d E F E p 1 E q 1 = 1 p 1 F p 2 F q 2 = 1 p 2 E F p 1 p 2 E F p 1 q 2 a, b, c, d f(a, b, c, d) f(a, b, c, d) = = N! a! b! c! d! (p 1p 2 ) a (q 1 p 2 ) b (p 1 q 2 ) c (q 1 q 2 ) d n! a! b! c! d! pa+c 1 q1 b+d p a+b 2 q2 c+d p 1, p 2 a + c, b + d, a + b, c + d a, b, c, d f(a, b, c, d) = K a! b! c! d! K = n! p a+c 1 q b+d 1 p a+b 2 q c+d 2 K a, b, c, d E F a + c, b + d, a + b, c + d a, b, c, d f(a, b, c, d) = a+b C a c+d C c NC a+c
111 14.5. Fisher 105 ( ) 1 1 K = a+c,b+d,a+b,c+d 1 a! b! c! d! a + b = k, a + c = l 2 kc a x a = (x + 1) k n kc c x c = (x + 1) n k 0 a k 0 c n k x l (x+1) n x l n! nc l = (a + c)! (b + d)! a+c=l 1 K = kc a n k C c = a+c,b+d,a+b,c+d a+c,b+d,a+b,c+d (a + b)! (c + d)! a! b! c! d! 1 a! b! c! d! = N! (a + b)! (c + d)! (a + c)! (b + d)! f(a, b, c, d) ( ) a + b = R, c + d = W, a + c = n N F R W N R W F ( ) E ( ) F ( ) F ( ) n ( ) a c RC a W C c NC n
112 a, b, c, d χ 2 ( ) 2 = χ 2 = N (ad bc)2 (a + b)(c + d)(a + c)(b + d) ( ) Np 1 p 2, Nq 1 p 2, Np 1 q 2, Nq 1 q 2 p 1, p 2 p 1, p 2 ˆp 1 = a + b N, ˆq 1 = c + d N ( ) ˆχ 2 = [ (a + b)(a + c) a N (a + b)(a + c) N [ (c + d)(a + c) c N + (c + d)(a + c) N, ˆp 2 = a + c N, ˆq 2 = b + d N ] 2 [ (a + b)(b + d) b N + (a + b)(b + d) N [ (c + d)(b + d) d N + (c + d)(b + d) N ( ) ] 2 ] 2 ] 2 a = 4, b = 3, c = 1, d = 7 a + b = 7, c + d = 8, a + c = 5, b + d = 10
113 14.5. Fisher ( ad bc) T 0 T 1 T 2 T 3 T 4 T 5 ad bc ad bc T 2, T 3, T 1, T 4, T 0, T 5 T 4 p = P {T 5 } + P {T 0 } + P {T 4 } = = ( p = ) Fisher s Tea Drinker
114 a 0 a a 60 a a a A1:A61 0, 1,, B1:B61 B1 =COMBIN(60,A1)*COMBIN(84,72-A1)/COMBIN(144,72) COMBIN(n, k) n C k 4. C1:C61 ad bc C1 =ABS(A1*(A1+12)-(60-A1)*(7 ABS(x) x 5. a = 36 ad bc = 864 ad bc 864 =SUM(B1:B25)+SUM(B37:B61) 6. p = Chadwick and Dudley Can malt whisky be discriminated from blended whisky? The proof. A modification of Sir Ronald Fisher s hypothetical tea tasting experiment., Br. Med. J. (Clin Res Ed) 1983, 287, Glendiddch, Springbank, Glenmorangic White Horse, Bells, Haig Glendiddch 30 7 White Horse 18 2
