2 2.1. : : 2 : ( ): : ( ): : : : ( ) ( ) ( ) : ( pp.53 6 2.3 2.4 ) : 2.2. ( ). i X i (i = 1, 2,..., n) X 1, X 2,..., X n X i (X 1, X 2,..., X n ) ( ) n (x 1, x 2,..., x n ) (X 1, X 2,..., X n ) : X 1, X 2,..., X n 1
3 3.3. I 3.3.2. [ ] N(µ, σ 2 ) σ 2 (X 1,..., X n ) X := 1 n (X 1 + + X n ): µ X N(µ, σ 2 /n) 1.8.4 Z = X µ σ/ n N(, 1) 1.8.2 < α < 1/2 Φ(z) =.5 α z α z z(α) α α z(α) II : z(.5) = 1.6449 z(.25) = 1.96 z(.5) = 2.5758 Z = X µ σ/ n N(, 1) II z(α/2) P ( Z < z(α/2)) = 1 α P ( X µ σ/ n ) < z(α/2) = 1 α 2
X µ X σ n z(α/2) < µ < X + σ n z(α/2) 1 α 1 α ( ) 1 (1 α)% µ µ 1 (1 α)% [ : ( )] σ 2 N(µ, σ 2 ) n (X 1,..., X n ) X µ 1(1 α)% X σ z(α/2) < µ < X + σ z(α/2) n n 1. (p.91) N(µ, (5.54) 2 ) 5 169.8cm µ 95% ( ) n = 5 σ = 5.54 X = 169.8 α =.5 II z(α/2) = z(.25) = 1.96 : 169.8 5.54 5 1.96 < µ < 169.8 + 5.54 5 1.96 168.3 < µ < 171.3 95% : 1 5 95% 1 1 95 µ 5 µ 3.4. II 3.4.1. [ ] (n 5) 3.3.2 ( 1.8.5) σ 2 S 2 ( ) (X 1,..., X n ) σ 2 σ : S 2 := 1 n 1 {(X 1 X) 2 + + (X n X) 2 } : S := S 2 2.2.1. ( ) S 2 = 1 n 1 {(X2 1 + + X 2 n) n X 2 } 3
( ) (X 1 X) 2 + + (X n X) 2 = X 2 1 + + X 2 n 2(X 1 + + X n ) X + n X 2 = X 2 1 + + X 2 n 2n X 2 + n X 2 = X 2 1 + + X 2 n n X 2 2.5.5. ( ) S 2 E(S 2 ) = σ 2 σ 2 n ( ) (X 1,..., X n ) n µ E(X i ) = µ (i = 1, 2,..., n) V (X i ) = σ 2 (i = 1, 2,..., n) E(X 2 i ) = V (X i ) + {E(X i )} 2 = σ 2 + µ 2 (1) X = (X 1 + + X n )/n X 1,..., X n E( X) = 1 n {E(X 1) + + E(X n )} = 1 n (nµ) = µ 2.2.1 (1), (2) (3) E(S 2 ) = 1 n 1 V ( X) = 1 n 2 {V (X 1) + + V (X n )} = 1 n 2 (nσ2 ) = σ2 n E( X 2 ) = V ( X) + {E( X)} 2 = σ2 n + µ2 (2) E(S 2 ) = 1 n 1 {E(X2 1) + + E(X 2 n) ne( X 2 )} (3) { ( σ n(σ 2 + µ 2 2 ) n n + µ2 )} = 1 n 1 (n 1)σ2 = σ 2 2.5.5 n n 1 [ : ( )] ( ) n ( 5) X S 2 µ 1 (1 α)% X S n z(α/2) < µ < X + S n z(α/2) 2. (p.94) 12 348g 23g 4
