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1 June 2016

2

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

4 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

5 iii URL PDF URL

6

7 1 i

8 χ A 147 A A A A A A A B 157 B B B.3 χ B.4 Student t C 163 C C

9 3 C C D 167 D D D.3 R E

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11 5 1 (descriptive statistics) 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)

12 x, µ (mean *1 ) 1 ( ) = x x µ *2 x = 1 n (x 1 + x x n ) = 1 n n x i (1.2) i= (deviation) (1.3) *3 δx i = x i x (1.3) δx 1 + δx δx n = (x 1 x) + (x 2 x) + + (x n x) = (x 1 + x x n ) n x = n 1 n (x 1 + x x n ) n x = 0 *1 average *2 µ mean m *3 δ

13 σ 2, σ 2 σ 2 *4 σ 2 = 1 ( (x1 x) 2 + (x 2 x) (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)

14 8 1 (standard error) 84 (representative value / descriptive statistics) (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 x 2 n) 2 n n xx i = xx 1 + xx xx n = x x i = x nx = nx 2 i=1 i=1

15 1.1 9 n n { }} { x 2 = x 2 ( ) = nx 2 i=1 x (1.6) n 1 p (Chebyshev s inequality) µ σ µ ± aσ a 1 a µ = 54.46, σ = 4.22 ( 1 1) a = 2 ± = = /2 2 = 1/4 25 6

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

17 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 (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 x 9 + x 10 ) Q 1 M 3 Q n = 100 4m n/4 = 25, n/4 + 1 = 26, n/2 = 50, n/2 + 1 = 51, 3n/4 = 75, 3n/4 + 1 = 76

18 12 1 M = 1 2 (x 50 + x 51 ) = 1 2 ( ) = Q 1 = 1 4 (x x 26 ) = 1 4 ( ) = Q 3 = 1 4 (3x 75 + x 76 ) = 1 4 ( ) = 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, ) 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, ) (percentile) 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

19 (five number summary) , , 54.75, , 64.6 (box and whiskers plot, box plot) min Q 1 M Q 3 Max Q 1 M 3 Q 3 min Max (IQR) 3 1 (IQR *7 ) *7 Interquartile Range

20 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 *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, A 2 B A: B: *8 *9 (2012)

21 A B 29,172 25,836 3,300 8,920 22,027 1 (robust)

22 (frequency distribution table) 1.2 (class) (frequency) (x i ) (f i ) (F i )

23 (histogram) = {}}{ { }} { { }} { (1.7) = = = (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

24 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 ) k σ 2 i=1 = f ix 2 i x 2 (1.11) n 1 3 (1.11) ( ) =

25 (mode) ( ) %

26 = , % * 11 *11

27 * , 17, , 8, (2012 ) *12 h14/index.htm

28 Math Kokugo

29 (element) (set) *1 *2 *1 *2

30 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

31 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

32 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

33 A, B S A B = A B 2 2 A, B A, B (exclusive or) A, B A B

34 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) ( A) 36 (B) 9 1. S 2.

35 p p = 0 p = /3 1/3 (Bayesian probability)

36 30 2 IT 1 1/ , 6 6 S A S A n, m S A P (A) P (A) = m n (2.11) * Venn S S *5

37 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 ) = 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)

38 32 2 A B P (A) + P (B) A B 2

39 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) (2.18) P (A B) = P (A)P (B A) (2.19)

40 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 * (A) (B) *7 (2.20) A B P (A) = P (A B) A, B *8 2 A B (2.21)

41 *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

42 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

43 /5 20/80 = = (double blind test) (Bayez s theorem) * 10 A, B S A, B S = A + B A, B E P (E A), P (E B) *10 18

44 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 (2.18) 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)

45 P (A E) = P (A)P (E A) P (E) (2.26) P (E) P (E A) P (E B) A, B A 800 5% B % 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) = ( 2.25) 100 * 12 2 * *12 100

46 A = = B = = (2.27) A = = B = = (2.28) A = (2.29) * : 1.5% 0.5% 2% *13

47 /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 /10

48

49 % 70% 1. 70% 2. 70% 3. 70% 1. 70% % 3. 70% 70% 70% 70%

50 /

51 * (2.17) 52 1 *14

52 /2

53 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)

54 48 3 (stochastic variable) X, Y X x 1, x 2,... (discrete probability variable) 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

55 (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

56 50 3 i F (x i ) = f(x k ) (3.7) k=1 F ( ) = 0 (3.8) F ( ) = 1 (3.9) F (x) 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

