( 28) ( ) ( 28 9 22 ) 0 This ote is c 2016, 2017 by Setsuo Taiguchi. It may be used for persoal or classroom purposes, but ot for commercial purposes.
i (http://www.stat.go.jp/teacher/c2epi1.htm ) = statistics ( ) statistik ( ) status ( = ) IBM ICT (1956-1742) (1820-1910) () ( ) = ( ) + ( ) () () ( ) PC http://www.artsci.kyushu-u.ac.jp/ se2otgc/
CONTENTS ii Cotets 1 1 1.1............................. 1 1.2............................. 1 1.3.................. 2 1.4 (sample variace)..................... 3 1.5............................. 3 2 4 2.1................................ 4 2.2.................. 4 2.3................................. 6 2.4.................................. 8 3 9 3.1............................. 9 3.2.............................. 10 4 11 4.1................................ 11 4.2................................ 11 4.3............................. 12 5 14 5.1.................................. 14 5.2......................... 14 5.3......................... 15 5.4............................. 16 6 17 6.1................................. 17 6.2................................ 17 6.3........................... 18 7 19 7.1.............................. 19 7.2............................ 21 8? 25 8.1............................. 25 8.2............................. 25 9? 30 9.1......................... 30 9.2 χ 2 ()................ 31 9.3.............................. 32
1 1 1. 17 19 21 20 19 19 21 20 20 21 19 21 19 18 20 20 19 19 20 21 21 21 20 19 19 18 21 19 19 21 20 21 20 19 21 20 19 21 19 20 21 20 20 21 20 21 21 20 18 20 20 20 19 22 22 20 21 20 20 19 21 20 22 19 21 19 22 20 20 20 21 20 19 20 19 20 20 20 18 21 18 21 20 19 20 20 20 22 19 19 22 20 18 19 18 21 21 19 22 19? 1.1. 1.1.1. 1.1.2. 1.1.3. 1.1.4. ( ) + 1 2 2 1.2. 95 78 85 56 69 89 87 72 80 2 + 1 1.2.1. 1.2.2. (a) COUNT
1 2 (b) (i) [ ] [] (ii) MAX (c) (i) [ ] [] (ii) MIN (d) (i)sum (ii) AVERAGE (e) (i) [ ] [] (ii) MEDIAN 1.2.3.? 1.3. 1.3.1. 1.3.2. 1.3.3. 1.3.4. 1.3.5. 1.3.6. 1.3.7. 1.3.8. 1.3.9.
1 3 1.4. (sample variace) 1.4.1. ( 225 ) ( ) 2 ( ) 1.4.2. 1.4.3. 95 78 85 56 69 89 87 72 80 (a) ( )- (b) VAR.P (variace) VAR.P VAR.S () 1.4.4. (85,95) (58,64,66,72) 1.5. 65 73 88 76 83 94 84 77 85 76 85 82 74 78 63 81 69 97 96 74 1.5.1. 60 3 70 7 80 7 90 3 1.5.2. COUNTIF 1.5.3. [ ] [ ] 1.5.4. 65 66 93 77 79 94 95 98 85 66 84 82 68 75 63 83 65 93 95 74