99% ( ) (n = 12) X = 348 S = 23 α =.1 II z(α/2) = z(.5) = 2.5758 : 348 23 12 2.5758 < µ < 348 + 23 12 2.5758 342.6 < µ < 353.4 3.5 [ : χ 2 ] 2.6.1. χ 2 X 1, X 2,..., X n N(, 1) χ 2 := X 2 1 + X 2 2 + + X 2 n n χ 2 1 p χ 2 n (x) = 2 n/2 Γ(n/2) xn/2 1 e x/2 (x > ) (x ) Γ(s) := e x x s 1 dx (s > ) χ 2 n (p.64 2.6.1 ) < α < 1 P (χ 2 x) = α x χ 2 n(α) χ 2 α α χ 2 ( 4, p.194 ) 5
1. (χ 2 α ) χ 2 a, b (1) χ 2 1 χ 2 P (χ 2 a) =.1 (2) χ 2 15 χ 2 P (χ 2 < 25.) = b ( ) (1) χ 2 a = χ 2 1 (.1) = 15.99 (2) χ2 15 (α) = 25. α χ2 α =.5 b = 1 α = 1.5 =.95 2.6.1. (χ 2 ) E(χ 2 ) = n V (χ 2 ) = 2n n χ 2 χ 2 ( ) E(χ 2 ) = 1 2 n/2 Γ(n/2) x x n/2 1 e x/2 dx x x n/2 1 e x/2 dx = E(χ 2 ) = E((χ 2 ) 2 ) = (2y) n/2 e y (2dy) (y = x/2 ) ( ) n + 2 = 2 n/2+1 y (n+2)/2 1 e y dy = 2 n/2+1 Γ 2 = 2 n/2 2 n ( n ) 2 Γ 2 1 2 n/2 Γ(n/2) 2n/2 nγ(n/2) = n 1 2 n/2 Γ(n/2) x 2 x n/2 1 e x/2 dx 6
x 2 x n/2 1 e x/2 dx = E((χ 2 ) 2 ) = (2y) n/2+1 e y (2dy) (y = x/2 ) = 2 n/2+2 y n/2+1 e y dy = 2 n/2+2 y (n+4)/2 1 e y dy ( ) n + 4 = 2 n/2 4Γ = n + 2 2 2 ( n ) = 2 n/2 n(n + 2)Γ 2 ( n + 2 Γ 2 ) = n + 2 2 1 ( n ) 2 n/2 Γ(n/2) 2n/2 n(n + 2)Γ = n(n + 2) 2 V (χ 2 ) = E((χ 2 ) 2 ) {E(χ 2 )} 2 = n 2 + 2n n 2 = 2n n ( n ) 2 Γ 2 2.6.3. ( n χ 2 ) n n χ 2 (X 1, X 2,..., X n ) N(µ, σ 2 ) χ 2 := 1 n σ 2 (X i µ) 2 i=1 2.6.4. ( ) n χ 2 := (X 1, X 2,..., X n ) N(µ, σ 2 ) X := 1 n S 2 := 1 n 1 n X i, i=1 n (X i X) 2 i=1 (n 1)S2 σ 2 = 1 σ 2 n (X i X) 2 n 1 χ 2 µ 1 3.5.1 [ ] (X 1, X 2,..., X n ) N(µ, σ 2 ) (µ ) n χ 2 = 1 n σ 2 (X i µ) 2 n χ 2 2.6.3 i=1 i=1 χ 2 χ 2 n(α/2) χ 2 n(1 α/2) P ( χ 2 n(1 α/2) < χ 2 < χ 2 n(α/2) ) = 1 α 7
P ( ) χ 2 n(1 α/2) < 1 n σ 2 (X i µ) 2 < χ 2 n(α/2) = 1 α i=1 n i=1 (X i µ) 2 σ 2 n i=1 (X i µ) 2 n χ 2 < σ 2 i=1 < (X i µ) 2 n(α/2) χ 2 n(1 α/2) 1 α [ : ( )] N(µ, σ 2 ) (µ ) n (X 1,..., X n ) σ 2 1 (1 α)% n i=1 (X i µ) 2 χ 2 n(α/2) < σ 2 < n i=1 (X i µ) 2 χ 2 n(1 α/2) 1. (p.97) 1 2.5mg 1 98% 2.46, 2.51, 2.52, 