115 14.6. R R 2 > x <- > x [,1] [,2] [1,] [2,] > Pearson s Chi-squared test with Yates continuity correction data: x X-squared = 3.225, df = 1, p-value = > fisher.test(x) Fisher s Exact Test for Count Data data: x p-value = alternative hypothesis: true odds ratio is not equal to 1 95 percent confidence interval: sample estimates: odds ratio > x <- matrix(c(261,163,342,402,229,219),ncol=3,byrow=t) > x [,1] [,2] [,3] [1,] [2,] > chisq.test(x) Pearson s Chi-squared test data: x X-squared = , df = 2, p-value = 1.350e-14
116 > fisher.test(x) Fisher s Exact Test for Count Data data: x p-value = 1.162e-14 alternative hypothesis: two.sided > x <- matrix(c(315,108,101,32), ncol=2, byrow=t) > x [,1] [,2] [1,] [2,] > chisq.test(x) Pearson s Chi-squared test with Yates continuity correction data: x X-squared = , df = 1, p-value = > x <- matrix(c(56,6759,272,11396), ncol=2, byrow=t) > x [,1] [,2] [1,] [2,] > chisq.test(x) Pearson s Chi-squared test with Yates continuity correction data: x X-squared = , df = 1, p-value = 9.978e-14 > x <- matrix(c(10,3,2,15), ncol=2, byrow=t) > x [,1] [,2] [1,] 10 3 [2,] 2 15 > chisq.test(x)
117 14.6. R Pearson s Chi-squared test with Yates continuity correction data: x X-squared = , df = 1, p-value = > fisher.test(x) Fisher s Exact Test for Count Data data: x p-value = alternative hypothesis: true odds ratio is not equal to 1 95 percent confidence interval: sample estimates: odds ratio
118
119 X, Y X, Y X Y X, Y X Y A B C D E F G H I J K L Excel A1:C B2:C13
120 B2:C13 r =
121 X Y X Y X Y X Y r 1 r 1 r > 0 r < 0 r = 1 r =
122 X Y X, Y X Y Z X, Y Y, Z Z, X X, Y, Z ( Excel X, Y, Z 2 3 X, Y, Z 1
123 X Y Z X Y Z X Y Z X Y Z H23-center.xls 23 X Y Z 200, 100, 200 X, Y Y, Z Z, X X, Y, Z (X, Y ) n (x 1, y 1 ), (x 2, y 2 ),, (x n, y n ) µ X = 1 n n x i, i=1 µ Y = 1 n n y i (15.1) i=1 X Y x i µ X y i µ Y x i µ X y i µ Y (x i µ X )(y i µ Y )
124 X Y x i µ X y i µ Y x i µ X y i µ Y (x i µ X )(y i µ Y ) s XY = 1 n n (x i µ X )(y i µ Y ) (15.2) i=1 s XY X, Y s XY X Y mm s XY cm s XY 10 2 = 100 s XY cm s XY 1/ X Y s X = 1 n (x i µ X ) n 2, s Y = 1 n (y i µ Y ) n 2 (15.3) i=1 mm s X, s Y cm 10 r = i=1 s XY s X s Y (15.4) (15.4) r (15.4) r = 1 n n i=1 x i µ X s X yi µ Y s Y.