2 4 2. 2.1. 2.1.1.? 1 10 0 1 2.1.2. X: a {X = a} P(X = a)! a {X a} P(X a) 2.1.3. (a) X; P(X = i) = 1 6 (i = 1,,..., 6) (b) 5 X; P(X = i) = 5 C i ( 1 2) 5 (i = 0,..., 5) (c) y X; P(X a) = a 2π (0 a 2π) 2.1.4.? (a) ( )=( ) (b)? (c) = 2.2. 2.2.1. 2.2.2. P(X = a i ) = p i (i = 1,..., ) (a) X = 0 X = 1 P(X = 0) = P(X = 1) = 1 2
2 5 (b) X P(X = 1) = P(X = 2) = P(X = 3) = P(X = 4) = P(X = 5) = P(X = 6) = 1 6 (c) 2 P(X = 0) = 2 3, P(X = 1) = 1 3 2.2.3. P(X a) = a f(x)dx a f(x)dx? (a) (1) y = f(x) a : b f(x)dx = (b) (2)b k=1 f( (a b)k ) a b a b a a f(x)dx b f(x)dx b f(x)dx
2 6 2.3. (c) b?? 2.3.1. E[X]; 100m 1m 1cm 100 1cm/ 100 =10cm=1m 1m 1 1cm/ 1 =1cm=0.01m 0.01m 0.01 1cm/ 0.01 =1cm=0.0001m 1 0.01 0.0001 0.000001 1 1.01 1.0101 1.010101 01 100 99 (1) P(X = a i ) = p i (i = 1,..., ) ( ) E[X] = a i p i. i=1 (2) P(X a) = a f(x)dx ( ) E[X] = xf(x)dx. 2.3.2. (1) 6 i 1 6 = 7 2 i=1
2 7 (2) (3) 5 1 i 5 C i 32 = 1 32 i=0 x 1 2π 1 [0,2π]dx = π F (a) = 0 (a < 0) = a 2π ( 0 1 + 1 5 + 2 10 + 3 10 + 4 5 + 5 1 = 5 2 (0 a 2π) = 1 (a > 2π) 2.3.3. X=E[X] 2014 9 19 10 10 13 1 =10 100 =10,00 2.3.4. Quiz? (a) 4 100? (b)?
2 8 Quiz (1) 4 1 2 4 C 2 = 6 2 16 = 3 4 8 (1) 4 1 4C 4 = 1 2 16 = 1 16 (??) 2 4 0 1 3 8 1 6 = 9 16 ( 0 0!) E[X] E[X] = 0 9 16 + 200 3 8 + 400 1 = 75 + 25 = 100 16 2.4. 2.4.1. V(X) = E [ (X E[X]) 2] (X E[X]) 2 X E[X] 2 = 0 1 X P(X = 1) = 1 P(X = i) = 0 (i = 2,..., 6) E[X] = 1 1 + 6 i 0 = 1, i=2 V(X) = (1 1) 2 1 + V(X) = 0 6 (i 1) 2 0 = 0. 2.4.2. X( ) = 1 X( ) = 0 p P(X = 1) = p X i=2 E[X] = 1 p + 0 (1 p) = p, V(X) = (1 p) 2 p + (0 p) 2 (1 p) = p(1 p)
3 9 3. 3.1. 3.1.1. (a) A, B P(A B) = P(A)P(B) (b) A, B, C A, B B, C C, A P(A B C) = P(A)P(B)P(C) (c) A 1,..., A ( 1) P(A 1 A ) = P(A 1 ) P(A ) (d) X, Y {X a}, {Y b} (e) X, Y, Z {X a}, {Y b}, {Z c} (f) X 1,..., X {X 1 a 1 },..., {X a } 3.1.2. X X 1, X 2,... X 1, X 2,... ( a) P(X = a) = P(X 1 = a) = P(X 2 = a) =... P(X a) = P(X 1 a) = P(X 2 a) =... 3.1.3. X 1 X 1 2 X 2 X 1, X 2,... X 3.1.4. 2 2 1 2 1 X 2 Y X, Y 3.1.5. (for whom?) (a) A, B B, C A, C ( ) A = {, } B = {, } C = {, } (b) A, B B, C A, C A, B, C A = {, } B = {, } C = {, } (c) i. X, Y Y, Z X, Z? ii. X, Y Y, Z Z, X X, Y, Z?
