1 1 3 1.1 (Frequecy Tabulatios)................................ 3 1........................................ 8 1.3.....................................

1 1 3 1.1 (Frequecy Tabulatios)................................ 3 1........................................ 8 1.3........................................... 1 17.1................................................ 17 3 3 3.1........................................... 3 3................................................. 6 3.3................................................ 9 3.4 ( ).................................. 33 3.5 ( ).................................. 35 3.6............................................ 37 3.7 χ................................................ 38 3.8 t................................................. 40 3.9 F................................................. 41 4 43 4.1......................................... 43 4................................... 47 4.3.............................................. 51 4.4.............................................. 54 4.5......................................... 59 5 63

3 1 1.1 (Frequecy Tabulatios) (samplig) (sample) 1.1 1.1: 13.6 14.8 13.7 14. 11.5 11.9 13.8 14.6 14. 1.7 13.4 11.5 11.9 14.8 1.7 1.4 15.3 15. 13.5 15.0 1.4 1.0 13.8 11.7 10.0 13. 15.5 14.0 13.5 15.0 1.7 1.9 13.7 15.1 13.5 15.7 1.7 15.7 10.9 14.0 14.8 14.0 13.8 1.7 11.9 1.0 11.4 11.1 13.7 13. (frequecy table) x i x i (frequecy) f i f 1 + f + + f k =

4 1 x 1 f 1 x f.. x k f k x i a b a b (class iterval) a b b a (class iterval width) a + b (midpoit) f i / (relative frequecy) f i F i = f 1 + f + + f i (frequecy distributio) 10.0 15.7 0.9 7 9.95 1.: 9.95 10.85 10.4 1 0.0 1 0.0 10.85 11.75 11.3 6 0.1 7 0.14 11.75 1.65 1. 7 0.14 14 0.8 1.65 13.55 13.1 1 0.4 6 0.5 13.55 14.45 14.0 1 0.4 38 0.76 14.45 15.35 14.9 9 0.18 47 0.94 15.35 16.5 15.8 3 0.06 50 1.00 (histogram) (cumulative distributio fuctio)

1.1. (Frequecy Tabulatios) 5 1.1: 1.: Sturges = 1 + log 10 log 10 (1.1) 50 k k = 1 + log 10 50 log 10 = 1 + 3.3 log 10 50 = 1 + 3.3(1.699) = 6.64 7 ( - )/ (15.7-10.0)/6.64 = 0.86 0.9 7 : x x 1, x,..., x (total) T = x 1 + x + x = x i x = x 1 + x + x = 1 x i = T (mea) x x 1, x,..., x k f 1, f,..., f k T = x 1 f 1 + x f + + x f

6 1 x = x 1f 1 + x f + + x f = 1 k x i f i (media) + 1 + 1 x i m i (mode) (1.1) ( 1.1)

1.1. (Frequecy Tabulatios) 7 1 1. (g/cm ) 30 380 340 410 380 340 360 350 30 370 350 340 350 360 370 350 380 370 300 40 370 390 390 440 330 390 330 360 400 370 30 350 360 340 340 350 350 390 380 340 400 360 350 390 400 350 360 340 370 40 40 400 350 370 330 30 390 380 400 370 390 330 360 380 350 330 360 300 360 360 360 390 350 370 370 350 390 370 370 340 370 400 360 350 380 380 360 340 330 370 340 360 390 400 370 410 360 400 340 360

8 1 1. (dispersio) 5 f i f i / F i F i / 0 0.0 0.0 1 13 0.13 15 0.15 33 0.33 48 0.48 3 35 0.35 83 0.83 4 16 0.16 99 0.99 5 1 0.01 100 1.00 T x fi x i s s 100 53.53 741 1.01 1.00 s s : x i (i = 1,,..., k) f i x x : s = (x 1 x) f 1 + (x x) f + + (x k x) f k = 1 k (x i x) f i (variace) : s = s = 1 k (x i x) f i (stadard deviatio) s = (x 1 x) f 1 + (x x) f + + (x k x) f k 1 1 k = (x i x) f i 1

1.. 9 x x 1, x,..., x s = (x 1 x) + (x x) + + (x k x) = 1 (x i x) 1. s = 1 s = 1 x i (x) (x i x) = 1 (x i x i x + x ) ( = 1 x i x x i + ( = 1 ) x i xx + x = 1 x i x ) x 10 30 60 x s s x 1.3 x 67 s x 8.5 ȳ 53 s y 1.6 A 75 68 A z i = x i x s

10 1 {z i } 0 s 1 z eglish = z math = 75 67 = 0.94 8.5 68 53 = 1.19 1.6 A. (x, y) (x 1, y 1 ), (x, y ),, (x, y ) (covariace) (correratio coefficiet) s xy = 1 (x i x)(y i y) = 1 r = s xy s x s y s x x s y y x i y i xy 1. p/100 x p 5% Q 1 1, 50% Q, 75% Q 3 3 0 1 67 54 54 66 56 65 46 35 45 45 83 7 54 58 47 60 43 8 76 9

1.. 11 1. 1.3: x y x y 1 3 43 13 38 1 1 8 14 51 37 3 18 17 15 109 65 4 17 16 16 90 65 5 17 16 17 78 50 6 15 10 18 75 58 7 13 5 19 34 4 8 14 5 0 33 5 9 16 8 1 9 55 10 17 13 31 55 11 17 11 3 5 55 1 35 8 4 5 51

1 1 1.3 1 x i, y i x y (x i, y i ) (correratio diagram) x i y i x i y i (positive correratio) x i y i x i y i (egative correratio) x x 1, x,..., x y y 1, y,..., y : (x 1, y 1 ), (x, y ),..., (x, y ) y ( (liear regressio)): y = ax + b x i y ( ŷ i ) ŷ i = ax i + b x i ( ) y i y i ŷ i d i d i = y i y i = y i ax i b d i = (y i ax i b)

