June 2016

i (statistics) F Excel Numbers, OpenOffice/LibreOffice Calc

ii *1 VAR STDEV 1 SPSS SAS R *2 R R R R *1 Excel, Numbers, Microsoft Office, Apple iwork, *2 R GNU GNU R

iii URL http://ruby.kyoto-wu.ac.jp/statistics/training/ PDF http://ruby.kyoto-wu.ac.jp/~konami/text URL E-mail: konami@kyoto-wu.ac.jp

1 i 1 5 1.1..................... 5 1.2.................................. 16 1.3.......... 19 2 23 2.1............................... 23 2.2................................. 29 2.3................................. 41 3 47 3.1............................ 47 3.2................. 49 3.3.......................... 49 3.4...................... 51 3.5...................... 53 4 55 4.1.................................. 55 4.2.................................. 57 4.3................................ 59 5 63 5.1................... 63 5.2.......................... 69 5.3............................. 71 5.4................................ 77

2 6 79 6.1............................... 79 6.2............................... 83 6.3............................... 84 6.4................................. 86 6.5 χ 2........................... 91 7 97 7.1............................. 97 7.2................................. 101 7.3................................ 102 8 111 8.1......................... 111 8.2................................ 124 9 131 9.1................................ 131 9.2............................ 134 A 147 A.1............................. 147 A.2................................ 148 A.3........................... 148 A.4................................ 150 A.5....................... 152 A.6...................... 152 A.7................................. 153 B 157 B.1.......................... 157 B.2................................. 158 B.3 χ 2.................................. 160 B.4 Student t-............................ 161 C 163 C.1....................... 163 C.2.................................. 164

3 C.3............................... 165 C.4.................... 165 D 167 D.1............................... 167 D.2....................... 170 D.3 R................... 173 E 177 187

5 1 (descriptive statistics) 1.1 100 1.1 1.1 100 kg 43.6, 45.2, 45.4, 45.8, 47.2, 47.8, 48.2, 48.7, 48.8, 48.9, 49.0, 49.0, 49.4, 49.5, 49.8, 50.4, 50.5, 50.9, 50.9, 51.2, 51.2, 51.2, 51.3, 51.3, 51.6, 51.7, 51.7, 51.8, 52.0, 52.0, 52.1, 52.1, 52.1, 52.2, 52.3, 52.7, 52.7, 52.8, 52.9, 52.9, 53.1, 53.1, 53.8, 54.0, 54.5, 54.5, 54.6, 54.7, 54.7, 54.7, 54.8, 54.9, 55.1, 55.1, 55.2, 55.3, 55.4, 55.4, 55.4, 55.6, 55.7, 55.8, 55.9, 56.1, 56.3, 56.3, 56.3, 56.4, 56.5, 56.7, 56.8, 57.0, 57.1, 57.1, 57.2, 57.3, 57.6, 57.7, 57.8, 58.1, 58.4, 58.6, 58.7, 58.7, 58.7, 58.7, 59.1, 59.3, 59.9, 60.0, 60.1, 60.3, 60.5, 60.6, 60.6, 60.7, 61.3, 62.7, 64.2, 64.6 x n x = {x 1, x 2,..., x n } (1.1)

6 1 1.1.1 x, µ (mean *1 ) 1 (43.6 + 45.2 + 45.4 + 45.8 + + 64.6) = 54.46 100 x x µ *2 x = 1 n (x 1 + x 2 + + x n ) = 1 n n x i (1.2) i=1 1.1.2 (deviation) (1.3) *3 δx i = x i x (1.3) δx 1 + δx 2 + + δx n = (x 1 x) + (x 2 x) + + (x n x) = (x 1 + x 2 + + x n ) n x = n 1 n (x 1 + x 2 + + x n ) n x = 0 *1 average *2 µ mean m *3 δ

1.1 7 1.1.3 σ 2, σ 2 σ 2 *4 σ 2 = 1 ( (x1 x) 2 + (x 2 x) 2 + + (x n x) 2) n = 1 n (x i x) 2 (1.4) n i=1 σ 2 (variance) σ (standard deviation) *5 σ = σ 2 (1.5) m 2 m 2 10 m 100 m 2 *4 σ *5 SD RMS (Root Mean Square)

8 1 (standard error) 84 (representative value / descriptive statistics) 1.1.4 (1.4), (1.5) σ 2 = 1 n (x i x) 2 n i=1 = 1 n (x 2 i 2xx i + x 2 ) n i=1 ( = 1 n ) n x 2 i 2x x i + nx 2 n i=1 i=1 ( = 1 n ) x 2 i 2nx 2 + nx 2 n = 1 n i=1 n x 2 i x 2 = x 2 x 2 (1.6) i=1 x 2 1 n (x2 1 + x 2 2 +... + x 2 n) 2 n n xx i = xx 1 + xx 2 +... + xx n = x x i = x nx = nx 2 i=1 i=1

1.1 9 n n { }} { x 2 = x 2 ( 1 + 1 +... + 1) = nx 2 i=1 x (1.6) 1 1 1.1 4 1 2 0 1 n 1 p (Chebyshev s inequality) µ σ µ ± aσ a 1 a 2 1.1 µ = 54.46, σ = 4.22 ( 1 1) a = 2 ±2 4.22 54.46 2 4.22 = 46.02 54.46 + 2 4.22 = 62.90 100 1/2 2 = 1/4 25 6

10 1 1.1.5, median / quartile 4 1/4, 2/4, 3/4 (quartile) 3 1 (first quartile) 2 (second quartile) 3 (third quartile) *6 2 1 3 n (median) 2 4 n n 4m, 4m + 1, 4m + 2, 4m + 3 (m = 0, 1, 2,...) 1.1 x 1, x 2,..., x n n 12 15 4m 4m + 3 4 1. n 12 4m 2. Q 1 n/4 n/4 + 1 x 3 x 4 3. Q 1 x 3 x 4 3 : 1 1 4 (x 3 + 3 x 4 ) 4. M n/2 = 6 n/2 + 1 = 7 2 *6 1/4 2/4 3/4

1.1 11 n/4 n/4+1 n/2 n/2+1 3n/4 3n/4+1 (4m) x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 n = 12 Q 1 M Q 3 (4m+1) x 1 x 2 x 3 (n+3)/4 (n+1)/2 (3n+1)/4 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 x 13 n = 13 Q 1 M Q 3 (n+2)/4+1 (3n 2)/4+1 (4m+2) x 1 x 2 x 3 (n+2)/4 n/2 n/2+1 (3n 2)/4 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 x 13 x 13 n = 14 Q 1 M Q 3 (n+1)/4+1 (3n 1)/4+1 (4m+3) x 1 x 2 x 3 (n+1)/4 (n+1)/2 (3n 1)/4 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 x 13 x 13 x 13 n = 15 Q 1 M Q 3 1.1 (M), 1 (Q 1), 3 (Q 3) 1 2 (x 6 + x 7 ) 5. Q 3 3n/4 = 9 3n/4 + 1 = 10 1 : 3 1 4 (3 x 9 + x 10 ) 1 1 1.1 1 Q 1 M 3 Q 3 1.1 n = 100 4m n/4 = 25, n/4 + 1 = 26, n/2 = 50, n/2 + 1 = 51, 3n/4 = 75, 3n/4 + 1 = 76

12 1 M = 1 2 (x 50 + x 51 ) = 1 2 (54.7 + 54.8) = 54.75 Q 1 = 1 4 (x 25 + 3x 26 ) = 1 4 (51.6 + 3 51.7) = 51.675 Q 3 = 1 4 (3x 75 + x 76 ) = 1 4 (3 57.2 + 57.3) = 57.225 1 2 1 Q 1 M 3 Q 3 ( ) 1. {3.2, 4.8, 14.0, 17.2, 22.8} (4.8, 14.0, 17.2) 2. {20.5, 30.5, 39.0, 46.5, 57.5, 59.0, 70.5, 80.5} (36.875, 52.0, 61.875) 3. {10.1, 10.7, 10.8, 11.2, 11.8, 12.5, 12.5, 12.8, 13.3, 13.8, 14.0, 14.7, 15.5, 16.3} (11.35, 12.65, 13.95) 4. {80.0, 80.0, 88.0, 92.8, 100.0, 108.8, 118.4, 129.6, 136.0, 144.8, 146.4, 161.6, 176.0, 185.6, 192.0} (96.4, 129.6, 154.0 ) 1.1.6 4 100 100 (percentile) 1 25 50 3 75 Q 1, Q 3 2 Q 1, Q 3 (lower hinge) (upper hinge) x = {1, 2, 3, 4}, = 1.5, = 3.5 x = {1, 2, 3, 4, 5}, = 2, = 4

1.1 13 1 3 5 (five number summary) 1 3 1.1 1 1 1 3 43.6, 51.675, 54.75, 57.225, 64.6 (box and whiskers plot, box plot) 1.1.6 1.1 43.6 51.675 54.75 57.225 64.6 min Q 1 M Q 3 Max 1.2 1 Q 1 M 3 Q 3 min Max (IQR) 3 1 (IQR *7 ) *7 Interquartile Range

14 1 (outlier) *8 x x < Q 1 k(q 3 Q 1 ) x > Q 3 + k(q 3 Q 1 ) Q 3 Q 1 = IQR 1 3 IQR k k 1.5 3 1.1.7 1.3 25 *9 ( 100 ) 87, 143, 149, 163, 180, 186, 186, 212, 222, 247, 251, 255, 257, 261, 271, 274, 277, 281, 287, 296, 306, 347, 406, 449, 1300 2 9 13 2 A 2 B 1 3 1.3 A: B: *8 http://people.richland.edu/james/lecture/m170/ *9 (2012)

1.1 15 A B 29,172 25,836 3,300 8,920 22,027 1 (robust) 1.1.8 10 15 20 2 1 8 12 8

16 1 1.2 1.2.1 (frequency distribution table) 1.2 (class) (frequency) 1.2 100 1.1 (x i ) (f i ) (F i ) 43.0 45.0 44.0 1 1 45.0 47.0 46.0 3 4 47.0 49.0 48.0 6 10 49.0 51.0 50.0 9 19 51.0 53.0 52.0 21 40 53.0 55.0 54.0 12 52 55.0 57.0 56.0 19 71 57.0 59.0 58.0 15 86 59.0 61.0 60.0 10 96 61.0 63.0 62.0 2 98 63.0 65.0 64.0 2 100 25 20 15 10 5 0 43 45 47 49 51 53 55 57 59 61 63 65 1.4 100

1.2 17 (histogram) 1.2 1.4 53 55 1.2.2 1.2 = + 1 3 {}}{ { }} { 44.0 + 46.0 + 46.0 + 46.0 6 { }} { 48.0 + 48.0 + 48.0 + 48.0 + 48.0 + 48.0 + (1.7) = 44.0 1 + 46.0 3 +... + 64.0 2 1 + 3 +... + 2 = 5446 100 = 54.46 (1.8) (1.8) (1.8) k x 1, x 2,..., x k f 1, f 2,..., f k * 10 x = 1 n k x i f i = i=1 k i=1 (x i f i n ) (1.9) *10 x i

18 1 n k i=1 f i (1.9) 2 2 = (i i ) (1.4) k σ 2 i=1 = (x i x) 2 f i n 2 = k i=1 (x i x) 2 f i n (1.10) = ( i 2 i ) 1.1.4 2 2 k σ 2 i=1 = f ix 2 i x 2 (1.11) n 1 3 (1.11) 1.2 50 51 (53.0 55.0) 41 52 12 53.0 55.0 12 12 50 51 41 52 53.0 55.0 53.0 + 2.0 12 10 = 54.666...