3 10 3.2. 3.2.1. X 1, X 2,... X 1 X 1 + + X E[X] 3.2.2. 0.7 { { 1 (1 ) X 1 = 0 (1 ), X 1 (2 ) 2 = 0 (2 ),... 1 2 ( 1) 1 0... 0 0 X 1 + + X = X 1 + + X = 0.7 (E[X] = 0.7) 3.2.3. 1 2 (E[X] = 1 2 ) 2 1 1 5 5 2989? 2,548? 3.2.4. 1,000 700 Yes 7!! 3.2.5. / ( )=( )+( ) ( )= 0 (?) = ( ) +
4 11 4. 4.1. 4.1.1. 100 94 1,000 936 94% = 94 936 93.6% = E[X] ( 100 1000 ) 94% 4.2. 4.2.1. g(x; µ, σ) = P(X a) = a 1 (x µ)2 e 2σ 2 2πσ 2 g(x; µ, σ)dx X µ σ 2 ( X N(µ, σ 2 )) 4.2.2. e x x ( ) 1 ( 1 + x ) () e x! (EXP(x) ) x 1 0.1 0.3 4 exp(x) 1 5. 100,000,000
4 12 4.2.3. Quiz y = g(x; µ, σ 2 ) g(x; µ, σ 2 ) A1 = µ, B1 = σ, C1 = x (a) (1/SQRT(2 PI() $B$1^2)) EXP( (C1 $A$1)^2/(2 $B$1^2)) ($A$1 ) (b) NORM.DIST(a,µ,σ,False) NORM.DIST(C1, $A$1, $B$1, False) 1 σ 2 σ ( ) 2 True False 4.2.4. Quiz a g(x; µ, σ)dx NORM.DIST(a,µ,σ,True) 1 True True 4.2.5. X N(µ, σ 2 ) X µ N(0, 1) σ ( ) ( ) X µ µ+σa P a = P(X µ + σa) = σ = a 1 ( e y2 2 dy 2π 1 2πσ 2 y = x µ σ ). (x µ)2 e 2σ 2 dx 4.2.6. X 1, X 2,... X N(µ, σ 2 ) X 1 + + X µ σ N(0, 1) 4.3. 4.3.1. (1) E[X] = µ, V(X) = σ 2 X 1, X 2,... X 4.3.2. Φ(a) = a X 1 + + X µ σ 1 2π e x2 2 dx N(0, 1)
4 13 4.3.3. (2) ( ) X1 + + X µ P σ a Φ(a) (4.1) ( ) X1 + + X µ P σ a Φ(a) (4.2) X 1 + + X µ σ a ( X 1) + + ( X ) ( µ) σ (4.2) (4.1) X a 4.3.4. (3) X = X 1 + + X ( P X aσ ) µ Φ(a) (4.3) ( P X + aσ ) µ Φ(a) (4.4) 4.3.5. y = Φ(x) 0, 0.2, 0.4,..., 5.8, 6, 6.2 (0.2 ) (a) NORM.DIST(x, 0, 1,TRUE) (b) NORM.S.DIST(x,TRUE) 4.3.6. Φ(x) = 0.99 x? NORM.S.INV(x) (a) 0, 0.2,..., 6.2? (b) z = 0, 0.6, 0.7, 0.8, 0.9, 0.95, 0.99, 0.995, 0.999, 0.9995, 0.9999 Φ(x) = z x
5 14 5.? 5.1. 5.1.1. 1.4.2 p E[X] = p, V(X) = p(1 p) X = 1 X = 0 i X i = 1 X i = 0 X 1, X 2,... X 5.1.2. X 1 + + X X = X 1 + + X 5.1.3. (4.3) (4.4) ( P X a ) p(1 p) p Φ(a) ( P X + a ) p(1 p) p Φ(a) p(1 p) 1 4 ( P X a ) 2 p Φ(a) (5.1) ( P X + a ) 2 p Φ(a) (5.2) 5.2. 5.2.1. p X (5.1) p X a 2 Φ(a) Φ(2.326) = 0.99
5 15 p X 2.326 2 99% 5.2.2.. (a) 100 94 Yes 94 100 2.326 2 100 0.824 Yes 82.4% 99% (b) 1,000 940 Yes? 940 1000 2.326 2 1000 0.903 Yes 90.3% 99% (c) 10,000 9,400 Yes? 9400 10000 2.326 2 10000 0.928 Yes 92.8% 99% 5.3. 5.3.1. (5.2) p X + a 2 Φ(a) Φ(2.326) = 0.99 p X + 2.326 2 99% 5.3.2.. (a) 100 48 Yes 48 100 + 2.326 2 100 0.596