1.3. 13 a, b (method of least square) (y i ax i b) a, b a, b (y i ax i b) F (a, b) = (y i ax i b) y = f(x) y = f (x) = 0 a b F a F b = = [(y i ax i b)( x i )] = (y i ax i b)x i [(y i ax i b)( 1)] = (y i ax i b) x = P xi x i y i a x i b x i = 0 y i a x i b = 0 P, y = yi x i y i a x i bx = 0 y ax b = 0 y ax b = 0 x i y i a x i (y ax)x = 0 ( ) x i y i xy = a x i x 1 ( 1 x i y i xy = a ) x i x s xy a x as x s xy = as x

14 1 a a = s xy as x y ax b = 0 b b = y ax = y s xy as x x x y y y = s xy s (x x) x {(x i, y i ) (i = 1,,..., ) y x a yx = s xy s x = T xy T x T y x i T x y x l y y = a xx (x x) x,y y y 3 x 1 : x : x 3 : y 3 x 1, x, x 3 y x 1, x, x 3 y = b 0 + b 1 x 1 + b x + b 3 x 3

1.3. 15 3 1.4: x y x y 1 3 43 13 38 1 1 8 14 51 37 3 18 17 15 109 65 4 17 16 16 90 65 5 17 16 17 78 50 6 15 10 18 75 58 7 13 5 19 34 4 8 14 5 0 33 5 9 16 8 1 9 55 10 17 13 31 55 11 17 11 3 5 55 1 35 8 4 5 51

17.1 x 1, x,..., x X X = x i p i X X P (X = x i ) X x F (x) X.1 F (x) = P (X x) 6 E = 1 P (E) = 1 6 X = E X 0 6 7 ( ) ( ) i ( ) 6 i 6 1 5 p i = P (X = i) = i 6 6 X (biomial distributio) X B(6, 1 6 ) 1 3 1.. 3. 1 X p i X P (X = i) = ( ) p i (1 p) i i X X B(.p) X X X X x 1, x,..., x (X = x i ) p 1, p,..., p P (X = x i ) = p i (i = 1,,..., ) pi = 1, (p i 0) X f

18 X x i x 1 x x P (X = x i ) = p i = f(x i ) p 1 p p X x 1 < x < < x F (x r ) f F 1. 0 p i = f(x i ) 1 (i = 1,,..., ) F (x r ) = P (X x r ) = p 1 + p + + p r =. F (x ) = P (X x ) = p 1 + p + + p = 1 3. P (a < X b) = F (b) F (a) 4. a < b = F (a) < F (b) X ( ) µ = E(X) = k x i p i r σ = V (X) = E ( (X µ) ) = E(X ) E(X). E(X) = k x ip i, E(Y ) = l j=1 y jq j E(X + Y ) = E(X) + E(Y ) P (X = x i, Y = y j ) p ij { l j=1 p k ij = p i p ij = q j k l j=1 p ij = k p i = l j=1 q j = 1 p i E(X + Y ) = = = k l (x i + y j )p ij j=1 k l l k (x i p ij ) + (y j p ij ) j=1 j=1 k l x i p i + y j q j = E(X) + E(Y ) j=1.3 E((X µ) ) = E(X ) (E(X))

.1. 19 E((X µ) ) = E(X Xµ + µ ) = E(X ) µe(x) + µ E(1) = E(X ) E(X)E(X) + E(X) = E(X ) E(X) a, b (a < b) P r (a X b) P r (a X b) = b a f(x)dx f(x) (, ) f(x) (probability desity fuctio) f(x) 0 X < X x f(x)dx = 1 F (x) = P r (X x) F (x) X (probability distributio) X ( ) X µ = E(X) = σ = V (X) = E ( (X µ) ) = g(x) = 1 πσ EXP [ xf(x)dx (x µ) f(x)dx ] (x µ) σ, < x < X X N(µ, σ ) X N(3, )

0 (ormalizatio) X E(X) 0 V (X) 1 Z = X E(X) V (X) E(Z) = 0, V (Z) = 1 P r (Z z) P r (Z z) = P r (Z 0) + P r (0 < Z z) P r (Z 0) 0.5 P r (0 < Z z).1:.4 X N(60.9,.9 ) (1) P (X 63.8) () P (6.3 < X 63.0)

.1. 1 (1) () P (X 63.8) = P ( X 60.9.9 = P (Z.9.9 63.8 60.9 ).9 ) = P (Z 1) = P (Z 0) + P (0 Z 1) = 0.5 + 0.3413 = 0.8413 6.3 60.9 P ( < X 60.9 63.0 60.9 ).9.9.9 = P ( 1.4.9 < Z.1.9 = P (0.48 < Z 0.7) = P (0 Z 0.7) P (0 Z 0.48) = 0.64 0.1844 = 0.0798

4 1. X N(80, 6 ) (a) P r (X 90) (b) P r ( X 80 1). A N(10, 1 ) ( ) 50(l/ )

3 3 3.1 6 6 (populatio) 6 (sample) 6 Π 6 X (Π, X) (x 1, x,..., x ) x i X X i (X 1, X,..., X ) (X 1, X,..., X ) X i (Π, X) X 6 1 5 X 1, X,..., X X i (i = 1,,..., ) E(X i ) = µ, V (X i ) = σ X 1, X,..., X X 1, X,..., X X = 1 S = 1 X i (X i X) X µ S σ