2.48, 2.49, 2.5, 2.54, 2.53, 2.49, 2.52 (mg) ( ) n = 1 α =.2 χ 2 χ 2 1 (α/2) = χ2 1 (.1) = 23.2 χ2 1 (1 α/2) = χ 2 1 (.99) = 2.56 µ = 2.5 1 i=1 (X i µ) 2 =.56 :.56 23.2 < σ 2 <.56 2.56.241 < σ 2 <.219 3.5.2. [ ] µ µ X χ 2 = 1 n σ 2 (X i X) 2 = n 1 σ 2 S 2 n 1 χ 2 2.6.4 i=1 [ : ( )] N(µ, σ 2 ) (µ ) n (X 1,..., X n ) X S 2 σ 2 1 (1 α)% n i=1 (X i X) 2 χ 2 n 1 (α/2) < σ 2 < (n 1)S 2 χ 2 n 1 (α/2) < σ2 < n i=1 (X i X) 2 χ 2 n 1 (1 α/2) (n 1)S 2 χ 2 n 1 (1 α/2) 2. (p.98) 1 2.5mg 98% 8
( ) 1 1 = 9 α =.2 χ 2 χ 2 9 (α/2) = χ2 9 (.1) = 21.7 χ 2 9 (1 α/2) = χ2 9 (.99) = 2.9 X = 2.54 1 i=1 (X i X) 2 =.544 :.544 21.7 < σ 2 <.544 2.9.25 < σ 2 <.26 3.6. [ : ] 2.9.1. C C 2 C C p q N = n C P = N/n p 2.9.1. p n (1) N B(n, p) : P (N = k) = n C k p k (1 p) n k (k =, 1,..., n) (2) n np 5 nq 5 Z = N np npq = P p (pq)/n N(, 1) ( ) ( ) (1) (2) 1.9.2 1. (p.74) A 38% K 78 A 39 ( ) 78 A N N 9
B(78,.38) n = 78 p =.38 np = 78.38 29.6 5 nq = 78.62 48.4 5 N N(np, npq) Z = (N np)/ npq N(, 1) npq = 78.38.62 18.4 P (N 39) B(78,.38) = P (N 39.5) N(29.6,18.4) ( ) N 29.6 38.5 29.6 = P = P (Z 2.7) 18.4 18.4 =.5 Φ(2.7) =.5.488 =.192 3.6.1. [ ] C C C ( ) p n N: i.e. C P = N/n: n np 5 nq 5 Z = n (P p) p (1 p) N(, 1) 2.9.1 II z(α/2) P ( Z < z(α/2)) = 1 α P ( ) n (P p) < z(α/2) = 1 α p (1 p) p 1 (1 α)% P p (1 p) n z(α/2) < p < P + 1 p (1 p) n z(α/2)
p p P [ : ( )] C p n P n np 5 nq 5 p 1 (1 α)% : P P (1 P ) n z(α/2) < p < P + P (1 P ) n z(α/2) 1. (p.11) M 1 37 M p 95% ( ) n = 1 P = 37/1 =.37 np np = 1.37 = 37 5 nq = 1 37 = 63 5 α =.5 z(α/2) = z(.25) = 1.96 :.37 (1.37).37 (1.37).37 1.96 < p <.37 + 1.96 1 1.37.95 < p <.37 +.95.275 < p <.465 11
4 4.3. [ ] N(µ, σ 2 ) (σ 2 ) µ µ [ ] (1) H : µ = µ (2) H 1 3 1 (a) µ > µ (b) µ < µ (c) µ µ (a) (b) (c) (3) : n X H X N(µ, σ 2 /n). Z = X µ σ/ n N(, 1) ( ) (4) : (= ) 1 α % H 1 (a) µ > µ P (Z > z(α)) = α Z > z(α) (b) µ < µ P (Z < z(α)) = α Z < z(α) (c) µ µ P ( Z > z(α/2)) = α Z > z(α/2) ( ) ( ) ( ) Z Z Z > z(α) Z < z(α) Z > z(α/2) 2 : 12