57 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= (1.6) V [X] = E[X 2 ] E[X] 2 (3.12) = , 0.05, , 200, = , 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

58 52 3 (3.10) E[X] = f(x 1 )x 1 + f(x 2 )x 2 + = ( 2000) = µ = = 3 2 = E[X 2 ] = = 3 E[X 2 ] {E[X]} 2 = 3 ( ) 2 3 = = n µ = 1 1 n n n 1 n = n(n + 1) 2 1 n = n σ 2 = E[X 2 ] {E([X]} 2 = n n n2 1 (n + 1)2 n 2 2 = n n k = k=1 n k 2 = k=1 n(n + 1) 2 n(n + 1)(2n + 1) 6

59 *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 B *4 *5

60 (3.13) = 1.92 A, B

61 A 40% 10 4 A 10 4 A (1 0.4) 6 A 2 6 A (1 0.4) A 4 6 A C 4 = 10! = 210 4! 6!

62 C (1 0.4) 6 = (1 0.4) 6 = 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] * % 1/5 = ! 5! 0! 0.25, 5! 4! 1! , 5! 3! 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

63 n = 5 f(x) n = 15 n = 30 p = x 4.1 p = p = 0.05 p = 0.1 n = f(x) p = 0.25 p = 0.5 p = 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% (polynomial distribution)

64 A 4 O 2 B 2 AB ! 2!4!2!2! E 1, E 2,..., E k, p 1, p 2,..., p k, p 1 + p 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 px px k k (4.4)

65 (Poisson distribution) B[60, 0.05] n = 60, p = C x 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) * µ (4.3) *4 Windows

66 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 = µ x f(x) = µx x! e µ µ = f(0) f(4) 1 1 (f(0) + f(1) + f(2) + f(3) + f(4)) ( ) = 1 e ! ! ! ! ! = % n p np n > 50 n 50 np 5

67 f(x) = µx x! e µ n p µ 1 5 n p n p µ = np µ = ,2

68 /

69 kg kg * n 1/n 5.1 *1

70 64 5 P (X) X = 1, 2,..., 6 1/ m X 5.2 [0, 2] *2 p * m 0.2 P(X) X 5.1 *2 1 2 [1, 2] *3 p

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

72 *4 f(x) dx = 1 (5.3) f(x) = 0.5, a = 0, b = 0.3 P (0 X 0.3) = dx = [0.5x] = 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) Φ(z) = z 0.5 dx = 0.5z, (0 z 2) (5.8) *4 [, ] X

73 m 1.5 m f(x) = 1/2 x = 0.8 x = [ x ] dx = = (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)

74 [0, 1] , , , , , , , , , 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)

75 B[n, p] n 5.5 p = 0.4 n 0.3 n = 4 p = n = 12 p = 0.4 f(x) f(x) x x 0.1 n = 60 p = n = 120 p = 0.4 f(x) 0.05 f(x) x x 5.5 B[n, p] n n (normal distribution) f(x) = 1 ] (x µ)2 exp [ 2πσ 2σ 2 (5.17)

76 70 5 exp(a) e a *5 µ, σ 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 = p = 0.4 n = 24 f(x) 0.2 f(x) x x 5.6 σ µ ( ) n = 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

77 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) 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)

78 z Φ(z) z Φ(z) z Φ(z) z Φ(z) 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) = Φ(z) φ(x) 0 z x 5.8 ϕ(x) = 1/ 2π exp( x 2 /2) Φ(z)

79 f(z) 2 f(z) 1 f(z) N[0, 1] 1 z 1 z < z > = z < 1 * = N[µ, σ 2 ] 1. x z 2. z % (5.19) 90 z = (90 65)/12.5 = 2.0 Φ(2.0) % 5.2 n *7 2Φ(z) 1

80 A 40% 10 A 4 6 µ = np, σ = np(1 p) µ = 4, σ = = 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) = Φ(1.61) = B[10, 0.4] x = 4, 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 ]

81 z 1 = x µ σ z 2 = x 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% kg 5 2 (standard score) µ, σ µ 50 σ, 2σ, 50, 60, A 40% 24 A /2 4 49% 51% % ±2.5%

82 ± 2.5 %

83 (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= (68 ) 1/12 ( (5.16) ) 2 *8 2 3

84 N (1/2,1/144) [0, 1] 12 ( (3.14)) [0, 1] 12 6

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

86 (population) *1, µ, σ 2 (population parameters) *2 *3 (sample) (size) (sampling) (random sampling) *4 *1 *2 *3 *4