4 3 3.1 V (X) = E(X ) E(X) = E( 1 ( Xi ) µ = 1 E(X 1 + + X + (X 1 X + + X 1 X )) µ = 1 E(Xi ) + E(X i X j ) µ 1 i,j = 1 (σ + µ ) + ( )µ µ = σ σ 3.1 ( ) X µ σ λ > 1 3. P ( X µ λσ) 1 λ P ( X µ < λσ) 1 1 λ µ, σ (Π, X) X X µ σ 1 5 0.9 P ( X µ < σ 5 ) 0.9 X σ P ( X µ < λσ ) 1 1 λ λ = 1 5 P ( X µ < σ 5 ) 1 5 0.9 50 X 1, X, X 3,..., X

3.1. 5 X = 1 [X 1 + X + X 3 + + X ] S = 1 [(X 1 X) + (X X) + + (X X) ( ) ( ) 1 θ {x 1, x,..., x } T (x 1, x,..., x ) (X 1, X,..., X ) ˆθ = T (X 1, X,..., X ) θ θ ˆθ = T (X 1, X,..., X ) ˆθ θ E(ˆθ) = E(T (X 1, X,..., X )) = θ N(µ, σ ) U 3.3 X = 1 X E(X + Y ) = E(X) + E(Y ) E( X) = µ. =1 X i, U = 1 1 (X i X) E( X) = E(X 1 ) + E(X ) + + E(X ) = µ σ U S = 1 (X i X) = 1 U U S σ

6 3 3. x 1, x,..., x X 1, X,..., X (maximum likelihood) 3.4 3 x 1, x, x 3 µ x 1, x, x 3 X 1, X, X 3 P (X 1 = x 1, X = x, X 3 = x 3 ) L X 1, X, X 3 L = P (X 1 = 1)P (X = x )P (X 3 = x 3 ) = e µ µx 1 x 1! µx µx3 e µ e µ x! x 3! = e 3µ µx1+x+x3 x 1!x!x 3! µ L µ x 1, x, x 3 µ L = L(µ) dl dµ = 3e 3µ µx1+x+x3 x 1!x!x 3! dl dµ = 0 3µ µx1+x+x3 1 + (x 1 + x + x 3 )e x 1!x!x 3! = 3L + µ 1 (x 1 + x + x 3 )L = L µ ( 3µ + x 1 + x + x 3 ) = 0 µ = 1 3 (x 1 + x + x 3 ) L 3.5 N(µ, σ ) x 1, x,..., x σ µ N(µ, σ ) f(x) = 1 (x µ) exp{ πσ σ } ( 1 L = P (X 1 = x 1 ) P (X = x ) = exp { (x 1 µ) + + (x µ) }) πσ σ x 1, x,..., x, σ dl dµ = 1 σ {(µ x 1) + (x µ) + + (µ x )}L = 1 σ {µ (x 1 + x + + x )}L = 0

3.. 7 µ = 1 (x 1 + x + + x ) = x

8 3 5 1. 110, 11, 133, 14, 16, 118, 11, 15, 131, 10(cm)

3.3. 9 3.3 θ [θ 1, θ ] θ θ 1, θ α (0 < α < 1) P r (θ 1 < θ < θ ) = 1 α (θ 1, θ ) θ θ 1, θ 100(1 α)% [θ 1, θ ] θ [θ 1, θ ] θ 1 α N(µ, σ ) σ µ X 1, X,..., X X i N(µ, σ ) µ (σ ) S = 1 S = E(X) = µ, V (X) = σ X N(µ, σ ) (X i X) E(S ) = 1 σ 1 S E(S ) = σ α = 0.05 95% X N(µ, σ ) Z = X µ N(0, 1) σ / P r ( Z z α ) = 1 α = 0.95 z α P r (Z z α ) = α

30 3 z α α = 0.05 z α z α = 1.96 X µ Z = σ / z α µ σ σ X z α µ µ X + z α 3.1: 3.6 8, 4, 31, 7,. 95%. σ = 6.5 X i X i N(µ, 6.5) X N(µ, σ /5) X X = 1 13 [8 + 4 + 31 + 7 + ] = 5 5 = 6.4 Z = X µ N(0, 1) σ /5 95% P r ( Z z α ) = 0.95. z 0.05 = 1.96. X z α σ 5 µ X + z α σ 5

3.3. 31 6.4 1.96 6.5/5 µ 6.4 + 1.96 6.5/5 4.1 µ 8.59 σ (σ ) α = 0.05 95% µ, σ σ σ S σ T = X µ S / 1 t t 1,α/ P r ( T t 1,α/ )) = 1 α P r (T t 1,α/ ) = α t 1,α/ t α = 0.05 = 10 t 9,0.05/ µ X t 1,α/ S t 9,0.05/ =.6 X µ t 1,α/ S / µ X + t 1,α/ S

3 3 6 1 BOD(ppm) σ = 6.5(ppm) = 15 X = 7.ppm 95% 145.3, 145.1, 145.4, 146. 146 95%

3.4. ( ) 33 3.4 ( ) A p A p (X 1,..., X ) { 1 A X i = 0 Ā X = X 1 + + X X A X A X p p (X 1,..., X ) X = X 1 + + X X B(, p) X ( ) N(p, p(1 p)) X = ˆp N p, p(1 p) ˆp p p(1 p) P ˆp p p(1 p) N(0, 1) z α = 1 α p(1 p) p(1 p) ˆp z α p p + z α p ˆp p 100(1 α)% ( ˆp z α ˆp(1 ˆp), ˆp + z α ˆp(1 ˆp) ) 3. ( ) X B(, p) X N(p, p(1 p)) 3.7

34 3 600 1 108 1 p 95% ˆp = 108 600 = 0.18 z α = z 0.05 = 1.96 ( ) (0.18)(0.8) (0.18)(0.8) 0.18 1.96, 0.18 1.96 600 600 (0.15, 0.1)