1.3 19 54.67 54.75 52.0 (mode) 52.0 1.3 2 1.3.1 1.5 538 ( ) 100 427 300 400 350 300 427 538 1.5 100 638 0.1%

20 1 1.5 23 538 0.999 + 638 0.001 = 538.1 1 1000 1 10,538 0.1% 548.1 10 * 11 *11

1.3 21 1.3.2 1.3 14 * 12 15.9, 17, 19 23 8.0, 8, 15 1.3 (2012 ) 0 54 54 420 420 1 14 68 730 1150 2 27 95 983 2133 3 52 147 1118 3251 4 76 223 1122 4373 5 118 341 1045 5418 6 149 490 950 6368 7 169 6 59 993 7361 8 286 945 948 8309 9 343 1288 889 9198 10 424 1712 937 10135 11 518 2230 907 11042 12 678 2908 873 11915 13 737 3645 943 12858 14 927 4572 999 13857 15 1023 5595 1551 15408 16 1171 6766 17 1268 8034 18 1426 9460 19 1450 10910 20 1278 12188 21 1030 13218 22 569 13787 23 151 13938 1.6 1 4 *12 http://www.nier.go.jp/kaihatsu/katei h14/index.htm

22 1 1600 Math Kokugo 1400 1200 1000 800 600 400 200 0 0 5 10 15 20 1.6 1.3 2 3 4 5 6 22 25 46 49 70 75 15 28 31 15 1 1 1 1 1 1

23 2 2.1 2.1.1 (element) (set) *1 *2 *1 *2

24 2 2.1.2 A, x 1, x 2,... A = {x 1, x 2,...} (2.1) x A A x, x A (2.2) x1 x2 x3... (Venn diagram) (empty set/null set) 0 *3 A B B A B A (subset) A B, B A (2.3) A A A B, A B (2.4) B A (proper subset) *3 ϕ

2.1 25, A A S S A A (complementary set) Ā S, A, Ā S A A A A A S 2 A, B A, B A B A B A B A B (union) A B A B A B A B (intersection) *4 A B 2 *4

26 2 A B A B A B A B A, B A B = (2.5) A, B A B A + B A, B A B A B A B A B A + B = A B A B

2.1 27 2 1 A, B S A B = A B 2 2 A, B A, B (exclusive or) A, B A B

28 2 2.1.3 A, B, C A B = B A A B = B A (2.6) A (B C) = (A B) C A (B C) = (A B) C (2.7) A (B C) = (A B) (A C) A (B C) = (A B) (A C) (2.8) A B = Ā B A B = Ā B (2.9) A B A B 3 A B C = Ā B C A B C = Ā B C (2.10) 2 3 50 27 ( A) 36 (B) 9 1. S 2.

2.2 29 2.2 2.2.1 p p = 0 p = 1 8 3 1/3 1/3 (Bayesian probability)

30 2 IT 1 1/6 1 2..., 6 6 S A S A n, m S A P (A) P (A) = m n (2.11) 2.2.2 *5 2.1.2 Venn S 2.1 2.1 S *5

2.2 31 2.2.3 A A 1, A 2,... S P (A) S 1 P (S) = 1 (2.12) A 1, A 2,... P (A 1 A 2...) = P (A 1 ) + P (A 2 ) + (2.13) P (A 1 + A 2 +...) = P (A 1 ) + P (A 2 ) + (2.14) P ( ) = 0 (2.15) *6 S = A + Ā P (Ā) = 1 P (A) (2.16) A B P (A B) = P (A) + P (B) P (A B) (2.17) *6 P ( ) > 0 S (2.13) P (S ) = P (S) + P ( ) > 1 S = S P (S) > 1 (2.12)

32 2 A B P (A) + P (B) A B 2

2.2 33 2.2.4 A B P (B A) (conditional probability) P (B A) (2.18) 2.1 P (B A) = P (A B) P (A) (2.18) A B A B A B A 2.1 Venn n A n 1 B m A B m 1 Ā B m 2 (2.18) A B P (B A) = m 1 n 1 A A B P (A) = n 1 n P (A B) = m 1 n (2.18) 2.2.5 (2.18) P (A B) = P (A)P (B A) (2.19)

34 2 2 P (B) = P (B A) (2.20) (2.19) P (A B) = P (A)P (B) (2.21) P (B) = P (B A) A B P (B A) A B P (B) *7 2 ( A) ( B) 2 2 A B A B A B 2 (2.21) A, B *8 2.2.6. (A) (B) *7 (2.20) A B P (A) = P (A B) A, B *8 2 A B (2.21)

2.2 35 *9 (2.18) P (B A) > P (B) (2.22) P (A B) P (A) > P (B) P (A B) > P (A)P (B) A B 2 (2.22) P (A B) = P (A)P (B) 2 A B 2.2 a, b, c, d *9

36 2 a c b d A B 2.2 A B, A B, A B, A B a, b, c, d P (A) = P (B) = P (A B) = P (B A) = a + c a + b + c + d a + b a + b + c + d a a + b a a + c A B (2.23) P (B A) = P (B) (2.23) a c = b d (2.24) a c a c (odds) b d ( ) (odds ratio) = a c b d = ad bc 1 1 1

2.2 37 2 1 100 20 1/5 20/80 = 0.25 1 20 80 11 87 = 1.98 20 11 80 87 (double blind test) 8 2.2.7 (Bayez s theorem) * 10 A, B S A, B S = A + B A, B E P (E A), P (E B) *10 18

38 2 E A P (A E) A, B, C,... P (A E) = P (A)P (E A) P (A)P (E A) + P (B)P (E B) (2.25) A, B E P (E A) P (E B) P (E A) > P (E B) P (A E) A S B E 2.3 2.3 (2.18) Ā B B E 2 P (A E) = P (A)P (E A) (A E ) P (B E) = P (B)P (E B) (B E ) 2.3 (2.26) P (A E), P (B E) 2 P (A E) E A P (E A) (2.25) (2.25)

2.2 39 P (A E) = P (A)P (E A) P (E) (2.26) P (E) P (E A) P (E B) 2.2.8 A, B A 800 5% B 2000 12% 1 A A P (A), A P (E A), A P (A E) B * 11 P (A E) = = P (A)P (E A) P (A)P (E A) + P (B)P (E B) 800 800+2000 = 0.143 800 800+2000 0.05 2000 0.05 + 800+2000 0.12 2.2.9 ( 2.25) 100 * 12 2 *11 800 2800 *12 100

40 2 800 A = 100 800 + 2000 = 28.571 B = 100 2000 = 71.429 (2.27) 800 + 2000 A = 28.571 0.05 = 1.4286 B = 71.429 0.12 = 8.5715 (2.28) A 1.4285 = 0.143 (2.29) 1.4285 + 8.5715 * 13 2 4 : 1.5% 0.5% 2% *13

2.3 41 2.3 2.3.1 2 5 3 1/4 H O 50 O H H O O H O O O O O O H H H O O H H O O H O O O H O H O O O H H H O O H O O O O O O H O H O O O O H O O H O O O O O O O O H O O O O O O O O O H O O O O O O O O O O O H H O O O O O H O O O O O O O H 2 5 0 9 50 2.4 1/10

42 2 2.4 9 1 0.25 3 2 0.25 4 1 3 0.25 0.25 3 3 23 10

2.3 43 2.3.2 70% 70% 1. 70% 2. 70% 3. 70% 1. 70% 2. 1 1 0 70% 3. 70% 70% 70% 70% 3. 2.3.3

44 2 10 1/1024 2.3.4

2.3 45 2 * 14 2 6 (2.17) 52 1 *14

46 2 2 7 1/2

47 3 3.1 2 2 A, B E 1, E 2, E 3 E 1 : 2 A E 2 : 1 A, 1 B E 3 : 2 B 3 1/4, 2/4, 1/4 P (E 1 ) = 1/4 P (E 2 ) = 2/4 P (E 3 ) = 1/4 (3.1) E i 1 B X P (X = 0) = 1/4 P (X = 1) = 2/4 P (X = 2) = 1/4 (3.2)

48 3 (stochastic variable) X, Y X x 1, x 2,... (discrete probability variable) 1 0 0 1 1 1 20 0 20 3000 0 3000 X = x 1, x 2,... (probabilty function) (probability density) f(x i ) P (X = x i ) = f(x i ) (3.2) f(0) = 1/4 f(1) = 2/4 f(2) = 1/4 (3.3) *1 F (X) (distribution function) *2 F (0) = 1/4 F (1) = 3/4 F (2) = 4/4 (3.4) *1 P (X = x) f(x) x P (X = x) = f(x) *2

3.2 49 3 1 2 2 2 12 3.2 (discrete uniform distribution) 1/2 1/6 f(1) = 1/6 f(2) = 1/6 f(6) = 1/6 (3.5) n f(x) = { 1/n (x = 1, 2,..., n) 0 ( ) 3.3 X = x 1, x 2,..., x n n x i f(x), F (x) n f(x i ) = f(x 1 ) +... + f(x n ) = 1 (3.6) i=1

50 3 i F (x i ) = f(x k ) (3.7) k=1 F ( ) = 0 (3.8) F ( ) = 1 (3.9) F (x) 3 2 3 X X X = 0, 1, 2, 3 1/8, 3/8, 3/8, 1/8 f(0) = 1/8 f(1) = 3/8 f(2) = 3/8 f(3) = 1/8 F (0) = 1/8 F (1) = 4/8 F (2) = 7/8 F (3) = 1

3.4 51 3.4 X (mean, average) (expectation value) *3 E[X] µ n E[X] = µ = x i f(x i ) i=1 = x 1 f(x 1 ) +... + x n f(x n ) (3.10) 1 p.17 (1.9) X V [X] σ 2 n V [X] = σ 2 = (x i µ) 2 f(x i ) (3.11) i=1 2 1.1.4 (1.6) V [X] = E[X 2 ] E[X] 2 (3.12) = 2 2 3 3 0.1, 0.05, 0.01 100, 200, 1000 0.001 2000 2000 2000 0 1 0.1 0.05 0.01 0.001 = 0.839 2000, 0, 100, 200, 1000 x 1, x 2,..., x 5 f(x) f(x 1 ) = 0.001, f(x 2 ) = 0.839, f(x 3 ) = 0.1, f(x 4 ) = 0.05, f(x 5 ) = 0.01 *3

52 3 (3.10) E[X] = f(x 1 )x 1 + f(x 2 )x 2 + = 0.001 ( 2000) + 0.839 0 + 0.1 100 + 0.01 1000 = 28 28 3 4 3 3 µ = 1 8 0 + 3 8 1 + 3 8 2 + 1 8 3 = 3 2 = 1.5 1.5 E[X 2 ] = 1 8 02 + 3 8 12 + 3 8 22 + 1 8 32 = 3 E[X 2 ] {E[X]} 2 = 3 ( ) 2 3 = 3 2 4 = 0.75 3 5 n µ = 1 1 n + 2 1 n +... + n 1 n = n(n + 1) 2 1 n = n + 1 2 σ 2 = E[X 2 ] {E([X]} 2 = 1 2 1 n + 22 1 n +... + n2 1 (n + 1)2 n 2 2 = n2 1 12 n k = k=1 n k 2 = k=1 n(n + 1) 2 n(n + 1)(2n + 1) 6

3.5 53 3.5 *4 2 E[X + Y ] = E[X] + E[Y ] (3.13) V [X ± Y ] = V [X] + V [Y ] (3.14) * 5 V [X] = E[X 2 ] (E[X]) 2 (3.15) E[aX + b] = ae[x] + b (3.16) V [ax + b] = a 2 V [X] (3.17) E[XY ] = E[X]E[Y ] (3.18) 3 6 A, B A 27.6 1.5 B *4 *5

54 3 16.2 1.2 (3.13) 2 1.5 2 + 1.2 2 = 1.92 A, B

55 4 4.1 4.1.1 A 40% 10 4 A 10 4 A 6 0.4 4 (1 0.4) 6 A 2 6 A 2 0.4 2 (1 0.4) 6 0.4 2 A 4 6 A 4 6 10C 4 = 10! = 210 4! 6!