87 ,8,4,7,4,3,7,2,5,2,4,3,2,1,5, ,6,4,5,1,0,1,9,3,6,... * *5

88 /kg( ) , 47, 47, 12, 79, 63, 08, 27, 88, 29, 42, 64, 81, 44, , 47, 47, 12, 79, 63, 08, 27, 88, , 54.7, 54.7, 49.4, 58.1, 56.1, 48.8, 51.8, 59.9, , 47, 12, 79, 63, 08, 27, 88, 29, , 54.7, 49.4, 58.1, 56.1, 48.8, 51.8, 59.9, 52.0, 53.8

89 n X 1, X 2,..., X n X 1, X 2 X 1, X 2,..., X n 2, X = 1 n (X 1 + X X n ) ( ) (6.1) s 2 = 1 ( (X1 X ) 2 ( + X2 X ) 2 ( + + Xn X ) ) 2 ( ) (6.2) n s n n n = 10 n = [0, 20] n = 10, 100 X, X X µ = 10

90 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) 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 ] = σ = 4.32 =

91 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) (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 ) (6 ) V [s 2 ] χ 2 - *6 *7 1 X 1 X 1 µ 2 σ 2

92 X X n X n = 1 2 n X n X n X % , p = p, 0 1 p (3.10) µ = 1 p + 0 (1 p) = p

93 (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 = µ * 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

94 88 6 Z = X µ n(x µ) σ/ n = σ (6.6) Z N[0, 1] 5 (p.71) 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 )

95 (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)

96 n = 5 n= s 6.5 N[0, 25] s n = 5 n = 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-

97 6.5 χ χ s 2 (6.5) E[s 2 ] s * 11 * 12 µ = 0, σ 2 = 1 N[0, 1] N[0, 1] n Z = 1 n (X2 1 + X 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 * *11 *12 *13

98 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) = π x5/2 e x/2 (6.19) 0.5 ν = 1 ν = 2 ν = 3 ν = 4 ν = 5 ν = 6 ν = 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) = π,...

99 6.5 χ 2 93 n Z = 1 σ 2 ( (X1 µ) 2 + (X 2 µ) (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) (X n X) 2) = ns2 σ 2 (6.21) Z n 1 χ 2 s 2 (6.2) s 2 σ χ 2 χ α t χ 2 T n (x) α t χ 2 α ν = ν = ν = ν = T 6 (x) α x = t N[0, 1] 6 (6.11) Z = 1 6 (X X 2 6 ) %

100 94 6 Tn(x) α 0 t x 6.8 χ 2 χ 2 1%, 5%, 90% α t 6 4 (χ 2 ) , 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, % χ 2 s 2 50% (6.21), Z = ns 2 /σ 2 n 1 = 9 χ Z = ns2 σ = /σ 2 = (µ ) N[3, σ 2 ] 6 50% (X 1 µ) 2 + (X 2 µ) (X n µ) 2 21 σ 2 6 χ 2 α = 0.5 t 5.35

101 6.5 χ 2 95 ( (6.20) Z = 1 σ 2 (X1 µ) 2 + (X 2 µ) (X 6 µ) 2) 5.35 α = 1 σ 21 σ 2 = (µ ) 11 90% 12.5 σ 2 µ (6.20) (6.21) n n 1 χ 2 ν = 10 α = t = (6.21) 4.87 = ns2 σ 2 n = 11, s 2 = 12.5 σ 2 = 28.1

102 (10000m 2 ) 10 m 1 (100m 2 ) g 0.22 g g 54.3 g (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, χ 2 1

103 X E[X] = µ V [X] = σ2 n (7.1) (7.2) 35% *1 X X ± α α *

104 % (point estimation) (point estimator) x 1 x 2 95% (interval estimation) (interval estimator) (confidence interval) N[0, 1] α z α 100(1 α) α = 0.05 z Φ(z) = 0.95 z p.12 N(0,1) α 0 zα 7.1. z α 100(1 α) %

105 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/ % 90% z = 1.28 Φ(z) = , z = 1.29 Φ(z) = Φ(z) = 0.9 z ( ) z = ( ) = z α z α/2 N[0, 1] 90% 95% % 5% 95 z , α z α

106 % ±1.960

107 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) (X n X) 2) 1 ( (X1 X) 2 + (X 2 X) (X n X) 2) (7.4) n 1 σ 2 (7.4) *3 1 (7.4) *2 unbiased *3 Excel VAR,STDEV