3.5. ( ) 35 3.5 ( ) p = P (A) A ( ) x p 100(1 α)% (p 1, p ) ( ) m 1 f, 1 f 1 + m 1 f + m 1 = ( x + 1), = x ( 1, ) F F f 1 F P (F > f 1 ) = α

36 3 7 1 900 180 95% 300 187 95%

3.6. 37 3.6 X, Y N(µ 1, σ1), N(µ, σ) ax + by N(aµ 1 + bµ, a σ 1 + b σ ) 3.3 X 1, X,..., X N(µ, σ ) X = X 1 + X + + X N(µ, σ ) X 1 + X + + X N(µ, σ ) 3.8 X = 1 (X 1 + X + + X ) N(µ, σ σ ) = N(µ, ) 10cm 4.5cm 50 10.6cm X i X i N(10, 4.5 ) 50 X 3.3 X N(10, 4.5 50 ) P (X > 10.5) = P (Z > 10.5 10 4.5 50 ) = P (Z > 0.948) = 0.5 P (0 < Z < 0.958) = 0.5 0.3710 0.173 X i [ ] X 1, X,..., X µ σ X = X 1 + X + + X > 100 N(µ, σ )

38 3 3.7 χ χ χ () χ f (x) Γ(x) 1 f (x) = Γ( ) x 1 e 1 x x > 0 0 x 0 Γ(x) = χ 0 t x 1 e t dt (x > 0) 3.4 X 1, X,..., X N(0, 1) χ = X 1 + X + + X χ E(χ ) =, V (χ ) = 3.5 (χ ) χ, χ m,m χ χ = χ +χ m + m χ S 3.6 N(µ, σ ) {X 1, X,..., X } Y = 1 σ (X i X) = S σ 1 χ 3.9 0 1.5 1 1.5 P (S 1.5) X i N(µ, 1) Y = 0S 1 = 0 (X i X)

3.7. χ 39 19 χ P (S 1.5) = P (0S 8.5) = P (χ 19 8.5) χ P (χ 19 > 7.0) = 0.10 P (χ 19 > 30.14) = 0.05 P (χ 19 8.5) = 0.05 + 8.5 7.0 (0.10 0.05) 0.07 30.14 7.0

40 3 3.8 t 3.1 f (x) f (x) = +1 γ( ) πγ( x +1 + )(1 ) ( 1) T t 3 E(T ) = 0, V (T ) = 3.7 X 1, X,..., X N(µ, σ ) U U = S = 1 1 (X i X) T = X µ U/ 1 t 3.8 Z χ χ Z χ T = Z χ m / t t t() t t P r (T c) P r (Z c) χ χ / 1 T T

3.9. F 41 3.9 F 3. f m, (x) f m, (x) = 0, γ( m+ ) γ( m )γ( )mm x m 1 (mx+) m+ (x > 0) F m, (m, ) F E(F m, ) = m ( > ) V (F m, ) = (m + ) m( ) ( 4) ( > 4) 3.9 X 11, X 1,..., X 11 N(µ 1, σ 1) 1 X 1 U 1 X 1,..., X N(µ, σ ) X U X 1 = X = 1 1 i X 1i 1 i X i, U 1 = 1 1 1, U = 1 1 (X 1i X 1 ) i (X i X ) i F = U 1 /σ 1 U /σ = σ U 1 σ 1 U F ( 1 1, 1) 3.10 F 1 (0.05) = F 5 10(0.05) F 5 10(0.05) = 3.358 F 1 (1 α) 3.11 F 1 (1 α) = 1 F 1 (α) F 5 11(1 0.05) F 5 11(1 0.05) = 1 5 (0.05) = 1 3.10 = 0.3 F 11

43 4 4.1 ESP 5 1 5 3 1. 5 3. 5 3 p 5 X X B(5, p) p = 0. p > 0. 5 3 H 0 : p = 0. ( ) P r (X 3) = P r (X = 3) + P r (X = 4) + P r (X = 5) ( ) ( ) ( ) 5 5 5 = (0.) 3 (0.8) + (0.) 4 (0.8) + (0.) 5 = 0.0579 3 4 5 1 H 0 H 0 H 0 (sigificace level) α α 0.05, 0.01 0.05 H 0 ( ) H 0 X P r (X 4) = 0.0067 X 4 (critical regio) H 0 : p = 0. (ull hypothesis) H 1 : p > 0. (alterative hypothesis)

44 4 θ H 0 : θ = θ 0 3 1.. α H 1 : θ > θ 0, H 1 : θ < θ 0, H 1 : θ θ 0 θ 3. 4. 5. (1) N(µ, σ ) (X 1, X,..., X ) X N(µ, σ ) µ (a) σ µ ( α) Z = X µ N(0, 1) σ / (b) σ µ ( α) () T = X µ t( 1) S / N(µ, σ ) (X 1, X,..., X ) σ (a) µ σ ( α) χ = 1 σ (X i µ) χ α, (b) µ σ ( α) χ = S σ χ α, 1

4.1. 45 (a) σ µ ( α) H 0 : µ = µ 0 3 (1) H 1 : µ > µ 0 () H 1 : µ < µ 0 (3) H 1 µ µ 0 ( µ = µ 0 ) Z 0 = X µ 0 N(0, 1) σ / α P r (Z 0 > z α ) = α z α α (1) Z 0 > z α (1) Z 0 > z α () Z 0 < z α (3) Z 0 > z α 4.1 64.5, 0 8 66 73 55 69 70 67 6 71 5% 1 µ H 0 : µ = 64.5 H 1 : µ > 64.5 X i X i N(µ, 0) X = 1 [66 + 73 + 55 + 69 + 70 + 67 + 6 + 71] = 533/8 = 66.65 8 X N(µ, 0/8). Z 0 = 66.65 64.6 0/8 =.05 1.58 = 1.8