56 4 10C 4 0.4 4 (1 0.4) 6 = 210 0.4 4 (1 0.4) 6 = 0.251 n p x f(x) = n C x p x (1 p) n x, (x = 0, 1,..., n) (4.1) (binominal distribution) B[n, p] *1 4 1 5 1 5 60% 1/5 = 0.2 5 5 4 3 5! 5! 0! 0.25, 5! 4! 1! 0.24 0.8, 5! 3! 2! 0.23 0.8 2, 0.058 *2 4.1.2 B[n, p] *3 µ = np (4.2) σ 2 = np(1 p) (4.3) *1 B[n, p] x n p *2 5% 10 *3 p n np

4.2 57 4.1.3 n = 5 f(x) 0.3 0.2 n = 15 n = 30 p = 0.5 0.1 0 0 5 10 15 20 25 30 x 4.1 p = 0.5 0.3 p = 0.05 p = 0.1 n = 50 0.2 f(x) p = 0.25 p = 0.5 p = 0.7 0.1 0 0 10 20 30 40 50 x 4.2 p 4.1 p = 1/2 n 5, 15, 30 n 4.2 p 50 np 4.2 A, O, B, AB 40%, 30%, 20%, 10% 10 2 4 2 2 (polynomial distribution)

58 4 2 A 4 O 2 B 2 AB 0.4 2 0.3 4 0.2 2 0.1 2 10! 2!4!2!2! E 1, E 2,..., E k, p 1, p 2,..., p k, p 1 + p 2 +... + p k = 1 n x 1, x 2,..., x k f(x 1, x 2,..., x k ) f(x 1, x 2,..., x k ) = n! x 1! x 2!... x k! px1 1 px2 2... px k k (4.4)

4.3 59 4.3 4.3.1 (Poisson distribution) 8 24 480 24 1 0.05 480 60 B[60, 0.05] n = 60, p = 0.05 60 C x 4.3.2 n, p 0, f(x) = lim n,p 0 n C x p x (1 p) n x = µx x! e µ (4.5) µ = np µ P [µ] n (4.5) *4 4.3.3 µ (4.3) *4 Windows

60 4 σ 2 = lim p 0 np(1 p) = np = µ (4.6) µ 0.5 µ = 0.25 µ = 0.5 µ = 1 µ = 2 µ = 4 µ = 8 µ = 12 0 0 5 10 15 20 4.3 µ x 4 2 60 2 60 5 f(x) = µx x! e µ µ = 2 5 4 f(0) f(4) 1 1 (f(0) + f(1) + f(2) + f(3) + f(4)) ( ) = 1 e 2 2 0 0! + 21 1! + 22 2! + 23 3! + 24 4! = 0.053 5% 4.3.4 n p np n > 50 n 50 np 5

4.3 61 f(x) = µx x! e µ n p µ 1 5 n p n p µ = np µ = 5 1 10 0.03 1 10 1,2

62 4 4 1 2/3 8 4 4 2 6 2 4 3 1 2.5

63 5 5.1 5.1.1 1 0.1 kg 58.32417081... kg *1 5.1.2 3.2 1 n 1/n 5.1 *1

64 5 P (X) X = 1, 2,..., 6 1/6 1 6 5.1.3 2 m 5.2 5.1 X 5.2 [0, 2] *2 p *3 5.1.4 5.3 2 m 0.2 P(X) 0.1 0 0 1 2 3 4 5 6 7 X 5.1 *2 1 2 [1, 2] *3 p

5.1 65 0 1 2 5.2 1 f(x) 0.5 0 0 0.3 1 2 x 5.3 2 m f(x) = 0.5, (0 x 2) f(x) = 0, (5.1) 0.5 1 [0, 2] 1 1 [0, 0.3] 0.3 0.5 = 0.15 [a, b] f(x) x P (a X b) P (a X b) = b a f(x) dx (5.2)

66 5 1 *4 f(x) dx = 1 (5.3) f(x) = 0.5, a = 0, b = 0.3 P (0 X 0.3) = 0.3 0 0.5 dx = [0.5x] 0.3 0 = 0.15 (5.4) x f(x) (probability density function) f(x) Φ(z) Φ(z) = z f(x) dx (5.5) 5.4 f(x) Φ( ) = 0 (5.6) Φ( ) = 1 (5.7) Φ(z) y = f(x) z x 5.4 Φ(z) f(x) 5.3 5.1 Φ(z) = z 0.5 dx = 0.5z, (0 z 2) (5.8) *4 [, ] X

5.1 67 5 1 5.1 0.8 m 1.5 m f(x) = 1/2 x = 0.8 x = 1.5 1.5 0.8 1 [ x ] 1.5 2 dx = = 0.35 2 0.8 5.3 5.1.5 (3.6) f(x) dx = 1 (5.9) 1 (normalizing condition), ( ) (3.10) E[X] = µ = (3.11) V [X] = σ 2 = xf(x) dx (5.10) (x µ) 2 f(x) dx (5.11) 3.4 (3.12) V [X] = E[X 2 ] {E[X]} 2 (5.12) E[X 2 ] = x 2 f(x) dx (5.13)

68 5 5.1.6 [0, 1] 10 0.2747, 0.2288, 0.6893, 0.1855, 0.9086, 0.1876, 0.9291, 0.5324, 0.3335, 0.8568 f(x) = 1, (0 x 1) f(x) = 0, ( ) (5.14) 1/2 (5.10) µ = 1 0 x 1 dx = 1 2 (5.15) (5.12) σ 2 = 1 0 x 2 1 dx µ 2 = 1 12 (5.16)

5.2 69 5.2 B[n, p] n 5.5 p = 0.4 n 0.3 n = 4 p = 0.4 0.2 n = 12 p = 0.4 f(x) 0.2 0.1 f(x) 0.1 0 0 2 4 x 0 0 5 10 x 0.1 n = 60 p = 0.4 0.06 n = 120 p = 0.4 f(x) 0.05 f(x) 0.04 0.02 0 0 20 40 60 x 0 0 50 100 x 5.5 B[n, p] n n (normal distribution) f(x) = 1 ] (x µ)2 exp [ 2πσ 2σ 2 (5.17)

70 5 exp(a) e a *5 µ, σ 2 *6 4.1.2 µ = np, σ 2 = np(1 p) n B[n, p] = n C x p x (1 p) n x n 1 ] (x µ)2 exp [ 2πσ 2σ 2 (5.18) 0.3 p = 0.4 n = 6 0.15 p = 0.4 n = 24 f(x) 0.2 f(x) 0.1 0.1 0.05 0 0 2 4 6 x 0 0 10 20 x 5.6 σ µ ( ) n = 6, 24 5.6 n = 24 np > 5 n(1 p) > 5 p = 0.5 n = 10 N[µ, σ 2 ] µ, σ 2 N[µ, σ 2 ] *5 exp *6 µ (5.17) x = µ x = µ

5.3 71 5.3 5.3.1 µ, σ 2 x (5.19) (5.20) z = x µ σ ϕ(z) = 1 2π e z2 /2 (5.19) (5.20) 1 (5.19) (standardization) (b) µ σ µ µ+σ x (a) 3 2 1 0 1 2 3 4 5 x 5.7 N[µ, σ 2 ] z = (x µ)/σ N[0, 1] 5.7 N[µ, σ 2 ] x z N[0, 1] (5.19) x = σz + µ (5.21)

72 5 5.1 z Φ(z) z Φ(z) z Φ(z) z Φ(z) 0.00 0.500000 1.00 0.841345 2.00 0.977250 3.00 0.998650 0.10 0.539828 1.10 0.864334 2.10 0.982136 3.10 0.999032 0.20 0.579260 1.20 0.884930 2.20 0.986097 3.20 0.999313 0.30 0.617911 1.30 0.903200 2.30 0.989276 3.30 0.999517 0.40 0.655422 1.40 0.919243 2.40 0.991802 3.40 0.999663 0.50 0.691462 1.50 0.933193 2.50 0.993790 3.50 0.999767 0.60 0.725747 1.60 0.945201 2.60 0.995339 3.60 0.999841 0.70 0.758036 1.70 0.955435 2.70 0.996533 3.70 0.999892 0.80 0.788145 1.80 0.964070 2.80 0.997445 3.80 0.999928 0.90 0.815940 1.90 0.971283 2.90 0.998134 3.90 0.999952 5.3.2 z Φ(z) 5.1 Φ(z) = z 1 2π e x2 /2 dx (5.22) 5.8 x = z [, z] [0, z] z = 0 Φ(z) = 0 0.5 Φ(z) φ(x) 0 z x 5.8 ϕ(x) = 1/ 2π exp( x 2 /2) Φ(z)

5.3 73 5.3.3 f(z) 2 f(z) 1 f(z) N[0, 1] 1 z 1 z < 1 0.841345 z > 1 1 0.841345 = 0.158655 5.8 z < 1 *7 1 0.158655 2 = 0.68269 N[µ, σ 2 ] 1. x z 2. z 3. 2 5 2 65 12.5 90 % (5.19) 90 z = (90 65)/12.5 = 2.0 Φ(2.0) 0.97725 100 97.7 90 2.3 % 5.2 n *7 2Φ(z) 1

74 5 5 3 A 40% 10 A 4 6 µ = np, σ = np(1 p) µ = 4, σ = 10 0.4 0.6 = 1.55 4 6 3.5 6.5 z 1 = (3.5 4)/1.55 = 0.32, z 2 = (6.5 4)/1.55 = 1.61 Φ(1.61) Φ( 0.32) 5.9 z Φ( 0.32) = 1 Φ(0.32) = 0.374 Φ(1.61) = 0.946 0.572 0 1 2 3 4 5 6 7 8 9 10 5.9 B[10, 0.4] x = 4, 5, 6 3.5 6.5 B[n, p] [x 1, x 2 ] x 1, x 2 x 1, x 2 z 1 = x 1 µ σ z 2 = x 2 µ σ (5.23) Φ(z 1 ) Φ(z 2 ) 5.9 x 1, x 2 1 [z 1, z 2 ]