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

109 % 95% Z ±1.645 X (7.5) Z = ±1.645 X [X σ n, X σ n ] (7.6) λ [X λ σ n, X + λ σ n ] (7.7) X σ n λ 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% λ Student t- Student t- 89 T n 1 t- T = n 1(X µ) s (7.9)

110 t- α α ν = ν = ν = ν = ν = ν = ν = ν = 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

111 L 10 (g) 65.1, 67.5, 71.5, 68.4, 70.1, 72.2, 68.7, 69.3, 70.6, g 2 90% 95% n = 10, σ 2 = % α = z α/2 = z 0.05 = (7.6) / 10 = / 10 = 70.1 µ 90% 68.0 < µ < % 67.8 < µ < X s kg, kg 2 90% = , = % 52.7 < µ < L 10 (g) 65.1, 67.5, 71.5, 68.4, 70.1, 72.2, 68.7, 69.3, 70.6, % 99%

112 X s X = ( )/10 = s 2 = ( )/10 X 2 = ν = n 1 = 9 s = = % ν = 9 t- 95 z [ , ] 9 9 µ 67.8 < µ < % 66.8 < µ < t- σ 2 X V [X] σ 2 /n ( (6.4) ) ns2 n 1 X X µ t- 6.6

113 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 : λ ±λ σ n ±λ s n 1 ±λ s n 1 t (6.8) T t-

114 108 7 n n s 2 = 1 ( (x1 X) 2 + (x 2 X) (x n X) 2) n X µ 1 ( (x1 µ) 2 + (x 2 µ) (x n µ) 2) n µ t % 95% , 4 µ 99%

115 µ 99%

116

117 111 8 (statistical test) (hypothetical test) (hypothesis) 8.1 * *1

118 112 8 X 1 10 X X X 10

119 X X I X X (hypothesis) II X X X II (= ) II

120 114 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] * (= 125.3) X N[1286, ] = 1.16 *2 *3 * /10

121 % (α = 0.1) % *5 α = 0.1 σ=11.2 (1.0) 1273 ( 1.16) 1286 (0) 8.1 N[1286, ] 1273 X 10 II X 0.1 (risk) A 0.01 A ( ) p X z = p 8.2 *5

122 116 8 α = 0.1 σ=11.2 (1.0) 1273 ( 1.16) 1286 (0) 8.2 z = 1.16 p 0.12 p p p

123 *6 ( ) *6

124 , 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, ( ) ( )

125 X X X = = X X s 2 = σ 2 /25( s = σ/5 = 0.925/5 = 0.185) X Z = Z = α = Z 25 *7 *7

126 X χ (p.93) I µ σ 2 N[µ, σ 2 ] n Z = 1 σ 2 ( (X1 µ) 2 + (X 2 µ) (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) (X n X) 2) = ns2 σ 2 (8.2) Z n 1 χ 2 s 2 (6.2) (8.1) Z µ σ 0.925

127 , 36.37,..., X 1 = 32.97, X 2 = 36.37,... Z Z = ( ( ) 2 + ( ) ( ) 2) = I Z 25 χ 2 25 Z 25 χ χ 2 χ 2 α ν = ν = ν = ν = α = ν = 25 χ 2 5% % 0.5% χ 2 0.5%

128 122 8 II II 25 X = s 2 = = Z Z = ns = σ = n 1 = 24 χ , 35.03, 35.11, 34.21, 35.08, 34.86, 35.13, 35.09, 34.36, 35.23, 35.24, , I Z Z = (( )2 + ( ) ( ) 2 ) = χ 2 α = 0.995, ,

129 % Z % χ 2 -

130 t- X χ 2 X t- n µ, X, s T T = n 1(X µ) T n 1 t- 10 X T 10 1( ) T = = α ν = ν = ν = ν = 9 X 0.1 s

131 T χ χ 2 χ (contingency table) *8 39,45,21,83,... (observed frequency) *8 contingency

132 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)

133 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 = ( /600) /600 ( /600) /600 + ( /600) / ( /600) /600 ( /600) /600 = 25.9 X A, B 6 (= (4 1) (3 1)) χ 2 X χ 2 A B X, χ 2 α α = 0.05 X (5% ) χ 2 5% X %, 5% * χ 2 α ν = ν = ν = ν = X χ 2 2 *9

134 128 8 α = X 8.3 n = 6 χ : % 1% g 11.78, 12.92, 7.55, 14.52, 12.05, 19.0, 11.29, 11.81, 15.38, 9.62, 14.19, g 1.9 g 1. 5% 2. (8.2) Z χ 2 5%