46 4 4.1: Z 0 1

4.. 47 8 1 7mm 0.0mm 16 7.09mm α = 0.05 µ 95% 1 ( mm) 0.0016 8 11.97 1.0 1.06 1.03 11.99 11.98 1.1 1.05 0.0016 α = 0.05 95% 4. X µ 1 σ 1 X 1, X,..., X 1 X S 1 Y µ σ Y Y 1, Y,..., Y Ȳ S 1. µ 1 µ (a) σ1, σ X Y N X N(µ 1, σ 1 1 ), Ȳ N(µ, σ ) ( µ 1 µ, σ 1 1 + σ ) Z = ( X Ȳ ) (µ 1 µ ) σ 1 / 1 + σ / N(0, 1) 4. A 30 B 50 148.cm 146.4cm 4.8cm 0.05 A N(µ 1, 4.8 ), B N(µ, 4.8 ) 1 = 30, = 50 X N(µ 1, 4.8 30 ), Ȳ N(µ, 4.8 50 ) H 0 : µ 1 = µ H 1 : µ 1 µ α = 0.05

48 4 H 0 z 0.05/ = 1.96 Z = ( X Ȳ ) (µ 1 µ ) σ 1 / 1 + σ / Z 0 = N(0, 1) 148. 146.4 4.8 /30 + 4.8 /50 = 1.638 Z 0 = 1.6 < z 0.05/ = 1.96 H 0 (b) σ 1, σ σ 1 = σ X 1,..., X 1, Y 1,..., Y S 1, S S 1 = 1 1 1 (X i X), S = 1 (Y i 1 Ȳ ) i ˆσ = ( 1 1)S 1 + ( 1)S 1 + S1 = 1 (X i X), S = 1 (Y i 1 Ȳ ) i ˆσ = 1S 1 + S 1 + ˆσ (a) T = ( X Ȳ ) (µ 1 µ ) ˆσ / 1 + ˆσ / t( 1 + ) i i 1 + t 4.3 A,B. 5 X A = 97.5%, XB = 95.3% S A = 1.3%, S B = 1.56% 0.05 µ A %, µ B % H 0 : µ 1 = µ H 1 : µ 1 µ

4.. 49 α = 0.05 T = ( X Ȳ ) (µ 1 µ ) ˆσ / 1 + ˆσ / t( 1 + ), ˆσ = 1S 1 + S 1 +. H 0 ˆσ = 5(1.3)+5(1.56) 5+5 = 1.748% T 0 = 97.5 95.3 1.7438/5 + 1.7438/5 =.634 t 0.05/,8 =.3060 T 0 = 1.748 < t 0.05/,8 =.3060 H 0 (c) σ 1, σ T = ( X Ȳ ) (µ 1 µ ) S 1 /( 1 1) + S /( 1) t(ϕ), 1 ϕ = c (1 c) + 1 1 1, 1 c = 1 + ( 1 1)S ( 1)S1. σ 1/σ 1 S 1 σ 1 χ ( 1 1), S σ χ ( 1) F = σ S 1 σ1 S F ( 1 1, 1) 1. H 1 : σ 1 σ. H 1 : σ 1 > σ 3. H 1 : σ 1 < σ W = {F : F > F 1 1 1 (α )} {F : F < F 1 1 1 (1 α )} W = {F : F > F 1 1 1 (α)} W = {F : F < F 1 1 1 (1 α)} F 1 1 1 (1 α) F 1 1 1 (1 α) = 1 F 1 1 1 (α)

50 4 4.4 A,B 10 16 5.3g.48g 5% A = 10, S A = 5.3, S = 10 9 S A B = 16, S B =.4, S = 16 15 S B H 0 : σ 1 = σ H 1 : σ 1 σ α = 0.05 H 0 F 9 15(0.05) = 3.17 F = σ S 1 σ1 S F ( 1 1, 1) F 0 = 10(5.3) 9 16(.4) 15 =.1967 F 0 =.1967 < F 9 15(0.05/) = 3.17 H 0

4.3. 51 9 1 A B A 10 B 1 A 71 79 9 91 87 79 77 89 71 84 B 63 84 71 81 80 84 71 84 64 84 69 77 A B N(µ 1, σ 1 ), N(µ, σ ) H 0 : µ 1 = µ 5% A B A 10 B 1 A 71 79 9 91 87 79 77 89 71 84 B 63 84 71 81 80 84 71 84 64 84 69 77 A B N(µ 1, σ 1 ), N(µ, σ ) 5% H 0 : σ 1 = σ 4.3 ( ) A p A p (X 1,..., X ) { 1 A X i = 0 Ā X = X 1 + + X X A X A p p 0 (0 p 0 1) H 0 : p = p 0 H 1 : p p 0 p (X 1,..., X ) X = X 1 + + X X B(, p) X ( ) N(p, p(1 p)) X = ˆp N p, p(1 p) Z = ˆp p p(1 p) N(0, 1)

5 4 4.5 600 1 108 1 p 1 6 5% H 0 : 1 p = 1 6 H 1 : p 1 6 α = 0.05 H 0 ˆp = 108 600 = 0.18 Z = ˆp p p(1 p) N(0, 1) Z 0 = 0.18 1 6 1 6 (1 1 6 ) 600 = 0.088 Z 0 < Z 0.05 = 1.96 H 0 A, B 1 C p 1, p 1, X 1, X H 0 : p 1 = p H 1 : p 1 p p 1, p p 1 = p = p, 1 p = q 1, X 1, X (N( 1 p, 1 pq), N( p, pq) X 1 = X 1 1 X = X (N(p, pq 1 ) (N(p, pq ) X 1 X N(0, ( 1 + 1 )pq) 1 1 Z = X 1/ 1 X / N(0, 1) ( 1 1 + 1 )p(1 p)