5 16 Yes 59.6% 99% (b) 1,000 480 Yes? 480 1000 + 2.326 2 1000 0.517 Yes 51.7% 99% (c) 10,000 4,800 Yes? 4800 10000 + 2.326 2 10000 0.491 Yes 49.1% 99% 5.4. 5.4.1. 95% 90% 5.4.2. 1 5,400? 5,150? 5,400 5400 10000 2.326 2 10000 0.528 5400 10000 + 2.326 2 10000 0.552 98%( 99%?) 0.528 0.552 5,150 5150 10000 2.326 2 10000 0.503 5150 10000 + 2.326 2 10000 0.527 98%( 99%?) 0.503 0.527 5,050 5050 10000 2.326 2 10000 0.493 5050 10000 + 2.326 2 10000 0.517 98%( 99%?) 0.493 0.517
6 17 6. 6.1. 6.1.1. 1 9.9 6.6 3.5 780 2 11.1 7.4 4.1 950 3 14.4 10.4 6.7 1280 4 19.5 15.1 11.2 1255 5 23.7 19.4 15.6 1290 6 26.9 23 19.9 1650 7 30.9 27.2 24.3 2000 8 32.1 28.1 25 2430 9 28.3 24.4 21.3 1200 10 23.4 19.2 15.4 1150 11 17.8 13.8 10.2 1210 12 12.6 8.9 5.6 1045 6.1.2. 6.2. 6.2.1. X 1,..., X S X S X = 1 i=1 (X i X ) 2. X = X 1 + + X 6.2.2. X 1,..., X Y 1,..., Y S XY S XY = 1 (X i X )(Y i Y ). i=1 6.2.3. X 1,..., X Y 1,..., Y r XY r XY = S XY SX SY 6.2.4. S XX = S X 6.2.5. (a) X 1 = Y 1,..., X = Y r XY = 1. (b) Y 1 = X 1,..., Y = X r XY = 1
6 18 (c) Y i = ax i + b r XY = { 1 (a > 0) 1 (a < 0). 6.2.6. X Y r XY = 1 X 1,..., X 10, Y 1,..., Y 10, Y i = X 4 i X 9 8 7 10 13 Y 6561 4096 2401 10000 28561 X 11 9 8 12 9 Y 14641 6561 4096 20736 6561 r XY = 0.972! 6.2.7. 0.980 0.974 0.958 6.3. 1 2 3 21 47.1 86.8 111.3 68 106.6 62.4 38.7 54 89 36.3 29 10 26.4 103.9 28.9 47 143.8 57.2 72.5 78 177.5 37.9 34 53 155.5 76.5 78.2 46 81.7 35.4 43.9 14 30.8 108.5 3.4 34 116.4 67.7 7.2 12 34.7 137 48.2 72 167.3 8.5 109 78 138.3 26.3 30.9 29 48.2 115.7 81 59 102.7 40.1 58.5 17 44.3 121.5 114.3 89 211.5 7.7 60.5 57 87 15.5 74.7 63 95.6 72.5 67 27 73.1 85 82.5
7 19 7. 7.1. 7.1.1. 20? 6.6 780 7.4 950 10.4 1280 15.1 1255 19.4 1290 23 1650 27.2 2000 28.1 2430 24.4 1200 19.2 1150 13.8 1210 8.9 1045 7.1.2. y = ax + b x = 20! a, b? 7.1.3. (x 1, y 1 ),..., (x, y ) x = x 1 + + x, y = y 1 + + y, S xy = (x i x)(y i y), S xx = (x i x) 2, i=1 â = S xy S xx, b = y âx i=1