135 %

136

137 (correlation) 9.1.1

138 132 9 *1 2 (causality) (scatter diagram) 9.1 X, Y 2 4 X Y X Y 9.1(2) *1 WHO 12.7% (2005 )

139 (3)(4) ρ xy (1) ρxy = (2) ρxy = Y Y X X (3) ρxy = (4) ρxy = Y Y X X 9.1 : (1) (2) (3) (4)

140 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 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) S 1, S 2, S 3, S S 1 S 2

141 x 4 ρxy = 0 _ y Y s1 h h s2 w w w w s3 h h s4 _ x X 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

142 136 9 I, II, III, IV 4 ρxy = ρxy = I II I II Y Y IV III IV III X 9.4 X 9.4 II IV (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

143 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 x, y ρ xy = ±1 0 y i = ax i + b 0.9

144

145 ( ) *3 (i) Y (ii) (iii) X (ii) (linear regression) (least-square method) (xi,yi) Y (xi,axi + b) (x2,y2) h2 hi y = a x + b h1 (x1,y1) X : h h a, b *3

146 (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 (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

147 ρxy = ρxy = Y Y X ρxy = X ρxy = Y Y X X 9.8 [ ] 2 n ρ xy (9.13) T n 2 t

148 142 9 T = (n 2)ρ 2 xy 1 ρ 2 xy (9.13) n ρ xy ρ xy T n 2 t- Y Y X X (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 = ρ xy

149 (9.13) T n = 6 T = (6 2) = 3.00 t α = 0.01 α = T T α = x i y i (9.4) (a) 9.10(b) (b) (a) (b) 9.10 (a) (b)2 2

150 y = ax 2 + bx + c a, b, c 2 2 2

151 A 8 ( x g) ( y g) x y x, y x y 2. x, y ρ xy 3. y = ax + b a, b 4. (9.13) T 0.01

152

153 147 A A 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 Q 1 n n 1 4 = n 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

154 148 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 (3.10) n µ = x n C x p x q n x = np, (q = 1 p) x=0 (A.5) (p + q) n

155 A 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 (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 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)

156 150 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 ) q (A.16) q q *1 n 0 n (x 1) 0, 1,..., (x 1) x

157 A (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

158 152 A A.5 E[X] = E[ 1 n (X 1 + X 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 X n ) µ = 1 n 2 E [ (X 1 + X 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) (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 σ 2 ) = σ2 n (A.22) X 1, X 2,... (3.18) *4 A.6 (6.5 ) s 2 *4

159 A s 2 = 1 n = 1 n ( (X1 X) 2 + (X 2 X) (X n X) 2) ( X X 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 h a, b A.2 (x 1, y 1 ), (x 2, y 2 ),..., (x n, y n ) y = ax + b i

160 154 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 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)

161 A 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

162

163 157 B B α z α

164 158 B B.2 Φ(z) = z 1 2π e x2 /2 dx Φ(z) φ(x) 0 z x ( )

165 B ( )

166 160 B B.3 χ 2 ν 5 α 0.05 X (X α ) e X 0 α ν = e e e ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = α

167 B.4 Student t- 161 B.4 Student t- ν 5 α 0.05 t (z α ) z α α t α ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν = ν =

168

169 163 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)

170 164 C C.2 C kg 52.7 x x < C ± (significant ) 5,2, kg kg 0.1 kg 0.01 kg ( C.1) kg C.2.2 C C.1 : * *

171 C , 810, C.3 a = 0.505, b = ab = = a = 0.51, b = 1.1 a b = = % 2 3 C.4 12C 4 ( 1 7 1/7 = , ) 4 ( ) = ( 1 7 ) 4 ( ) /7 =

172 166 C 1/ , 6/ * *1

173 167 D D M+ MR M+ (memory plus) MR (memory recall) MC (memory clear) MRC ( MR,MC ) + : : 1 M+ 2 M+ 3 M+ 10 M+ MR 55 MC : = M = M = M = M+ MR 609.5

174 168 D 2 : 12 2 : 12 = : : 1 = M+ 2 = M+ 3 = M+ 10 = M+ = * = 25, 6 2 = = = = = = = , 41.2, 50.1, M M M M+ MR 4 = : = M = M+ MR 4 = : = : *1 ( )

175 D = MC 5 2 = M = M = M+ MR 20 = = 2 = M+ 15 = 4 = M+ 45 = 3 = M+ MR 20 = = 815 = 139 = = 11.79

176 170 D D.2 Microsoft Excel 1 CSV 1 1.1(p.5) Windows Mac * txt.csv A A1 A A B C *2 Mac

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