4.3. 53 p 4.6 p = X 1 + X 1 + 400 10 500 180 5% p 1 p p 1 = 10 400, p = 180 500 H 0 : p 1 = p H 1 : p 1 p α = 0.05 Z = X 1/ 1 X / N(0, 1) ( 1 1 + 1 )p(1 p) H 0 p = 10+180 400+500 = 1 3 Z 0 = 10/400 180/500 ( 1 400 + 1 500 ) 1 3 (1 1 3 ) = 1.897 Z 0 < Z 0.05 = 1.96 H 0

54 4 10 1 8% 3,000 5% 5% 00 5 300 0 5% 4.4 (goodess of fit test) X (1) k A 1, A,..., A k A i P (A i ) P (A i ) = p i p 1 + p + + p k = 1 A 1, A,..., A k 1,,..., k ( ) p 1 1 1,,..., p p k k =! k 1!! k! p 1 1 p p k k k = (multiomial distributio) A i X i 4.7 P (X 1 = 1, X =,..., X k = k ) =! 1!! k! p1 1 p p k k 6 1 6 1 1 6 1 6 1 ( 6 1,1,1,1,1,1) ( ) 6 1, 1, 1, 1, 1, 1 ( 1 6 )6 = 6! 6 6 = 6 5 4 3 1 6 6 6 6 6 6 = 5 34 A 1, A,..., A k P (A i ) = p i A i p i = m i A i X i m 5 χ = (X 1 m 1 ) m 1 + (X m ) m + + (X k m k ) m k

4.4. 55 χ (k 1) m i X i i χ χ m i H 0 : p 1 = p 10, p = p 0,..., p k = p k0 (p i0 p 10 + p 0 + + p k0 = 1 ) H 1 : p 1 = p 11, p = p 1,..., p k = p k1 (p 11, p 1,..., p k1 ) (p 10, p 0,..., p k0 ) H 0 A i m i m i = p i0 i m i 5 A i x i k χ (x i p i0 ) = > χ α,k 1 H 0 p i0 4.8 600 0.05 1 3 4 5 6 10 89 87 106 115 101 600 H 0 : (p 1, p, p 3, p 4, p 5, p 6 = 1 6, 1 6, 1 6, 1 6, 1 6, 1 6 ) H 1 : (p 1, p, p 3, p 4, p 5, p 6 1 6, 1 6, 1 6, 1 6, 1 6, 1 6 ) α = 0.05 H 0 χ = 6 (X i p i ) p i = 6 X i p i χ 0 = 10 100 + 89 100 + 87 100 + 106 100 + 115 100 + 101 100 600 = 104.04 + 79.1 + 75.69 + 11.36 + 13.5 + 10.01 600 = 5.56 χ 0.05,6 1 = 1.83 χ 0 = 5.56 < χ 0.05,5 = 11.07 H 0

56 4 11 1 A : B : C : D = 9 : 3 : 3 : 1 5% A B C D 43 7 78 15 408 () H 0 : D D θ 1, θ,..., θ i µ, σ A 1, A,..., A k (X 1, X,..., X k ) (x 1, x,..., x k ) θ i θ i = ˆθ i (x 1, x,..., x k ) (i = 1,,..., l) θ i A 1, A,..., A k m 1, m,..., m k m i = p i0. k χ (X i m i ) = χ χ k l 1 H 0 4.9 10 1 10 0 1 3 4 109 65 3 1 00 5% H 0 : P (λ) α = 0.05 P (λ) λ k p k kp k = E(X) = λ k=0 m i

4.4. 57 k 0 1 3 4 f k 109 65 3 1 00 kf k 0 65 44 9 4 1 p k 0.5435 0.3313 0.1011 0.006 0.0031 m k 108.7 66.3 0. 4.1 0.6 p k f k k kf k λ λ λ 1 k kf k = 1 00 = 0.61 k 0 1 3 4 x k 109 65 3 1 00 m k 108.7 66.3 0. 4.1 0.6 k 3 m k 5 χ m i 5 k 1 H 0 χ = (x i m i ) i=0 χ 0 = (109 108.7) + 108.7 = 0.066 m i (65 66.3) 66.3 + (6 5) 5 χ 0.05,3 1 1 = 3.84 χ 0 = 0.066 < χ 0.05,1 = 3.84 H 0 λ 1 3 1 1 = 1 (3) A, B A, B A 1,..., A k B 1,..., B l A i B j x ij B 1 B B l A 1 x 11 x 1 x 1l x 1 A x 1 x x l x A 3.... A k x k1 x k x kl x k

58 4 x i., x.j k l (cotigecy table) A B A i, B j X ij A i, B j p i, q j A i, B j P ij A, B A, B H 0 P ij = P r (A i B j ) = P r (A i )P r (B j ) = p i q j p i, q j ˆp i = x i., ˆq j = x.j H 0 χ = k j=1 l (X ij P ij ) (k 1)(l 1) x ij χ P ij χ 0 = = k j=1 k j=1 l (x ij ˆp i ˆq j ) l ˆp i ˆq j { x ij ˆp i ˆq j x ij + ˆp i ˆq j } k = l j=1 x ij 1 x i. x.j

4.5. 59 1 1 10 1 10 0 1 3 4 14 99 46 11 3 300 5% 350 A 1, A, A 3 3 B 1, B, B 3, B 4 4 5% B 1 B B 3 B 4 A 1 39 54 49 17 159 A 7 43 40 9 119 A 3 14 3 15 0 7 80 10 104 46 350 4.5 N(µ, σ ) σ Z = X µ N(0, 1) σ / N(µ, σ ) σ σ S T = X µ t( 1) S / N(µ, σ ) µ χ = 1 σ (X i µ) χ () N(µ, σ ) µ χ = S σ χ ( 1)