7 20 a = â b = b a, b y = âx + b 7.1.4. (1) Q(a, b) = {y i (ax i + b)} 2 ( ) i=1 ( ) S xx = i {x 2 i 2xx i + (x) 2 } = i x 2 i 2x 2 + x 2 = i x 2 i x 2, S xy = i = i S yy = i {x i y i xy i yx i + xy} = i x i y i x y, y 2 i y. x i y i xy xy + xy {y i (ax i + b)} 2 = (y i ax i b) 2 i i = {yi 2 + x 2 i a 2 + b 2 2x i y i a + 2x i ab 2y i b} i ( ) ( = b 2 + 2(xa y)b + a 2 2 ( = {b + (xa y)} 2 (xa y) 2 + ( ) = {b + (xa y)} 2 + x 2 i x 2 a 2 i i x 2 i i i x 2 i ( ) 2 x i y i x y a + i i x i y i ) a + i ) ( a 2 2 i y 2 i y 2 y 2 i x i y i ) a + i y 2 i = {b + (xa y)} 2 + S xx a 2 2S xy a + S yy ( = {b + (xa y)} 2 + S xx a S ) 2 xy S2 xy + S yy. S xx S xx ( ) 2 0
7 21 7.1.5. (2) Y = ax + b + Z (Z N(0, σ 2 )) (!) Y i = ax i + b + Z i Z i Z â = S xy S xx b = Y âx â, b (a) ; (b) ; L(a, b) = a, b i=1 E[â] = a, ( 1 2πσ 2 exp 1 2σ 2 E[ b] = b ) (y i (a + bx i )) 2 (y i (a + bx i )) 2! (c) ; 7.1.6. a, b ã = i=1 c i Y i, b = d i Y i i=1 V(ã) V(â), i=1 V( b) V( b). 1) S xx, S xy ( ) 2) SLOPE INTERCEPT 3)! 7.1.7. 7.2. 7.2.1. () 1 2 3 4 5 6 7 8 9 10 117 208 335 433 538 701 803 857 966 1176 y = 113.4545x 10.6 11 1237.4 11 1237.4 1176 = 61.4
7 22 1 2 3 4 5 6 7 8 9 10 239 476 719 953 1259 1500 1528 1673 2235 2375 7.2.2. ( ) 100% 1 y = ( ) 1 + eax+b 1 K
7 23 1 2 3 4 5 0.006425725 0.011929039 0.013588838 0.031338532 0.040913811 6 7 8 9 10 0.05212115 0.080896291 0.135832283 0.15159588 0.237454253 11 12 13 14 15 0.297813165 0.341376055 0.520958623 0.545791293 0.627789932 16 17 18 19 20 0.719939233 0.802076416 0.859232933 0.908140584 0.942957833 21 22 23 24 25 0.950664594 0.965184638 0.979148139 0.986842793 0.992209042 26 27 28 29 30 0.992894015 0.996895595 0.997354583 0.998558411 0.998587263 ( 1 ) z = l y 1 z = ax + b l x x = e z z LN LOG log 10 a = 0.404, b = 5.291 1 y = 1 + e 0.404x+5.291 7.2.3. 1 2 3 4 5 0.064790694 0.090423841 0.111418814 0.142571342 0.211331758 6 7 8 9 10 0.288737166 0.355482215 0.368020098 0.533691118 0.589594203 11 12 13 14 15 0.606063121 0.753918853 0.824012705 0.857759485 0.891809752 16 17 18 19 20 0.923783022 0.892751872 0.938201537 0.97154412 0.981719392 10 0.5 1 1.5 2 2.5 3 3.5 5.06 6.23 8.65 13.32 17.98 23.13 30.19 4 4.5 5 5.5 6 6.5 7 36.47 50.27 59.37 73.24 86.27 101.47 103.06 7.5 8 8.5 9 9.5 10 125.92 139.58 160.08 184.41 192.1 225.37