5.3 75 z 1 = x 1 1 2 µ σ z 2 = x 2 + 1 2 µ σ (5.24) (5.24) 1/2 z 1, z 2 σ 1/2 n = 400 p = 0.5 σ = 100 1/2 0.5% 5 1 100 kg 5 2 (standard score) µ, σ µ 50 σ, 2σ, 50, 60, 12000 58 5 3 A 40% 24 A 9 12 5 4 1/2 4 49% 51% 5 5 1. 45.0 % 400 45.0±2.5%

76 5 2. 2000 45.0 ± 2.5 %

5.4 77 5.4 (central limit theorem) X 1, X 2,..., X n µ, σ 2 X = 1 n n X i (5.25) n Z = (X µ) (5.26) σ n Z N[0, 1] (5.20) X µ, σ 2 /n N[µ, σ 2 /n] Z n 10 *8 i=1 5 6 5.1.6 (68 ) 1/12 ( (5.16) ) 2 *8 2 3

78 5 4 N (1/2,1/144) 2 0 0.2 0.4 0.6 0.8 5.10 [0, 1] 12 ( (3.14)) [0, 1] 12 6

79 6 6.1 6.1 (random sampling) (population) (population mean) (population variance) (sample) µ σ 2 x 1 x 2 x 2... x n X s 2 6.1 µ, σ 2 X 1, X 2,..., X n X, s 2

80 6 6.1.1 (population) *1, µ, σ 2 (population parameters) *2 *3 (sample) (size) 6.1.2 (sampling) (random sampling) *4 *1 *2 *3 *4

6.1 81 6.1 98474 71279 63082 78829 42648 14443 69985 58505 73760 96835 37252 88586 62283 71713 61004 62979 29684 15151 41589 44958 43215 04177 61654 95413 43685 95877 61315 09869 46923 85614 76004 67425 09426 72476 52651 44729 98959 10064 09796 98117 60610 70770 57281 67053 19024 01629 41143 01965 07339 99938 29309 69622 63555 86700 03750 39202 84902 06042 74703 02108 80801 28750 82589 28729 15136 88027 03250 15225 78384 25588 22125 23483 80242 76254 93014 67361 03408 69128 47009 48339 09106 73507 67285 93722 35009 67651 95285 00497 76141 58511 84030 37979 89450 30578 64083 12380 12603 51943 37857 46401 6.1 5 9,8,4,7,4,3,7,2,5,2,4,3,2,1,5,.. 5 5 4,6,4,5,1,0,1,9,3,6,... *5 2 2 2 6 1 6.2 10 *5

82 6 6.2 100 /kg( ) 43.6 45.2 45.4 45.8 47.2 47.8 48.2 48.7 48.8 48.9 49.0 49.0 49.4 49.5 49.8 50.4 50.5 50.9 50.9 51.2 51.2 51.2 51.3 51.3 51.6 51.7 51.7 51.8 52.0 52.0 52.1 52.1 52.1 52.2 52.3 52.7 52.7 52.8 52.9 52.9 53.1 53.1 53.8 54.0 54.5 54.5 54.6 54.7 54.7 54.7 54.8 54.9 55.1 55.1 55.2 55.3 55.4 55.4 55.4 55.6 55.7 55.8 55.9 56.1 56.3 56.3 56.3 56.4 56.5 56.7 56.8 57.0 57.1 57.1 57.2 57.3 57.6 57.7 57.8 58.1 58.4 58.6 58.7 58.7 58.7 58.7 59.1 59.3 59.9 60.0 60.1 60.3 60.5 60.6 60.6 60.7 61.3 62.7 64.2 64.6 100 0 99 2 6.1 1 2 98, 47, 47, 12, 79, 63, 08, 27, 88, 29, 42, 64, 81, 44,... 47 2 3 47 98, 47, 47, 12, 79, 63, 08, 27, 88, 29 64.2, 54.7, 54.7, 49.4, 58.1, 56.1, 48.8, 51.8, 59.9, 52.0 2 47 98, 47, 12, 79, 63, 08, 27, 88, 29, 42 64.2, 54.7, 49.4, 58.1, 56.1, 48.8, 51.8, 59.9, 52.0, 53.8

6.2 83 6.2 6.2.1 6.1 n X 1, X 2,..., X n X 1, X 2 X 1, X 2,..., X n 2, X = 1 n (X 1 + X 2 + + X n ) ( ) (6.1) s 2 = 1 ( (X1 X ) 2 ( + X2 X ) 2 ( + + Xn X ) ) 2 ( ) (6.2) n s n n 6.2 18 16 n = 10 n = 100 14 12 10 8 6 4 2 0 0 2 4 6 8 10 12 14 16 18 20 6.2 [0, 20] 10 100 20 n = 10, 100 X, X X µ = 10

84 6 6.2.2 X X n σ 2 X E[X] = µ V [X] = σ2 n (6.3) (6.4) A.5 (p.152) 2 (6.4) (standard error) 6.3 6.3.1 X s 2 s 2 A.6 (p.152) 6 2 E[s 2 ] = n 1 n σ2 (6.5) 6 (4.32 kg) 2 (s 2 ) ( ) (6.5) E[s 2 ] σ 2 σ σ 2 = n n 1 E[s2 ] = 6 5 4.322 6 σ = 4.32 = 4.73 5

6.3 85 4.73 kg (6.5) *6 s 2 s 2 = 1 ( (X1 X ) 2 ( + X2 X ) 2 ( + + Xn X ) ) 2 n 1 ((X 1 µ) 2 + (X 2 µ) 2 + + (X n µ) 2) n X 1, X 2, µ 2 σ 2 *7 s 2 µ X µ X s 2 (X 1 X) 2 (X 1 µ) 2 s 2 σ 2 n t- X X µ 6.3 6 (6 ) (6 ) 6.3.2 V [s 2 ] χ 2 - *6 *7 1 X 1 X 1 µ 2 σ 2

86 6 6.4 6.4.1 X X n X n = 1 2 n X n X n X 6 3 30% 1000 1, 0 0 1 p = 0.3 1 1 p, 0 1 p (3.10) µ = 1 p + 0 (1 p) = p

6.4 87 (3.11) σ 2 = (1 µ) 2 p + (0 µ) 2 (1 p) = p(1 p) µ = 0.3, σ 2 = 0.21 µ = 0.3 σ 2 /n = 0.21/1000 = 0.00021 0.0145 µ *8 2 0 1 6.4.2 N[µ, σ 2 ] n (6.3),(6.4) X µ σ 2 /n ( σ/ n) 6.4 n X σ/ n +σ/ n µ X 6.4 N[µ, σ 2 ] n X 6.4 *8

88 6 Z = X µ n(x µ) σ/ n = σ (6.6) Z N[0, 1] 5 (p.71) 6.4.3 X (6.6) X (6.3) (6.4) µ σ 2 X (6.5) s 2 σ 2 n 1 n E[s 2 ] = n 1 n σ2 s 2 σ 2 n 1 n σ 2 = n n 1 s2 X X X σ n σ n n 1 s n n 1 s s = n n 1 X µ s 2 /(n 1) ( s n 1 )

6.4 89 (6.6) Z Z = X µ n 1(X µ) s = n 1 s (6.7) Z N[0, 1] t- (6.7) Z T T = n 1(X µ) s (6.8) s Z( T ) *9 6.3 s 2 n σ 2 s 6.5 (6.8) s t- t- * 10 t- (6.9) 6.6 ν f ν (T ) = c (1 + T 2 ν ) ν+1 2, (ν = 1, 2, 3,...) (6.9) c f ν (t) 1 ν *9 s T *10 Student (Willam S. Gosset)

90 6 1.2 1 n = 5 n=100 0.8 0.6 0.4 0.2 0 0 2 4 6 8 10 s 6.5 N[0, 25] s n = 5 n = 100 100 t- ν = 3 ν = 1 0 t 6.6 ν = 1, 3 t- : t- N[µ, σ 2 ] n X s 2 T = n 1(X µ) s (6.10) T n 1 t-

6.5 χ 2 91 6.5 χ 2 6.5.1 s 2 (6.5) E[s 2 ] s 2 84 1 * 11 * 12 µ = 0, σ 2 = 1 N[0, 1] N[0, 1] n Z = 1 n (X2 1 + X2 2 +... + X2 n ) (6.11) Z T n (x) = 1 2 n/2 Γ(n/2) xn/2 1 e x/2 (6.12) (6.12) n χ 2 n = 1, 2,..., 7 χ 2 * 13 6.7 *11 *12 *13

92 6 T 1 (x) = 1 2π x 1/2 e x/2 (6.13) T 2 (x) = 1 2 e x/2 (6.14) T 3 (x) = 1 2π x 1/2 e x/2 (6.15) T 4 (x) = 1 4 xe x/2 (6.16) T 5 (x) = 1 3 2π x3/2 e x/2 (6.17) T 6 (x) = 1 4 x2 e x/2 (6.18) T 7 (x) = 1 15 2π x5/2 e x/2 (6.19) 0.5 ν = 1 ν = 2 ν = 3 ν = 4 ν = 5 ν = 6 ν = 7 0 5 10 15 6.7 ν = 1, 2,..., 7 χ 2 (6.12) Γ(x) n! * 14 n ν µ σ 2 N[µ, σ 2 ] *14 x Γ(x) = (x 1)! Γ(1/2) = π, Γ(3/2) = 1 2 π, Γ(5/2) = 3 4 π, Γ(7/2) = 15 8 π, Γ(9/2) = 105 16 π,...

6.5 χ 2 93 n Z = 1 σ 2 ( (X1 µ) 2 + (X 2 µ) 2 +... + (X n µ) 2) (6.20) Z n χ 2 µ σ 2 µ σ 2 N[µ, σ 2 ] n Z = 1 σ 2 ( (X1 X) 2 + (X 2 X) 2 +... + (X n X) 2) = ns2 σ 2 (6.21) Z n 1 χ 2 s 2 (6.2) s 2 σ 2 6.5.2 χ 2 χ 2 6.8 α t χ 2 T n (x) α t χ 2 α 0.995 0.975 0.950 0.900 0.500 0.05 0.025 0.01 0.005 ν = 6 0.676 1.24 1.64 2.20 5.35 12.59 14.45 16.81 18.55 ν = 9 1.73 2.70 3.33 4.17 8.34 16.92 19.02 21.67 23.59 ν = 10 2.16 3.25 3.94 4.87 9.34 18.31 20.48 23.21 25.19 ν = 11 2.60 3.82 4.57 5.58 10.34 19.68 21.92 24.73 26.76 T 6 (x) α 0.995 x = t 0.676 N[0, 1] 6 (6.11) Z = 1 6 (X2 1 + + X 2 6 ) 0.676 99.5%

94 6 Tn(x) α 0 t x 6.8 χ 2 χ 2 1%, 5%, 90% α t 6 4 (χ 2 ) 82 6.2 10 30 7.02, 8.03, 8.53, 9.34, 13.12, 13.65, 14.17, 14.24, 15.77, 15.83, 16.13, 16.30, 16.52, 16.56, 16.89, 17.41, 17.47, 17.77, 18.25, 18.36, 19.21, 19.48, 20.68, 21.25, 22.91, 24.33, 25.54, 26.24, 26.80, 41.87 50% χ 2 s 2 50% 16.89 17.41 17.15 (6.21), Z = ns 2 /σ 2 n 1 = 9 χ 2 8.34 Z = ns2 σ = 10 17.15/σ 2 = 8.34 2 20.6 9 1 1 17.84 6 5 (µ ) N[3, σ 2 ] 6 50% (X 1 µ) 2 + (X 2 µ) 2 +... + (X n µ) 2 21 σ 2 6 χ 2 α = 0.5 t 5.35