125 x i µ X s X y i µ Y s Y X, Y r X, Y r x i x i µ X s X r r 1 1 n (x i µ X )(y i µ Y ) n 1 n (x i µ X ) n 2 1 n (y i µ Y ) n 2 i=1 2 n 2 ( n 2 ( n ) ( n ) (x i µ X )(y i µ Y )) (x i µ X ) 2 (y i µ Y ) 2 i=1 i=1 i=1 x i µ X = a i, y i µ y = b i (a 1 b 1 + a 2 b a n b n )) 2 ( ) ( a a a 2 n b b bn) 2 i=1 i=1 2. r = 1 n (x i, y i ) r = 1 k b 1 = ka 1, b 2 = ka 2,, b n = ka n i = 1, 2,, n y i µ Y = k(x i µ X ) n (x i, y i ) y µ Y = k(x µ X )
126 n (x i, y i ) y = ax + b l i l1 2 + l ln 2 a, b
127 N N N N N N N N N N N N N N N N N N N N N N N N N N N N N S S S S S S S S S S 0.5
128 ,667 40, ,333 52, ,167 54, ,167 56, ,833 59, ,500 60, ,167 63, ,667 66, ,500 65, ,500 71, ,000 71, ,833 72, ,167 76, ,333 76, ,500 83, ,500 87, ,333 88, , ,116 1 X 1 Y X, Y a206-1.xls 1 12
129 X, Y 3 (1) 150 X Y 0.6 Z X, Z Y, Z 0.8 Z X, Y (2) X Y Z Z X Y X, Y (3) X Y Z Z X Y X, Y 4 (1) X Y r r = 0.2 (2) X = 1, 2, 3 X = 4, 5, 6 Y X, Y r r = 0.6
130 X Y 1 = 2.54cm X Y (X, Y ) = (65.5, 63.2) 9 1. X B1:L1 Y A2:A15 B2:L15 2. A1:L15
131 Excel X, Y X, Y X, Y
132
133 (a) II B 20 kg A B E C F D
134 (1) 10 A. kg 20 M. kg. kg (2) 20 M S. (3) t 20 M S M ts M + ts 20 N(t) N(1) = N(2) = (4) 10 D kg B kg C kg B C E F (5) 20 (a) (b) 20 (c) (a) 0 4
135 129 (b) (c)
136 II B 10 I II A, B, C, D I 3 59 B C 80 1 A 48 D II E (1) 1 I A II 1 A (2) II (3) 1 I 3 B M B. (4) 2 I II 4.6 I 5 C II 1 D
137 131 (5) 1 (a) 2 (b) 1 r 1 2 r 2 (r1, r 2 ) (c) (a), (b) 0 3 (c) 0 3
138 (0.54, 0.20) 1 ( 0.54, 0.20) 2 (0.20, 0.54) 3 (0.20, 0.54) (6) (d) (e) (d), (e)
139 II B x y C 1 x y (1) 1 12 x 12, 9, 3, 3, 10, 17, 20, 19, 15, 7, 1, C. C (2) x 0 C A 0 C B A x. C. A y 6.0 C B y 21.5 C 1 12 y. C x y x 7 C y 30 C y 18 C
140 (3) y. C y (a) (a) (4) y (b) C 1 (c) C 1 (b), (c) (5) z = y x z. C x z (d) (d) 0 3
141 135 (6) 1 12 x z (e) (e) II B P 20 x
142 y x, y x, y x y x x (x x) 2 y y (y y) 2 (x x)(y y) A B (1) 1 x x 3.0 B. A (2) x. (3) z = x + y z z. z (z z) 2 = (x x) 2 + (y y) 2 + 2(x x)(y y) (z ) (a) {(x ) + (y )} (a) > 1 = 2 < (4) x y (b) 1 (b) 0 3
143 137 P 20 Q 25 P Q
144 (5) (c) (c) P 1 Q 2 P Q 3 (6) Q.. (1) P B (d) (d) P 1 Q 2 P Q 3 (7) (e) (e) Q 1 54 Q 2 65 Q 3 70 P
145 Excel X, Y X, Y Y X Y = ax + b Y = ax + b a b X Y J
146 F X Y B2:B19 C2:C19 D2:D Y B2:B19 X C2:C19