7 24 2 y = ax 2 + b 2 z = x 2 y = az + b a = 2.16, b = 4.56 y = 2.16x 2 + 4.56 1 2 3 4 5 6 7 7.24 22.85 47.88 82.42 132.77 185.15 256.24 8 9 10 11 12 13 14 324.32 410.69 521.63 618.1 757.69 888.11 1024.46 15 16 17 18 19 20 1137.92 1283.49 1460.3 1654.67 1844.63 2091.71 7.2.4. (a) y = bx a l y = l b + a l x (b) y = b exp(ax) l y = l b + ax (c) y = b + a l x (d) y = x bx + a 1 y = b + a 1 x (e) y = 1 ax + b 1 y = ax + b (f) y = a + b x exp(a + bx) (g) y = 1 + exp(a + bx) ( )
8? 25 8.? 8.1. 8.1.1. 2 X Y Y X 100 Y 80 X 1 Y 1 1 1 1 1 Y 8.1.2. (a) 0 (b) 95% (c) a 95% a (d) 11000 10000 = 1000 > a 5% 8.1.3. (a) H 0 (b) α 1 α (c) (d) 8.2. 8.2.1. X X 1,..., X 100 Y Y 1,..., Y 80 X i = () + ( ) i = µ X + e i Y j = () + ( ) j = µ Y + E j e 1,..., e 100, E 1,..., E 80 N(0, σ 2 ) 8.2.2.
8? 26 (a) U N(µ U, σ 2 U ) V N(µ V, σ 2 V ) U + V N(µ U + µ V, σ 2 U + σ2 V ) (b) W N(µ, σ 2 ) aw N(aµ, a 2 σ 2 ), W µ σ N(0, 1) (c) Z 1,..., Z ( N(µ, σ 2 )) Z = Z 1 + + Z 8.2.3. X Y X 100 N (µ X, σ2 100 X 100 Y 80 N (a) H 0 µ X = µ Y ) ), Y 80 N (µ Y, σ2 80 ( µ X µ Y, σ 2( 1 100 + 1 80 ( X 100 Y 80 N 0, σ 2( 1 100 + 1 )) 80 X 100 Y 80 ( 1 σ 2 100 + 1 ) N(0, 1) 80 [ ] Z N(0, 1) P( Z a) = Φ(a) Φ( a) = 2Φ(a) 1 Φ(a) = (b) α a a 1 2π e x2 2 dx. 1 α = 2Φ(a) 1 )). N ) (µ, σ2 Φ(a) = 1 α 2 (a = NORM.S.INV(1 α2 ) ) 5%=0.05 a = NORM.S.INV(0.975) = 1.96 95% (d) σ 2? X 100 Y 80 ( 1 σ 2 100 + 1 ) 1.96 80
8? 27 σ 2 = 100 X 100 Y 80 ( 1 100 100 + 1 ) 1.96 80 95% X 100 Y 80 = 1000 ( ) = 666.7 1.96 8.2.4. (σ 2 ) X X X Y Y Y H 0 X Y α X Y ( 1 σ 2 + 1 ) > NORM.S.INV(1 α 2 ) X Y X Y 8.2.5. = 0.05(5%) 5% 5% 8.2.6. A X A B Y B α σ 2 NORM.S.INV 100 220 200 219.6 0.05 1 100 220 200 219.6 0.1 1 100 220 200 219.6 0.01 1 100 220 200 219.7 0.05 1 100 220 200 219.7 0.1 1 100 220 200 219.7 0.01 1 100 220 200 219.76 0.05 1 100 220 200 219.76 0.1 1 100 220 200 219.76 0.01 1 ABS(x)
8? 28 8.2.7. σ 2 X X X Y Y Y H 0 A B α t X,Y = S XX = S Y Y = X i=1 Y j=1 X Y S XX + S Y Y X + Y 2 ( 1 X + 1 Y ) (X i X) 2 = X X (Y j Y ) 2 = Y Y. t X,Y > T.INV.2T(α, X + Y 2) X Y 8.2.8. What s behid! t- (a) t X,Y X + Y 2 t- (b) W t- P(W a) = a B(a, b) = (c) 1 1 B( 2, 1 2 ) (1 + x2 1 0 ) +1 2 x a 1 (1 x) b 1 dx. dx = T (a) P( t X,Y a) = T X + Y 2(a) T X + Y 2( a) = 2T X + Y 2(a) 1 1 α = 2T X + Y 2(a) 1 a ; a = T.INV(1 α 2, X + Y 2) = T.INV.2T(α, X + Y 2).