6.5 χ 2 95 ( (6.20) Z = 1 σ 2 (X1 µ) 2 + (X 2 µ) 2 +... + (X 6 µ) 2) 5.35 α 0.5 5.35 = 1 σ 21 σ 2 = 3.92 2 6 6 (µ ) 11 90% 12.5 σ 2 µ (6.20) (6.21) n n 1 χ 2 ν = 10 α = 0.900 t = 4.87 12.5 (6.21) 4.87 = ns2 σ 2 n = 11, s 2 = 12.5 σ 2 = 28.1

96 6 6 1 1 (10000m 2 ) 10 m 1 (100m 2 ) 100 5 6 2 1970 6 3 54.2 g 0.22 g 10 54.1 g 54.3 g 6 4 12 30 (g) 8.76, 9.47, 9.99, 11.85, 12.59, 13.23, 14.79, 18.83, 20.32, 20.74, 21.00, 21.11, 22.40, 23.43, 24.61, 26.14, 27.41, 29.53, 32.22, 33.51, 41.81, 50.57 6 4 χ 2 1

97 7 7.1 X E[X] = µ V [X] = σ2 n (7.1) (7.2) 35% *1 X X ± α α *1 1000 3000

98 7 95% (point estimation) (point estimator) x 1 x 2 95% (interval estimation) (interval estimator) (confidence interval) 7.1.1 7.1 N[0, 1] α z α 100(1 α) α = 0.05 z 0.05 95 Φ(z) = 0.95 z 95 1.645 1 p.12 N(0,1) α 0 zα 7.1. z α 100(1 α) 7.1 95 5 90%

7.1 99 z α z α/2 α/2 z α/2 α 1 α 7.2 N(0,1) α/2 α/2 -zα/2 0 zα/2 7.2 z α/2 7.1 90% 90% z = 1.28 Φ(z) = 0.899727, z = 1.29 Φ(z) = 0.901475 Φ(z) = 0.9 z ( ) z = 1.28 + 0.9 0.899727 (1.29 1.28) = 1.28156 1.282 0.901475 0.899727 z α z α/2 N[0, 1] 90% 95% 7.1 90% 5% 95 z 0.05 1.645, 7.1 90 95 97.5 99 99.5 α 0.10 0.05 0.025 0.01 0.005 z α 1.282 1.645 1.960 2.326 2.576

100 7 1.645 95% ±1.960

7.2 101 7.2 µ σ 2 θ X s 2 Θ θ = E[Θ] (7.3) Θ θ (unbiased estimation) *2 Θ (X s 2 ) θ E[X] = µ (7.1) µ X σ 2 (6.5) E[s 2 ] = n 1 n σ2 [ ] n σ 2 = E n 1 s2 n (n 1) s2 s 2 = 1 n ( (X1 X) 2 + (X 2 X) 2 +... + (X n X) 2) 1 ( (X1 X) 2 + (X 2 X) 2 +... + (X n X) 2) (7.4) n 1 σ 2 (7.4) *3 1 (7.4) *2 unbiased *3 Excel VAR,STDEV

102 7 7.3 X µ 6 (p.86, ) 3 *4 σ 2 (7.1) (7.2) X X 84 (6.5) X Student t- X 7.3.1 6.4.2 (p.87) X Z = X µ n(x µ) σ = n σ (7.5) (7.5) Z *4

7.3 103 90% 95% 7.1 1.645 Z ±1.645 X (7.5) Z = ±1.645 X [X 1.645 σ n, X + 1.645 σ n ] (7.6) λ [X λ σ n, X + λ σ n ] (7.7) X σ n λ 7.3.2 88 Z N[0, 1] Z = n 1(X µ) λ [ X λ s s n 1, X + λ ] s n 1 (7.8) 90% 95% 1.645 λ 7.3.3 Student t- Student t- 89 T n 1 t- T = n 1(X µ) s (7.9)

104 7 7.2 t- α 0 90 95 97.5 99 99.5 99.75 α 0.10 0.05 0.025 0.01 0.005 0.0025 ν = 1 3.078 6.314 12.706 31.821 63.657 ν = 2 1.886 2.920 4.303 6.965 9.925 14.089 ν = 5 1.476 2.015 2.571 3.365 4.032 4.773 ν = 6 1.440 1.943 2.447 3.143 3.707 4.317 ν = 7 1.415 1.895 2.365 2.998 3.499 4.029 ν = 8 1.397 1.860 2.306 2.896 3.355 3.832 ν = 9 1.383 1.833 2.262 2.821 3.250 3.690 ν = 10 1.372 1.812 2.228 2.764 3.169 3.581 z α t t- ( 7.2) (7.10) (7.7) (7.8) 90% 95% ( t- ) λ λ t- [ ] s s X λ, X + λ (7.10) n 1 n 1 n t- 7.2

7.3 105 7.3.4 7 1 L 10 (g) 65.1, 67.5, 71.5, 68.4, 70.1, 72.2, 68.7, 69.3, 70.6, 67.1 4.0 g 2 90% 95% 69.05 n = 10, σ 2 = 4.0 90% α = 0.05 7.1 z α/2 = z 0.05 = 1.645 (7.6) 69.05 1.645 4.0/ 10 = 68.0 69.05 + 1.645 4.0/ 10 = 70.1 µ 90% 68.0 < µ < 70.1 95% 67.8 < µ < 70.3 7 2 40 X s 2 53.8 kg, 18.81 kg 2 90% 18.81 18.81 53.8 1.645 = 52.6575..., 53.8 + 1.645 = 54.9424... 40 1 40 1 90% 52.7 < µ < 54.9 7 3 L 10 (g) 65.1, 67.5, 71.5, 68.4, 70.1, 72.2, 68.7, 69.3, 70.6, 67.1 90% 99%

106 7 105 X s X = (65.1 + 67.5 + 71.5 +... + 67.1)/10 = 69.05 s 2 = (65.1 2 + 67.5 2 + 71.5 2 +... + 67.1 2 )/10 X 2 = 4.1845 ν = n 1 = 9 s = 4.1845 = 2.045 90% ν = 9 t- 95 z 0.05 1.833 [ 69.05 1.833 2.045, 69.05 + 1.833 2.045 ] 9 9 µ 67.8 < µ < 70.3 99% 66.8 < µ < 71.3 7.3.5 t- σ 2 X V [X] σ 2 /n ( (6.4) ) ns2 n 1 X 7.3.2 X µ t- 6.6

7.3 107 7.3.6 7.3 λ t- n n 1 n n = 20 Student t- 20 n n 1 s2 s 2 1 n 100 n n 1 1 (7.11) s 2 7.3 : λ ±λ σ n ±λ s n 1 ±λ s n 1 t- 3 3 5 (6.8) T t-

108 7 n n s 2 = 1 ( (x1 X) 2 + (x 2 X) 2 + + (x n X) 2) n X µ 1 ( (x1 µ) 2 + (x 2 µ) 2 + + (x n µ) 2) n µ t- 7 1 100 30% 95% 7 2 12 16, 4 µ 99%

7.3 109 7 3 7 2 80 µ 99%

111 8 (statistical test) (hypothetical test) (hypothesis) 8.1 *1 8.1.1 *1

112 8 X 1 10 X 1 1 1250 1305 10 1273 29.5 X 1286 35.4 1286 1273 X 10

8.1 113 8.1.2 X X I X X (hypothesis) II X X 10 1273 1286 X II (= ) II

114 8 X (null hypothesis) *2 I (alternative hypothesis) H 0, H 1 *3 (test) µ σ n = 10 X µ X σ 2 /n 84 (6.3)(6.4) 10 N[µ, σ 2 /n] N[1286, 125.3] *4 1286 11.2 (= 125.3) 10 1273 X N[1286, 11.2 2 ] 1273 1273 1286 11.2 = 1.16 *2 *3 *4 125.3 35.4 2 /10

8.1 115 1.28 10% (α = 0.1) 1273 10% *5 α = 0.1 σ=11.2 (1.0) 1273 ( 1.16) 1286 (0) 8.1 N[1286, 11.2 2 ] 1273 X 10 II X 0.1 (risk) A 0.01 A ( ) 0.01 8.1.3 p X z = 1.16 0.12 p 8.2 *5

116 8 α = 0.1 σ=11.2 (1.0) 1273 ( 1.16) 1286 (0) 8.2 z = 1.16 p 0.12 p p 0.015 p 8.1.4 4 1000 25 25000 1000 25

8.1 117 *6 ( ) *6

118 8 25 25 32.97, 36.37, 35.24, 36.03, 34.84, 33.63, 37.94, 33.48, 34.09, 33.74, 34.53, 36.86, 31.79, 35.61, 34.14, 34.51, 35.13, 32.83, 34.89, 32.19, 36.67, 36.01, 37.04, 35.1, 33.73 ( ) 1000 35.03 0.925 34.77 ( )

8.1 119 8.1.5 25 25 25 X X X = 32.97 + 36.37 + 35.24 + + 33.73 25 = 34.77 X X s 2 = σ 2 /25( s = σ/5 = 0.925/5 = 0.185) X 35.03 34.77 Z = Z 34.77 35.03 0.185 = 1.405 α = 0.05 1.405 1.645 0 1.645 Z 25 *7 *7

120 8 0.1 X 8.1.6 χ 2 25 25 6 6.5 (p.93) I µ σ 2 N[µ, σ 2 ] n Z = 1 σ 2 ( (X1 µ) 2 + (X 2 µ) 2 + + (X n µ) 2) (8.1) Z n χ 2 X s 2 (p.93) II µ σ 2 N[µ, σ 2 ] n Z = 1 σ 2 ( (X1 X) 2 + (X 2 X) 2 + + (X n X) 2) = ns2 σ 2 (8.2) Z n 1 χ 2 s 2 (6.2) (8.1) Z µ 35.03 σ 0.925

8.1 121 25 25 1000 32.97, 36.37,..., 33.73 X 1 = 32.97, X 2 = 36.37,... Z Z = 1 0.925 2 ( (32.97 35.03) 2 + (36.37 35.03) 2 + + (33.73 35.03) 2) = 70.49 I Z 25 χ 2 25 Z 25 χ 2 70.49 χ 2 χ 2 α 0.995 0.975 0.95 0.9 0.5 0.05 0.025 0.01 0.005 ν = 23 9.260 11.688 13.090 14.848 22.337 35.172 38.076 41.638 44.181 ν = 24 9.886 12.401 13.848 15.659 23.337 36.415 39.364 42.980 45.558 ν = 25 10.520 13.120 14.611 16.473 24.336 37.652 40.646 44.314 46.928 ν = 26 11.160 13.844 15.379 17.292 25.336 38.885 41.923 45.642 48.290 α = 0.05 0 50 37.7 ν = 25 χ 2 5% 70.49 5% 0.5% χ 2 0.5%