1 α 2 H : µ = µ 2 (5) : X ( ) Z (6) : Z (4) H 1 α% H 1 Z H 1 α% H ( ) (n 5) (i) ( ) (ii) σ 2 S 2 ( ) 1 (p.121) 5mm 1 4.998mm.9mm 5% ( ) (n = 1) σ 2 S 2 [ ( )] µ µ = 5 n = 1 ( ) σ = S =.9 X = 4.998 (1) H : µ = 5 (2) H 1 : µ < 5 ( ) (3) : X H Z = X µ σ/ n N(, 1) (4) : 5% 1α = 5 α =.5 z(.5) = 1.6449 : Z < 1.6449 (5) : σ S Z = 4.998 5.9/ 1 2.22 (6) : Z H 5% ( 5%) 13
4.7 4.7.1 [ ] N(µ, σ 2 ) σ 2 σ 2 [ ] (1) H : σ 2 = σ 2 (2) H 1 3 1 (a) σ 2 > σ 2 (b) σ 2 < σ 2 (c) σ 2 σ 2 (3) : N(µ, σ 2 ) n S 2 H χ 2 = (n 1)S2 σ 2 n 1 χ 2 ( ) (4) : 1α% H 1 (a) σ 2 > σ 2 P (χ2 > χ 2 n 1 (α)) = α χ2 > χ 2 n 1 (α) ( ) (b) σ 2 < σ 2 P (χ2 < χ 2 n 1 (1 α)) = α χ2 < χ 2 n 1 (1 α) ( ) (c) σ 2 σ 2 P (χ2 > χ 2 n 1 (α/2) χ2 < χ 2 n 1 (1 α/2)) = α χ 2 < χ 2 n 1 (1 α/2) χ2 > χ 2 n 1 (α/2) ( ) (5) : S 2 ( ) χ 2 (6) : χ 2 (4) H 1 α% 14
1 (p.134) 2 S 2 = 13.6 5% H : σ 2 = 14.2 H 1 : σ 2 < 14.2 ( ) [ ] σ 2 σ 2 = 14.2 n = 2 S 2 = 13.6 (1) H : σ 2 = 14.2 (2) H 1 : σ 2 < 14.2 ( ) (3) : S 2 H χ 2 = (n 1)S2 σ 2 n 1 χ 2 (4) : 5% 1α = 5 α =.5 χ 2 19 (.95) = 1.12 : χ 2 < 1.12 (5) : χ 2 = (2 1) 13.6 14.2 = 18.2 (6) : χ 2 H 5% 14.2 ( 5%) 4.8 4.8.1 [ ] p p [ ] (1) H : p = p (2) H 1 3 1 (a) p > p (b) p < p (c) p p (3) : n P n np 5 nq 5 H P N(p, p (1 p )/n) Z = P p p (1 p )/n N(, 1) ( ) (4) : 1α% H 1 15
(a) p > p P (Z > z(α)) = α Z > z(α) ( ) (b) p < p P (Z < z(α)) = α Z < z(α) ( ) (c) p p P ( Z > z(α/2)) = α Z > z(α/2) ( ) (5) : P ( ) Z (6) : Z H 1 α% 1 (p.137) 12 1 28 1 5% ( ) ( ) [ ( )] p p = 1/6 n = 12 P = 28/12.2333 (1) H : p = 1/6 (2) H 1 : p > 1/6 ( ) (3) : P np = 12/6 = 2 > 5 nq = 12 2 = 1 > 5 H Z = P p p (1 p )/n N(, 1) (4) : 5% 1α = 5 α =.5 z(.5) = 1.6449 : Z > 1.6449 (5) : Z =.2333 1 6 ( ) 1.9586 1 6 1 1 6 /12 (6) : Z H 5% 1 ( 5%) 16