60 4 N(µ 1, σ 1),N(µ, σ ) σ 1, σ Z = ( X Ȳ ) (µ 1 µ ) σ 1 / 1 + σ / N(0, 1) N(µ 1, σ 1),N(µ, σ ) σ 1, σ T = ( X Ȳ ) (µ 1 µ ) ˆσ / 1 + ˆσ / t( 1 + ), ˆσ = 1S 1 + S 1 +. N(µ 1, σ 1),N(µ, σ ) F = σ S 1 σ1 S F ( 1 1, 1) A X X B(, p) X N(p, p(1 p)) ( ). p Z = X p p(1 p) N(0, 1) p = X 1+X 1 + A, B 1 C p 1, p 1, X 1, X Z = X 1/ 1 X / N(0, 1) ( 1 1 + 1 )p(1 p) A 1, A,..., A k P (A i ) = p i A i p i = m i A i X i m 5 χ = k (X i p i ) p i = k X i p i χ (k 1) H 0 : D D θ 1,..., θ l A 1, A,..., A k (X 1,..., X k ) (x 1,..., x k ) θ i θ i A i m 1,..., m k χ = k (X i m i ) χ (k l 1) p i

4.5. 61 A i, B j X ij A i, B j p i, q j A i, B j P ij P ij 5 χ = k l j=1 X ij P ij P ij χ ((k 1)(l 1)), P ij = p i q j

63 5 1 = 1 + log 100 log = 1 + 6.64 = 7.64 440 300 = 440 300 7.64 = 18.4 18 5.1: 300 318 309 0.0 0.0 318 336 37 10 0.1 1 0.1 336 354 345 5 0.5 37 0.37 359 37 363 31 0.31 68 0.68 37 390 381 8 0.08 76 0.76 390 408 399 18 0.18 94 0.94 408 46 417 5 0.85 99 0.99 46 444 435 1 0.01 100 1.00 x = 1 [318 + 336 10 + 354 5 + 363 31 + 381 8 + 399 18 + 417 5 + 435 1] 100 = 365.16 440 300 360 + 360 = 360 363

64 5 5.1: 5.:. T x = 841 T y = 806 x = 35.04 y = 33.58 4 4 T xx = x i = 45553 T yy yi = 36990 1 1 s x = 4 T xx (x) = 5.88 s y = 4 T yy (y) = 0.34 T xy = 4 x i y i = 3719 s xy = 1 T xy T x T y = 1 841 3719 4 4 806 4 = 37.85 r = s xy 37.85 = s x s y 5.88 0.34 = 0.71

65 3 = 1 + log log 4 log = 1 + log = 1 + 4.58 = 5.58 x 109 13 = 109 13 5.58 = 17.0 x 17 y 65 5 y 10 = 65 5 5.58 = 10.75 5.: x 10 7 7 44 44 61 61 78 78 95 95 11 y 18.5 35.5 5.5 69.5 86.5 103.5 0 10 5 3 3 10 0 15 6 6 0 30 5 1 3 30 40 35 1 1 40 50 45 1 1 50 60 55 3 7 60 70 65 1 1 13 7 1 3 1 4 T x = 841 T y = 806 x = 35.04 y = 33.58 4 4 T xx = x i = 45553 T yy yi = 36990 1 1 s x = 4 T xx (x) = 5.88 s y = 4 T yy (y) = 0.34 4 T xy = x i y i = 3719 s xy = 1 T xy T x = 1 4 T y 3719 841 4 806 4 = 37.85 r = s xy 37.85 = s x s y 5.88 0.34 = 0.71

66 5 x y y 33.58 = 37.85 (x 35.04) = 0.56(x 35.04) 670.4 y = 0.56x + 13.96 1. (a) P r (X 90) = P r ( X 80 6 ) 90 80 6 = P r (Z 1.67) = P r ( < Z < 0) + P r (0 Z 1.67) = 0.95 (b) P r ( X 80 1) = P r ( X 80 6 1 ) 6 = P r ( Z ) = P r (0 Z ) = (0.477) = 0.95 () X X N(10, 1 ) ( ) 50(l/ ) P r (X 50) ( ) X 10 50 10 P r (X 50) = P r 1 1 1. = P r (Z 1.90) = 1 P r(0 < Z < 1.96) = 0.5 0.475 = 0.05 X = 1 (110 + 11 + 133 + 14 + 16 + 118 + 11 + 15 + 131 + 10) = 1(cm) 10 U = 1 ( (110 1) + (11 1) + + 10 1) ) 9 = 55.111(cm) S = 9 10 U = 49.5999 S = 7.043 1 BOD X X N(µ, 6.5) 15 X = 7. X N(µ, 6.5 15 )

67 X Z = X µ 6.5 15 = 7.5 µ 6.5 15 P r ( Z z α ) = 0.95 z α z α = 1.96 95% Z = 7.5 µ 1.96 6.5 15 6.5 7.5 1.96 15 µ 7.5 + 6.5 15. σ = 6.5 X i X i N(µ, 6.5) X N(µ, σ /5) X X = 1 13 [8 + 4 + 31 + 7 + ] = 5 5 = 6.4 Z = X µ N(0, 1) σ /5 95% P r ( Z z α ) = 0.95. z 0.05 = 1.96. X z α σ 5 µ X + z α σ 5 6.4 1.96 6.5/5 µ 6.4 + 1.96 6.5/5 4.1 µ 8.59 3 µ = 146 σ X i X i N(146, σ ) X N(146, σ /4) X X = 1 58 [145.3 + 145.1 + 145.4 + 146.] = 4 4 = 145.5 S T = X µ t 1,α/ S /4