122 8 II II 25 X = 34.77 2 2 s 2 = 32.972 + 36.37 2 + + 33.73 2 25 34.77 2 = 2.347 Z Z = ns2 25 2.347 = σ2 0.925 2 = 68.57 n 1 = 24 χ 2 8.1.7 8 1 12 35.7, 35.03, 35.11, 34.21, 35.08, 34.86, 35.13, 35.09, 34.36, 35.23, 35.24, 35.67 35.03, 0.925 I Z Z = 1 0.925 2 ((35.7 35.03)2 + (35.03 35.03) 2 + + (35.67 35.03) 2 ) = 2.451 12 χ 2 α = 0.995, 0.005 3.074, 28.300

8.1 123 0.5% Z 2.451 1% χ 2 -

124 8 8.2 8.2.1 t- X χ 2 X 10 8.1.1 7.3.3 t- n µ, X, s T T = n 1(X µ) T n 1 t- 10 X T 10 1(1273 1286) T = = 1.322 29.5 α 0.1 0.05 0.025 0.01 0.005 0.0025 ν = 8 1.397 1.860 2.306 2.896 3.355 3.833 ν = 9 1.383 1.833 2.262 2.821 3.250 3.690 ν = 10 1.372 1.812 2.228 2.764 3.169 3.581 ν = 9 X 0.1 s

8.2 125 0.1 1.322 1.383 0 T 8.2.2 χ 2 1. 2. χ 2 χ 2 8.1 (contingency table) *8 39,45,21,83,... (observed frequency) *8 contingency

126 8 8.1 39 45 21 105 83 68 47 198 53 51 65 169 41 32 55 128 216 196 188 600 2 N A, B m n B 1, B 2, B 3,..., B n A 1 x 11, x 12, x 13,, x 1n a 1 A 2 x 21, x 22, x 23,, x 2n a 2,,,, A m x m1, x m2, x m3,, x mn a m b 1, b 2, b 3,, b n N A, B X (m 1)(n 1) χ 2 X = (x 11 a 1 b 1 /N) 2 a 1 b 1 /N + (x 21 a 2 b 1 /N) 2 a 2 b 1 /N (x 12 a 1 b 2 /N) 2 + (x 22 a 2 b 2 /N) 2 a 1 b 2 /N a 2 b 2 /N + (x 1n a 1 b n /N) 2 + (x 2n a 2 b n /N) 2 a 1 b n /N a 2 b n /N + + (x m1 a m b 1 /N) 2 a m b 1 /N + + (x m2 a m b 2 /N) 2 a m b 2 /N + + (x mn a m b n /N) 2 a m b n /N + + (8.3)

8.2 127 8.3 8.1 X N = 600, m = 4, n = 3, a 1 = 105, a 2 = 198,..., b 1 = 216, b 2 = 196, x 11 = 39, x 12 = 45,, x 43 = 55 X = (39 105 216/600) 2 105 216/600 (41 128 216/600) 2 128 216/600 + (83 198 216/600)2 198 216/600 + + + (55 128 188/600)2 128 188/600 (53 169 216/600)2 169 216/600 = 25.9 X A, B 6 (= (4 1) (3 1)) χ 2 X χ 2 A B X, 25.9 8.3 χ 2 α α = 0.05 X 12.59 (5% 12.59 ) χ 2 5% X 25.9 5%, 5% * 9 + 8.2 χ 2 α 0.995 0.975 0.950 0.900 0.500 0.05 0.025 0.01 0.005 ν = 6 0.676 1.24 1.64 2.20 5.35 12.59 14.45 16.81 18.55 ν = 9 1.73 2.70 3.33 4.17 8.34 16.92 19.02 21.67 23.59 ν = 10 2.16 3.25 3.94 4.87 9.34 18.31 20.48 23.21 25.19 ν = 11 2.60 3.82 4.57 5.58 10.34 19.68 21.92 24.73 26.76 X χ 2 2 *9

128 8 α = 0.05 0 12.59 X 8.3 n = 6 χ 2 8 1 2 1 2 3 : 1 144 119 25 1 5% 1% 8 2 12 g 11.78, 12.92, 7.55, 14.52, 12.05, 19.0, 11.29, 11.81, 15.38, 9.62, 14.19, 12.62 12.6 g 1.9 g 1. 5% 2. (8.2) Z χ 2 5%

8.2 129 8 3 1% 37 37 74 93 133 226 130 170 300

131 9 9.1 2 (correlation) 9.1.1

132 9 *1 2 (causality) 2 9.1.2 2 (scatter diagram) 9.1 X, Y 2 4 X Y X Y 9.1(2) *1 WHO 12.7% (2005 ) http://www.who.int/mediacentre/factsheets/fs310/en/

9.1 133 (3)(4) ρ xy 2 1 1 (1) ρxy = 0.010 (2) ρxy = 0.501 Y Y X X (3) ρxy = 0.944 (4) ρxy = 1.000 Y Y X X 9.1 : (1) (2) (3) (4)

134 9 9.2 9.2.1 2 2 x, y x = x 1, x 2, x 3,..., x n y = y 1, y 2, y 3,..., y n (9.1) σ xy x, y (covariance) σ xy = 1 n n δx i δy i i=1 = 1 n (δx 1δy 1 + δx 2 δy 2 +... + δx n δy n ) (9.1) δx i x i x i x (p.6) (9.1) = x i y i 1 p.7 (9.1) x i = y i 9 1 y i x i ( ) σ xy σ xy = 1 n xi y i x y (9.2) 9.2.2 4 S 1, S 2, S 3, S 4 9.2 S 1 S 2

9.2 135 x 4 ρxy = 0 _ y Y s1 h h s2 w w w w s3 h h s4 _ x X 9.2 2 x y x i, y i x i, y i 9.2 x, y 9.1 σ xy = 1 {( w) h + ( w) ( h) + w h + w ( h)} = 0 4 σ xy 9.3 Y y I IV II III x X 9.3 x y x II III I IV y I II III IV (9.1) (x i x)(y i y) II IV I III

136 9 I, II, III, IV 4 ρxy = 0.737 ρxy = 0.769 I II I II Y Y IV III IV III X 9.4 X 9.4 II IV 9.2.3 (correlation constant) ρ xy *2 (9.3) ρ xy = σ xy σ x σ y (9.3) σ x, σ y x, y σ xy 9.1(4) (9.4) y i x i 1 y i = ax i + b, (i = 1, 2,..., n) (9.4) *2 r xy ρ

9.2 137 b = 0 x y y i = ax i, (i = 1, 2,..., n) (9.5) σ xy δy i δy i = y i y = ax i ax = aδx i (9.6) y i a a (9.1) σ xy = 1 n = 1 n = a 1 n n δx i δy i i=1 n δx i aδx i i=1 n δx 2 i = aσx 2 (9.7) i=1 σ 2 y σ 2 y = 1 n = 1 n n (δy i ) 2 i=1 n (aδx i ) 2 = a 2 1 n i=1 n (δx i ) 2 i=1 = a 2 σx 2 (9.8) σ y = σy 2 = a σ x ρ xy = σ xy = aσ2 x σ x σ y a σx 2 = ±1, (a > 0 1, a < 0 1) (9.9) (9.5) y i x i 1 1 2 x, y ρ xy = ±1 0 y i = ax i + b 0.9

138 9 0.4 9.2.5

9.2 139 9.2.4 ( ) *3 (i) Y (ii) (iii) X 9.5 9.5 3 (ii) (linear regression) (least-square method) (xi,yi) Y (xi,axi + b) (x2,y2) h2 hi y = a x + b h1 (x1,y1) X 9.6 2 : h 2 1 + h 2 2 +... a, b *3

140 9 9.6 (x 1, y 1 ), (x 2, y 2 ),..., (x n, y n ) y = ax + b i h i h i = y i (ax i + b) (9.10) y = ax + b a b a, b p.153 a = xy x y x 2 x = σ xy 2 σx 2 (9.11) b = y ax (9.12) (9.11),(9.12) a, b 9 2 6 2 (24.5, 165.4), (28.0, 182.7), (26.0, 171.6), (25.5, 173.1), (25.0, 175.1), (24.0, 170.6) (x 1, y 1 ), (x 2, y 2 ),... (1.6) σ 2 x (9.2) σ xy x i y i x 2 i y2 i, x iy i x, ȳ, x 2, ȳ2, xy (9.12) Excel a, b 9.7 9.2.5 9.8

9.2 141 187.5 175.0 162.5 25 30 9.7 6 10 ρxy = 0.160 ρxy = 0.242 Y Y X ρxy = 0.131 X ρxy = 0.495 Y Y X X 9.8 [ ] 2 n ρ xy (9.13) T n 2 t

142 9 T = (n 2)ρ 2 xy 1 ρ 2 xy (9.13) 2 9.9 2 2 9.9 n ρ xy ρ xy T n 2 t- Y Y X X 9.9 9.2.4 9.7 6 9 3 6 (24.5, 165.4), (28.0, 182.7), (26.0, 171.6), (25.5, 173.1), (25.0, 175.1), (24.0, 170.6) (9.3) 2 ρ xy = 0.8323 ρ xy

9.2 143 (9.13) T n = 6 T = (6 2)0.8323 2 1 0.8232 2 = 3.00 t- 4 3.00 α = 0.01 α = 0.025 T 0.025 0.025 T α = 0.025 0.05 9.2.6 x i y i (9.4) 1 1 2 9.10(a) 9.10(b) 2 9.10(b) (a) (b) 9.10 (a) (b)2 2

144 9 2 2 y = ax 2 + bx + c a, b, c 2 2 2

9.2 145 9 1 A 8 ( x g) ( y g) x 62.2 42.8 61.8 79.3 63.1 51.4 60.9 69.9 y 36.7 28.7 32.0 37.7 31.8 31.5 32.3 34.8 1. x, y x y 2. x, y ρ xy 3. y = ax + b a, b 4. (9.13) T 0.01

147 A A.1 10 3 2 n = 4m (m = 1, 2, 3,...) 1. n/4 n/4+1 n/2 n/2+1 3n/4 3n/4+1. n Q 1 M Q 3 A.1 4 Q 1, M, Q 3 x 1, x n/2 x 1, x 2,..., x n 1 1 n 1 n 1 4 4 Q 1 n 1 4 1 + n 1 4 = n 4 + 3 4 x n/4 x n/4+1 3 : 1 Q 1 Q 3 3(n 1) 4 x 3n/4 x 3n/4+1 1 : 3 Q 3 M x n/2 x n/2+1 1 : 1

148 A A.2 A, B, C A, B, C E P (E A), P (E B), P (E C) P (A E) = P (A)P (E A) P (A)P (E A) + P (B)P (E B) + P (C)P (E C) (A.1) P (A E) (p.33, (2.18) ) P (A E) = P (A E) P (E) (A.2) P (B E), P (C E) P (E A) = P (A E) P (A) (A.3) A, B, C P (E) P (E) = P (E A) + P (E B) + P (E C) (A.4) (A.2) (A.3 ) (A.4) A.3 A.3.1 51 (3.10) n µ = x n C x p x q n x = np, (q = 1 p) x=0 (A.5) (p + q) n