8? 29 8.2.9. 0.01, 0.05, 0.1 X; 581, 700, 597, 534, 596, 582, 538, 588, 581, 539 Y; 543, 510, 580, 520, 506, 550 8.2.10. 0.01, 0.05, 0.1 X; 53, 59, 51, 58, 57, 55, 53, 56, 54, 51, 54, 60 Y; 58, 57, 58, 60, 58, 56, 58, 57, 56, 55
9? 30 9.? 9.1. 9.1.1.? 9.1.2. 20 32 24 28 28 18 30 20 9.1.3. 200 ( 104 96 ) 104 96 200 200 48 50 54 48 48 50 54 48 200 200 200 200 200 104 200 48 = 24.96 ( ) 200
9? 31 24.96 20 = 4.96 9.2. χ 2 () 9.2.1. M A 1,..., A M N B 1,..., B N B 1 B 2 B N A 1 S 11 S 12 S 1N. A M S M1 S M2 S MN p i = S i1 + + S in, q j = S 1j + + S Mj M N χ 2 (S ij p i q j ) 2 = p i p j i=1 j=1 p i, q j B 1 B 2 B N p i A 1 S 11 S 12 S 1N S1 A p 1. A M S M1 S M2 S MN SM A p M q j S1 B q 1 S2 B q 2 SN B q N α χ 2 > CHISQ.INV(1 α, (M 1)(N 1)) α 9.2.2. = B M = 2 N = 4 χ 2 = 6.94 CHISQ.INV(0.9, 3) = 6.25
9? 32 CHISQ.INV(0.95, 3) = 7.81 CHISQ.INV(0.99, 3) = 11.3487 0.1 0.05 0.01 9.2.3. What s behid! (a) A 1,..., A M B 1,..., B N ( B ) (b) χ 2 (M 1)(N 1) (c) k P(χ 2 a) = Γ(y) = a 0 1 k 2 x 2 k e x2 2 dx. 2 Γ( k 2 ) 0 x y 1 e x dx. (d) α a = CHISQ.INV(1 α, k) 9.2.4.! P(χ 2 a) = 1 α 10 25 48 33 28 30 35 35 27 36 50 40 28 43 22 9.3. 9.3.1. 3:2:3:2 1000 280 215 314 191? 9.3.2. ;
9? 33 9.3.3. 280 215 314 191 0.3 0.2 0.3 0.2 300 200 300 200 B 1,..., B N p 1,..., p N B i B 1 B 2 B N X 1 X 2 X N χ 2 = N (X i p i ) 2 i=1 N χ 2 N 1 α p i χ 2 CHISQ.INV(1 α, N 1)! p 1,..., p N ; a 2 = 4 a = 2 a 2 = 4 a = 2 9.3.4. N = 4 3 χ 2 = 3.51 CHISQ.INV(0.9, 3) = 6.25 CHISQ.INV(0.95, 3) = 7.81 CHISQ.INV(0.99, 3) = 11.34 9.3.5. 2:1:2:2:3 1000 210 90 190 190 320?