68 5 S S = 1 3 [(145.3 145.5) + (145.1 145.5) + (145.4 145.5) + (146. 145.5) ] = 1 (0.04 + 0.16 + 0.01 + 0.49) = 0.3 3 95% P r ( T t 1,α/ ) = 0.95. t 3,0.005/ = 3.18. X t 3,0.05/ S 4 µ X + t 3,0.05/ S 145.5 3.18 0.3/4 µ 145.5 + 3.18 0.3/4 144.73 µ 146.6 7 1 ˆp = 180 900 = 0. z α = z 0.05 = 1.96 X N(p, pq ) α P ( X p pq z α ) = 1 α 4 p(1 p) p(1 p) X z α X p p p X + z α p(1 p) p(1 p) ( p z α, p + z α ( ) (0.)(1 0.) (0.)(1 0.) 0. 1.96, 0. + 1.96 900 900 (0.174, 0.6) 1 ˆp = 187 300 = 0.63 z α = z 0.05 = 1.96 X N(p, pq ) α P ( X p pq z α ) = 1 α p(1 p) p(1 p) X z α X p p p X + z α p(1 p) p(1 p) ( p z α, p + z α

69 ( ) (0.63)(1 0.63) (0.63)(1 0.63) 0.63 1.96, 0.63 + 1.96 300 300 (0.568, 0.678) 1 X X N(µ, 0.0 ) 16 X X = 7.09, X N(µ, 0.0 /16) H 0 : µ = 7 H 1 : µ 7 α = 0.05 σ H 0 Z 0 = Z = X µ σ / N(0, 1) 7.09 7.00 0.0 /16 = 4(0.09) = 1.8 0. 5.3: z 0.05 = 1.96 H 0 95% 7.09 1.96 0.0 16 µ 7.09 1.96 0.0 16

70 5 6.99 µ 7.19 X X N(µ, 0.0016) 8 X X = 1 (11.97 + 1.0 + + 1.05) = 1.08 8 0.0016 α = 0.05 µ σ H 0 : σ = 0.0016 H 1 : σ > 0.0016 χ = S σ χ α, 1 H 0 S = 1 8 [(11.97 1.0) + + (1.05 1.08) ] = 0.001 χ 0 = 8(0.001) 0.0016 = 10.5 χ 0 > χ 0.05,7 = 14.07 H 0

71 5.4: 95% χ 1 0.05/,8 1 0.0168 σ χ 0.05/,8 1 1.690 0.0168 σ 16.01 0.0010 σ 0.0099 1 σ 1 = σ A = 10, X = 8, S A = 54.41 B = 1, Ȳ = 76, S B = 59.17 H 0 : µ 1 = µ H 1 : µ 1 µ α = 0.05 T = ( X Ȳ ) (µ 1 µ ) t A + B ˆσ A + ˆσ B

7 5 H 0 t 0.05/,0 =.3 T 0 = ˆσ = AS A + BS B A + B 8 76 6 = = 1.77 6.7/10 + 6.7/1 6.7 + 5.3 T 0 = 1.77 < t 0.05/,0 =.09 H 0 II. A = 10, X = 8, SA = 54.41, S A = 60.44 B = 1, Ȳ = 76, S B = 59.17, S B = 64.53 σ 1 < σ H 0 : σa = σ B H 1 : σa < σ B α = 0.05 H 0 F 1 0.05,10 1,1 1 = F = σ B S A σa S F A 1, B 1 B F 0 = 60.44 64.53 = 0.936 1 = 1 F 0.05,1 1,10 1 3.105 = 0.3 F 0 = 0.936 > F 1 0.05,10 1,1 1 = 0.3 H 0 F α,1, = 1 F 1 α,, 1 1 A : B : C : D = 9 : 3 : 3 : 1 H 0 : (p 1, p, p 3, p 4 = 9 16, 3 16, 3 16, 1 16 ) H 1 : (p 1, p, p 3, p 4 9 16, 3 16, 3 16, 1 16 ) α = 0.05

73 H 0 χ = 4 (X i p i ) p i = 4 X p i χ 0 = 43 9.5 + 7 76.5 + 78 76.5 + 15 5.5 408 = 57.94 + 67.764 + 79.59 + 8.84 408 = 5.411 χ 0.05,4 1 = 7.81 χ 0 = 5.411 < χ 0.05,3 = 7.81 H 0 1 H 0 : P (λ) α = 0.05 P (λ) λ k p k kp k = E(X) = λ k=0 k 0 1 3 4 f k 14 99 46 11 300 kf k 0 99 9 33 8 3 p k 0.473 0.33 0.153 0.036 0.0066 m k 141.9 99 45.9 10.8 1.98 p k f k k kf k λ λ λ 1 k kf k = 3 300 = 0.77 k 0 1 3 4 f k 14 99 46 11 300 m k 141.9 99 45.9 10.8 1.98

74 5 k = 4 m k 5 χ m i 5 k 3 1 H 0 χ 0 = (14 141.9) + 141.9 = 0.0040 χ = 3 (x i m i ) i=0 (99 99) 99 m i + (46 45.9) 45.9 + (13 1, 78) 1.78 χ 0.05,3 1 1 = 3.84 χ 0 = 0.0040 < χ 0.05,1 = 3.84 H 0 λ 1 3 1 1 = 1 H 0 : H 1 : α = 0.05 H 0 χ = 350 (X ij P ij ) P ij χ 0 = 1 350[ 159 (39 80 + 54 10 + 49 104 + 17 46 ) + 1 119 (7 80 + 43 10 + 40 104 + 9 46 ) + 1 7 (14 80 + 3 10 + 15 104 + 0 46 )] 1 = 367.55 (3 1) (4 1) = 6 χ 0.05,6 = 1.59. χ 0 = 367.55 > χ 0.05,6 = 1.59 H 0