A.3 149 n (p + q) n = nc x p x q n x x=0 = p n + np n 1 n(n 1) q + p n 2 q 2 + 2 (A.6) p n(p + q) n 1 = n x n C x p x 1 q n x x=0 (A.6) (A.7) p p + q = 1 n np = x n C x p x q n x x=0 (A.8) A.3.2 σ 2 = 2 2 (A.7) p n n(n 1)(p + q) n 2 = x(x 1) n C x p x 2 q n x x=0 (A.9) p 2 p + q = 1 n n n 2 p 2 np 2 = x 2 nc x p x q n x x n C x p x q n x x=0 x=0 (A.10) (A.5) np = µ 1 2 1 2 2 n σ 2 = x 2 nc x p x q n x n 2 p 2 = x=0 n x n C x p x q n x np 2 = np np 2 = npq x=0 (A.11)

150 A A.4 n p n, p 0 nc x p x (1 p) n x µx x! e µ (A.12) µ = np x n C x nc x = = n! x! (n x)! n (n 1) 2 1 x (x 1) 2 1 (n x) (n x 1) 2 1 (A.13) n n > x 0 n (n x) n! = n (n 1) (n x + 1) (n x) (n x 1) 2 1 (A.13) nc x = n (n 1) (n x + 1) x (x 1) 2 1 (A.14) n (n 1) (n x + 1) x *1 n n x n 1 n x + 1 n (A.14) n nc x nx x! (A.15) p x (1 p) n x ( ) e ( e = lim 1 + 1 ) q (A.16) q q *1 n 0 n (x 1) 0, 1,..., (x 1) x

A.4 151 (A.16) ( lim 1 1 ) q = 1 q q e (A.17) (A.12) *2 p x (1 p) n x = ( ) x p (1 p) n (A.18) 1 p (A.18) p 0 (1 p) 1 ( ) x p p x (A.19) 1 p *3 (1 p) n q = 1/p p 0 (1 p) n = (1 p) 1 p np ( = 1 1 q ) q µ e µ (A.20) 3 (A.17) (A.15) (A.19) (A.20) nc x p x (1 p) n x nx x! px e µ = (np)x e µ x! = µx x! e µ (A.21) *2 ( factor) ax(x 1) a, x, x 1 *3 p 0 1 p 1 p p 0 p 0.99999 1 0.00001

152 A A.5 E[X] = E[ 1 n (X 1 + X 2 +... + X n )] = 1 n (E[X 1] + E[X 2 ] +... + E[X n ]) = 1 (µ + µ +... + µ) = µ n 2 (3.13) E[X 1 ] µ 1 (6.4) V [X] =E[(X E[X]) 2 ] [ ( ) ] 2 1 =E n (X 1 + X 2 +... + X n ) µ = 1 n 2 E [ (X 1 + X 2 +... + X n nµ) 2] = 1 n 2 E [ ((X 1 µ) + (X 2 µ) +... + (X n µ)) 2] = 1 n 2 E [ (X 1 µ) 2 + (X 2 µ) 2 +... + (X n µ) 2] 2 n 2 E [((X 1 µ)(x 2 µ) + (X 1 µ)(x 3 µ) +...)] = 1 n 2 ( E[(X1 µ) 2 ] + E[(X 2 µ) 2 ] +... + E[(X n µ) 2 ] ) 0 = 1 n 2 (σ2 + σ 2 +... + σ 2 ) = σ2 n (A.22) X 1, X 2,... (3.18) *4 A.6 (6.5 ) s 2 *4

A.7 153 s 2 = 1 n = 1 n ( (X1 X) 2 + (X 2 X) 2 +... + (X n X) 2) ( X 2 1 + X2 2 + + Xn 2 ) 2 X (2 2 ) E[X 2 1 ] = E[X 2 2 ] = = σ 2 (A.23) E[X 2 ] = V [X] = σ2 n (A.24) E[s 2 ] = 1 n (σ2 + σ 2 + ) σ2 n = n 1 n σ2 (6.3), (6.4) A.7 (xi,yi) Y (xi,axi + b) (x2,y2) h2 h1 (x1,y1) hi y = a x + b X A.2 : h 2 1 + h 2 2 +... a, b A.2 (x 1, y 1 ), (x 2, y 2 ),..., (x n, y n ) y = ax + b i

154 A h i h i = y i (ax i + b) (A.25) y = ax + b a b a, b S S S = 1 n (h2 1 + h 2 2 +... + h 2 n) (A.26) 1 n (A.26) (A.25) S = y 2 + a 2 x 2 2by 2axy + 2abx + b 2 (A.27) a, b S 2 S a S b = 2ax 2 2xy + 2bx = 0 = 2y + 2ax + 2b = 0 (A.28) a, b x 2 a + xb = xy (A.29) xa + b = y (A.30) a = xy x y x 2 x = σ xy 2 σx 2 (A.31) b = y ax (A.32)

A.7 155 1 x, y f(x, y) = ax 2 + bxy + cy 2 + dy x, y d f(x, y) x f(x, y) y = 2ax + by = bx + 2cy + d x y f(x, y) cy 2 dy

157 B B.1 90 95 97.5 99 99.5 α 0.10 0.05 0.025 0.01 0.005 z α 1.282 1.645 1.960 2.326 2.576

158 B B.2 Φ(z) = z 1 2π e x2 /2 dx Φ(z) φ(x) 0 z x 0 1 2 3 4 5 6 7 8 9 0.0.50000.50398.50797.51196.51595.51993.52392.52790.53188.53585 0.1.53982.54379.54775.55171.55567.55961.56355.56749.57142.57534 0.2.57925.58316.58706.59095.59483.59870.60256.60641.61026.61409 0.3.61791.62171.62551.62930.63307.63683.64057.64430.64802.65173 0.4.65542.65909.66275.66640.67003.67364.67724.68082.68438.68793 0.5.69146.69497.69846.70194.70540.70884.71226.71566.71904.72240 0.6.72574.72906.73237.73565.73891.74215.74537.74857.75174.75490 0.7.75803.76114.76423.76730.77035.77337.77637.77935.78230.78523 0.8.78814.79102.79389.79673.79954.80233.80510.80784.81057.81326 0.9.81593.81858.82121.82381.82639.82894.83147.83397.83645.83891 1.0.84134.84375.84613.84849.85083.85314.85542.85769.85992.86214 1.1.86433.86650.86864.87076.87285.87492.87697.87899.88099.88297 1.2.88493.88686.88876.89065.89251.89435.89616.89795.89972.90147 1.3.90319.90490.90658.90824.90987.91149.91308.91465.91620.91773 1.4.91924.92073.92219.92364.92506.92647.92785.92921.93056.93188 1.5.93319.93447.93574.93699.93821.93942.94062.94179.94294.94408 1.6.94520.94630.94738.94844.94949.95052.95154.95254.95352.95448 1.7.95543.95636.95728.95818.95907.95994.96079.96163.96246.96327 1.8.96406.96485.96562.96637.96711.96784.96855.96925.96994.97062 1.9.97128.97193.97257.97319.97381.97441.97500.97558.97614.97670 2.0.97724.97778.97830.97882.97932.97981.98030.98077.98123.98169 2.1.98213.98257.98299.98341.98382.98422.98461.98499.98537.98573 2.2.98609.98644.98679.98712.98745.98777.98808.98839.98869.98898 2.3.98927.98955.98982.99009.99035.99061.99086.99110.99134.99157 2.4.99180.99202.99223.99245.99265.99285.99305.99324.99343.99361 2.5.99379.99396.99413.99429.99445.99461.99476.99491.99505.99520 2.6.99533.99547.99560.99573.99585.99597.99609.99620.99631.99642 2.7.99653.99663.99673.99683.99692.99702.99710.99719.99728.99736 2.8.99744.99752.99759.99767.99774.99781.99788.99794.99801.99807 2.9.99813.99819.99824.99830.99835.99841.99846.99851.99855.99860 ( )

B.2 159 ( ) 0 1 2 3 4 5 6 7 8 9 3.0.99865.99869.99873.99877.99881.99885.99889.99892.99896.99899 3.1.9 3 032.9 3 064.9 3 095.9 3 125.9 3 155.9 3 183.9 3 211.9 3 237.9 3 263.9 3 288 3.2.9 3 312.9 3 336.9 3 359.9 3 381.9 3 402.9 3 422.9 3 442.9 3 462.9 3 480.9 3 499 3.3.9 3 516.9 3 533.9 3 549.9 3 565.9 3 581.9 3 595.9 3 610.9 3 624.9 3 637.9 3 650 3.4.9 3 663.9 3 675.9 3 686.9 3 698.9 3 709.9 3 719.9 3 729.9 3 739.9 3 749.9 3 758 3.5.9 3 767.9 3 775.9 3 784.9 3 792.9 3 799.9 3 807.9 3 814.9 3 821.9 3 828.9 3 834 3.6.9 3 840.9 3 846.9 3 852.9 3 858.9 3 863.9 3 868.9 3 873.9 3 878.9 3 883.9 3 887 3.7.9 3 892.9 3 896.9 4 003.9 4 042.9 4 079.9 4 115.9 4 150.9 4 183.9 4 215.9 4 246 3.8.9 4 276.9 4 305.9 4 332.9 4 359.9 4 384.9 4 409.9 4 433.9 4 455.9 4 477.9 4 498 3.9.9 4 519.9 4 538.9 4 557.9 4 575.9 4 592.9 4 609.9 4 625.9 4 640.9 4 655.9 4 669 4.0.9 4 683.9 4 696.9 4 709.9 4 721.9 4 732.9 4 743.9 4 754.9 4 764.9 4 774.9 4 784 4.1.9 4 793.9 4 802.9 4 810.9 4 818.9 4 826.9 4 833.9 4 840.9 4 847.9 4 854.9 4 860 4.2.9 4 866.9 4 872.9 4 877.9 4 883.9 4 888.9 4 893.9 4 897.9 5 022.9 5 065.9 5 106 4.3.9 5 146.9 5 183.9 5 219.9 5 254.9 5 287.9 5 319.9 5 349.9 5 378.9 5 406.9 5 433 4.4.9 5 458.9 5 483.9 5 506.9 5 528.9 5 550.9 5 570.9 5 590.9 5 608.9 5 626.9 5 643 4.5.9 5 660.9 5 675.9 5 690.9 5 705.9 5 718.9 5 731.9 5 744.9 5 756.9 5 767.9 5 778 4.6.9 5 788.9 5 798.9 5 808.9 5 817.9 5 825.9 5 834.9 5 841.9 5 849.9 5 856.9 5 863 4.7.9 5 869.9 5 876.9 5 882.9 5 887.9 5 893.9 5 898.9 5 903.9 5 907.9 5 912.9 5 916 4.8.9 5 920.9 5 924.9 5 928.9 5 931.9 5 935.9 5 938.9 5 941.9 5 944.9 5 946.9 5 949 4.9.9 5 952.9 5 954.9 5 956.9 5 958.9 5 960.9 5 962.9 5 964.9 5 966.9 5 968.9 5 969 5.0.9 5 971.9 5 972.9 5 974.9 5 975.9 5 976.9 5 977.9 5 979.9 5 980.9 5 981.9 5 982.9 3 032 0.999032 9