147 17.1. Excel 141 a b B18, B17 Y = X % % ax + b
148 X Y Y = X
149 17.1. Excel 143 Y = ax + b a b r 1 2 (1) (2) (1) (2) X Y kion-kadai.xls (1) (2) 100 m 4 X Y X Y 5 X Y X Y 2 3
150 n x i (i = 1, 2,, n) µ n µ n x = 1 n x i n x i x (i = 1, 2,, n) n n (x i x) = 0 i=1 b n x i b b n Q = (x i b) 2 i=1 b ν n n n Q = x 2 i 2b x i + b 2 = x 2 i 2nxb + b 2 = i=1 i=1 i=1 n x 2 i x 2 + (b x) 2 i=1 i=1 Q b = x n n Q min = x 2 i x 2 = (x i i=1 i=1 n (x i µ) 2 ) 2 i=1 Q min /n =
151 Q = n [y i (ax i + b)] 2 i=1 a, b a, b 2 Q b a n n Q = [(y i ax i ) b] 2 = [(y i ax i ) 2 2b(y i ax i ) + b 2 ] = i=1 n (y i ax i ) 2 2b i=1 x = 1 n i=1 n (y i ax i ) + nb 2 i=1 n x i, i=1 y = 1 n b Q n = 1 n (y i ax i ) 2 2b(y ax) + b 2 n i=1 = [b (y ax)] n (y i ax i ) 2 (y ax) 2 n i=1 n i=1 y i Q n b = y ax (17.1) = 1 n n (y i ax i ) 2 (y ax) 2 n = (yi 2 2ax i y i + a 2 x 2 i ) (y 2 2axy + a 2 x 2 ) 1 n i=1 ( = a 2 1 n i=1 ) n x 2 i x 2 2a i=1 ( 1 n ) n x i y i xy + i=1 ( 1 n ) n yi 2 y 2 i=1
152 s 2 x = 1 n s xy = 1 n s 2 y = 1 n n x 2 i x 2 = 1 n (x i x) 2, n i=1 n x i y i xy = 1 n (x i x)(y i y), n i=1 n yi 2 y 2 = 1 n (y i y) 2 n i=1 i=1 i=1 i=1 = a 2 s 2 x 2as xy + s 2 y a ( = s 2 x a s ) 2 xy + s2 xs 2 y s 2 xy s 2 x s 2 x Q min n a = s xy s 2 x = s2 xs 2 y s 2 xy s 2 x (17.2) (17.3) 1. x, y s 2 x, y 2 2 s xy 2. a (17.2) y b (17.1) X Y 1 = 2.54cm
153 X Y X Y (63.5, 61.2) (68.5, 69.2) (73.5, 73.2)
154 Y = 0.645X cm 170 cm cm 6.45 cm 10 cm 6.45 cm X 10 Y kakei-kadai.xls (1) 10 Y = ax + b (2) 10
155 y t y = at + b y t y t fertility.xls A1:B27 3. Y B13:B24 X A13:A24 4. E1 5. F17 b X 1 F18 a a = , b = a CO 2 Mauna Loa CO 2 5 ppm (parts per million, ) 15 WEB Excel maunaloa.co2.xls source Excel ( ) source
156 = data CO t t t = t = t = 1/12, 6/12, 11/ t = 2 + 6/12 B t 1. B B3 =B2+1/12 3. B3 B4:B Y C2:C541 X B2:B E GDP GDP GDP 30 48
157 GDP GDP GDP GDP GDP GDP = 10 at+b log 10 (GDP ) = at + b GDP t = t = t = 1/ t = /4 A B GDP C t 1. C C2 =C1+1/4 3. C2 C3:C44 D GDP 1. D1 =LOG10(B1) 2. D1 D2:D44 F17 b X 1 F18 a GDP
158 E1 =F$18 * C1 + F$17 2. E1 E2:E GDP 17.5 y x y = 10 ax+b x t y log 10 y x log 10 y = ax + b log 10 y log 10 x log 10 y = a log 10 x + b y = 10 b x a y x a x y x y Doll,R. The Age Distribution of Cancer: Implicatins for Models of Carcinogenesis, Journal of the Royal Statistical Society,Series A,1971