160 B B.3 χ 2 ν 5 α 0.05 X (X α ) 11.071 3.93e-5 3.93 10 5 X 0 α 0.995 0.975 0.95 0.9 0.5 0.05 0.025 0.01 0.005 ν = 1 3.93e-5 9.82e-4 3.93e-3 0.0158 0.455 3.841 5.024 6.635 7.879 ν = 2 0.010 0.051 0.103 0.211 1.386 5.991 7.378 9.210 10.597 ν = 3 0.072 0.216 0.352 0.584 2.366 7.815 9.348 11.345 12.838 ν = 4 0.207 0.484 0.711 1.064 3.357 9.488 11.143 13.277 14.861 ν = 5 0.412 0.831 1.145 1.610 4.351 11.071 12.833 15.086 16.750 ν = 6 0.676 1.237 1.635 2.204 5.348 12.592 14.449 16.812 18.548 ν = 7 0.989 1.690 2.167 2.833 6.346 14.067 16.013 18.475 20.278 ν = 8 1.344 2.180 2.733 3.490 7.344 15.507 17.535 20.090 21.955 ν = 9 1.735 2.700 3.325 4.168 8.343 16.919 19.023 21.666 23.589 ν = 10 2.156 3.247 3.940 4.865 9.342 18.307 20.483 23.209 25.188 ν = 11 2.603 3.816 4.575 5.578 10.341 19.675 21.920 24.725 26.757 ν = 12 3.074 4.404 5.226 6.304 11.340 21.026 23.337 26.217 28.300 ν = 13 3.565 5.009 5.892 7.041 12.340 22.362 24.736 27.688 29.819 ν = 14 4.075 5.629 6.571 7.790 13.339 23.685 26.119 29.141 31.319 ν = 15 4.601 6.262 7.261 8.547 14.339 24.996 27.488 30.578 32.801 ν = 16 5.142 6.908 7.962 9.312 15.338 26.296 28.845 32.000 34.267 ν = 17 5.697 7.564 8.672 10.085 16.338 27.587 30.191 33.409 35.718 ν = 18 6.265 8.231 9.390 10.865 17.338 28.869 31.526 34.805 37.156 ν = 19 6.844 8.906 10.117 11.651 18.338 30.144 32.852 36.191 38.582 ν = 20 7.434 9.591 10.851 12.443 19.337 31.410 34.170 37.566 39.997 ν = 21 8.034 10.283 11.591 13.239 20.337 32.670 35.479 38.932 41.401 ν = 22 8.643 10.982 12.338 14.041 21.337 33.924 36.781 40.289 42.796 ν = 23 9.260 11.688 13.090 14.848 22.337 35.172 38.076 41.638 44.181 ν = 24 9.886 12.401 13.848 15.659 23.337 36.415 39.364 42.980 45.558 ν = 25 10.520 13.120 14.611 16.473 24.336 37.652 40.646 44.314 46.928 ν = 26 11.160 13.844 15.379 17.292 25.336 38.885 41.923 45.642 48.290 ν = 27 11.807 14.573 16.151 18.114 26.336 40.113 43.194 46.963 49.645 ν = 28 12.461 15.308 16.928 18.939 27.336 41.337 44.461 48.278 50.993 ν = 29 13.121 16.047 17.708 19.768 28.336 42.557 45.722 49.588 52.336 ν = 30 13.787 16.791 18.493 20.599 29.336 43.773 46.979 50.892 53.672 ν = 40 20.706 24.433 26.509 29.050 39.335 55.758 59.342 63.691 66.766 ν = 50 27.991 32.357 34.764 37.689 49.335 67.505 71.420 76.154 79.490 ν = 60 35.534 40.482 43.188 46.459 59.335 79.082 83.298 88.379 91.952 ν = 80 51.172 57.153 60.391 64.278 79.334 101.88 106.63 112.33 116.32 ν = 90 59.20 65.65 69.13 73.29 89.33 113.15 118.14 124.12 128.30 ν = 100 67.33 74.22 77.93 82.36 99.33 124.34 129.56 135.81 140.17 α

B.4 Student t- 161 B.4 Student t- ν 5 α 0.05 t (z α ) 2.015 0 z α α t α 0.1 0.05 0.025 0.01 0.005 0.0025 ν = 1 3.078 6.314 12.706 31.821 63.657 127.321 ν = 2 1.886 2.920 4.303 6.965 9.925 14.089 ν = 3 1.638 2.353 3.182 4.541 5.841 7.453 ν = 4 1.533 2.132 2.776 3.747 4.604 5.598 ν = 5 1.476 2.015 2.571 3.365 4.032 4.773 ν = 6 1.440 1.943 2.447 3.143 3.707 4.317 ν = 7 1.415 1.895 2.365 2.998 3.499 4.029 ν = 8 1.397 1.860 2.306 2.896 3.355 3.833 ν = 9 1.383 1.833 2.262 2.821 3.250 3.690 ν = 10 1.372 1.812 2.228 2.764 3.169 3.581 ν = 11 1.363 1.796 2.201 2.718 3.106 3.497 ν = 12 1.356 1.782 2.179 2.681 3.055 3.428 ν = 13 1.350 1.771 2.160 2.650 3.012 3.372 ν = 14 1.345 1.761 2.145 2.624 2.977 3.326 ν = 15 1.341 1.753 2.131 2.602 2.947 3.286 ν = 16 1.337 1.746 2.120 2.583 2.921 3.252 ν = 17 1.333 1.740 2.110 2.567 2.898 3.222 ν = 18 1.330 1.734 2.101 2.552 2.878 3.197 ν = 19 1.328 1.729 2.093 2.539 2.861 3.174 ν = 20 1.325 1.725 2.086 2.528 2.845 3.153 ν = 21 1.323 1.721 2.080 2.518 2.831 3.135 ν = 22 1.321 1.717 2.074 2.508 2.819 3.119 ν = 23 1.319 1.714 2.069 2.500 2.807 3.104 ν = 24 1.318 1.711 2.064 2.492 2.797 3.091 ν = 25 1.316 1.708 2.060 2.485 2.787 3.078 ν = 30 1.310 1.697 2.042 2.457 2.750 3.030 ν = 35 1.306 1.690 2.030 2.438 2.724 2.996 ν = 40 1.303 1.684 2.021 2.423 2.704 2.971 ν = 45 1.301 1.679 2.014 2.412 2.690 2.952 ν = 50 1.299 1.676 2.009 2.403 2.678 2.937 ν = 60 1.296 1.671 2.000 2.390 2.660 2.915 ν = 70 1.294 1.667 1.994 2.381 2.648 2.899 ν = 80 1.292 1.664 1.990 2.374 2.639 2.887 ν = 90 1.291 1.662 1.987 2.368 2.632 2.878 ν = 100 1.290 1.660 1.984 2.364 2.626 2.871 ν = 120 1.289 1.658 1.980 2.358 2.617 2.860

163 C C.1 y = f(x) 2 f 2 D f m E f 1 A B C x 1 x m x 2 C.1 C.1 f(x) x 1 x 2 f(x) x 1, x 2 f(x 1 ), f(x 2 ) f 1, f 2 x m f m ACD ABE x 2 x 1 f 2 f 1 = x m x 1 f m f 1 (C.1) f m = f 1 + (f 2 f 1 )(x m x 1 ) x 2 x 1 (C.2)

164 C C.2 C.2.1 52.6 52.7 52.7 kg 52.7 x 52.7 52.65 x < 52.75 C.2 56 55 56 57 56.0 55 56 57 56 56.0 56 56.0 52.7 ±0.05 0.7 (significant ) 5,2,7 3 3 56.0 kg 0 0 56 kg 0.1 kg 0.01 kg ( C.1) 56.0 56 kg C.2.2 C.1 0.01234 0 0 810 0 1 1 C.1 : * 2 3 12.3 3 12.30 4 0.01234 4 0.0012 2 813 3 810 * 810. 3 8.10 10 2 3 6.02 10 23 3

C.3 165 10 1 3 2 625, 810, 752 8.10 10 2 3 8.1 10 2 2 C.3 a = 0.505, b = 1.05 2 ab = 0.505 1.05 = 0.53025 3 a = 0.51, b = 1.1 a b = 0.561 0.561 0.53025 0.53025 = 0.0580 6% 2 3 C.4 12C 4 ( 1 7 1/7 = 0.14285714... 0.143, ) 4 ( ) 8 6 12 11 10 9 = 7 4 3 2 1 ( 1 7 ) 4 ( ) 8 6 7 6/7 = 0.85714285... 0.857 0.060227... 0.060067... 3

166 C 1/7 0.142857, 6/7 0.857412 0.060066.. *1 4 1 2 *1

167 D D.1 300 M+ MR M+ (memory plus) MR (memory recall) MC (memory clear) MRC ( MR,MC ) + : 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 : 1 M+ 2 M+ 3 M+ 10 M+ MR 55 MC 23.5 3 + 41.2 5 + 51.0 6 + 13.5 2 : 23.5 3 = M+ 41.2 5 = M+ 51.0 6 = M+ 13.5 2 = M+ MR 609.5

168 D 2 : 12 2 : 12 = 144 2 : 1 2 + 2 2 + 3 2 + + 10 2 : 1 = M+ 2 = M+ 3 = M+ 10 = M+ = 385 2 *1 30 3 5 2 = 25, 6 2 = 36 5.5 = 30.25 5.4 = 29.16 5.45 = 29.7025 5.47 = 29.9209 5.48 = 30.0304 5.475 = 29.975626 5.475 5.480 5.475 5.48 23.5, 41.2, 50.1, 32.3 23.5 M+ 41.2 M+ 50.1 M+ 32.3 M+ MR 4 = 36.775 2 2 : 2 23.5 = M+ 41.2 = M+ MR 4 = 1450.7475 2 2 : 36.775 = 1450.7475 98.347 : 98.35 *1 ( )

D.1 169 98.35 = 9.92 5 15 25 35 45 2 4 7 4 3 2 6 13 17 20 MC 5 2 = M+ 15 4 = M+ 45 3 = M+ MR 20 = 26 2 5 = 2 = M+ 15 = 4 = M+ 45 = 3 = M+ MR 20 = 815 2 26 = 815 = 139 = 139 139 = 11.79

170 D D.2 Microsoft Excel 1 CSV 1 1.1(p.5) Windows Mac * 2 1 1.txt.csv A A1 A100 1 43.6 2 45.2 3 45.4 4 45.8 5 47.2 99 64.2 100 64.6 A B C *2 Mac

D.2 171 A101 =SUM(A1:A100)/100 µ = (43.6 + 45.2 + + 64.5)/100 =AVERAGE(A1:A100) AVERAGE B1 =A1^2 B1 B2 B100 B101 =SUM(B1:B100)/100 C101 =B101-A101^2 C102, =SQRT(C101) =2 2 (p.8 ) 1 1.2(p.16), 1.csv A B C 1 44 1 2 46 3 3 48 6 4 50 9 10 62 2 11 64 2 C1 =A1*B1

172 D C1 C2 C11 C12 =SUM(C1:C11)/SUM(B1:B11) D1 =A1^2*B1 D1 D2 D11 D12 =SUM(D1:D11)/SUM(B1:B11)-C12^2 D13 =SQRT(D12) VAR A1 A100 =VAR(A1:A100) STDEV

D.3 R 173 D.3 R Microsoft Excel http://aoki2.si.gunma-u.ac.jp/hanasi/excel/ Beautiful Visualization R R R S R

174 D R Windows, MacOS, Linux OS R R R var R ggplot2 R R The R Project for Statistical Computing R http://www.r-project.org/ The R Manuals HTML PDF http://cran.r-project.org/manuals.html RjpWiki R Wiki http://www.okada.jp.org/rwiki/?rjpwiki