159 x y 1. A1:A5 x = 40, 50, 60, 70, 80 y B1:B5 2. C1:C5 x C1 =LOG10(A1) C2:C5 3. D1:D5 y 4. C X D Y 5. a, b x y 17.6 GDP GDP y t = A at b
160 y t y t w t = log 10 A y t w t = at + b w t t fukyuritsu.xls 1003fukyuritsu.xls fukyuritsu.xls mywork.xls B2:B t = t = 48 A2:A49 0, 1, 2,, 47
161 A2 0 A3 =A2+1 A4:A49 3. C2:C49 w t = log 10 ( yt 100 y t ) C2 =LOG10(B2/(100-B2)) C3:C49 4. w t w t
162
163 ( ) ( ) 4/ / : 2 = 2 1
164 p 1 p 2 ( ) rr rr = p 1 p 2 n 1, n 2 X 1 X 2 X 1 n 1 X 1 n 1 X 2 n 2 X 2 n 2 X 1 n 1 X 2 n 2 ( ) RR RR = X 1 n 1 : X 2 n 2 = X 1 n 1 X 2 n 2 1. s = 1 X 1 1 n X 2 1 n 2 2. k = e zs z 90% z = % z = % z = RR/k, RR k 95%
165 s = = k = e = 5.46 = 2/5.46 = 0.366, = = ( ) ( ) (2.5 km ) ( km) 2.5 km 12 ( ) ( ) Sv 90%
166 % Sv 90% Sv 90% % msv % 5 msv
167 , , , % 1.97 ( ) 1.50 ( ) 1.57 ( ) 1.57 ( ) 2.66 ( ) 1.89 ( ) 1.97 ( ) 1.23 ( ) 3.39 ( ) 2.22 ( ) 1.90 ( ) 3.59 ( ) 1.51 ( ) 1.28 ( ) 1.22 ( ) 1.47 ( ) 1.81 ( ) 1.63 ( ) 1.73 ( ) 1.23 ( ) 1.58 ( ) 1.19 ( ) 1.81 ( ) 1.96 ( ) 5.47 ( ) 3.03 ( ) 0.00 () 0.00 () 4.79 ( ) 2.41 ( ) 3.88 ( ) 2.63 ( ) 2.32 ( ) 1.00 ( ) 1.57 ( ) 1.46 ( ) 0.60 ( ) 1.55 ( ) 5.35 ( ) 2.76 ( ) 1.86 ( ) 0.00 () 1.45 ( ) 2.13 ( ) 0.96 ( ) 0.96 ( ) 1 Katanoda et al. Population Attributable Fraction of Mortality Associated with Tobacco Smoking in Japan: A Pooled Analysis of Three Large-scale Cohort Studies, J. Epidenmiology (2008)
168
169 T N T p 1 p 2 p 1, p 2 T α β p 1 /p 2 αp 1 N β(1 p 1 )N αp 2 N β(1 p 2 )N = = αp 1 N β(1 p 1 )N = αp 1 β(1 p 1 ) αp 2 N β(1 p 2 )N = αp 2 β(1 p 2 )
170 = = αp 1 β(1 p 1 ) : αp 2 β(1 p 2 ) p 1 p 2 : 1 p 1 1 p 2 = p 1 p 2 1 p 2 1 p 1 p 1, p 2 1 p 1, 1 p X 1, X 2 B(n 1, p 1 ), B(n 2, p 2 ) n 1, n 2 Z 1 = X 1 n 1 p 1 n1 p 1 q 1, Z 2 = X 2 n 2 p 2 n2 p 2 q 2 X 1 n 1 = p 1 + p1 q 1 n 1 Z 1, X 2 n 2 = p 2 + p2 q 2 n 2 Z 2 RR = p 1 + p1 q 1 n 1 Z 1 p 2 + p2 q 2 n 2 Z 2 = p 1 + q1 1 p 1 n 1 Z 1 = rr p q2 p 2 n 2 Z q1 p 1 n 1 Z q2 p 2 n 2 Z 2 ( ) q1 log RR = log rr + log 1 + Z 1 p 1 n 1 q1 q2 rr + Z 1 Z 2 p 1 n 1 p 2 n 2 ( ) q2 log 1 + Z 2 p 2 n 2 Y = q1 q2 Z 1 Z 2 p 1 n 1 p 2 n 2 Y 0 q 1 + q ( 2 1 = 1 ) ( ) p 1 n 1 p 2 n 2 n 1 p 1 n 1 n 2 p 2 n 2
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