Armitage 1 1.1 2 t *1 α β 1.2 µ x µ 2 2 2 α β 2.1 1 α β α ( ) β *1 t t 1
α β *2 α α β β α = α 1 β = 1 β 2.2 α 0 β 1 0 0 1 1 5 2.5 *3 2.3 *4 3 3.1 1 1 1 *2 *3 *4 (µ A ) (µ P ) (µ A > µ P ) 10 (µ A = µ P + 10) 15 (µ A = µ P + 15) 2
, ( A) 5 130 ( B) 1,000,000 130 ( A) 130 130 ( B) 130 130 3.2 H 0 : µ = 0 H 1 : µ > 0 1 1 *5 3.3 2.5 *6 2.5 (i) (ii) (iii) (ii) 2.5 (i) 3 N(120, 20 2 ) *7 A B A B *8 A µ A, B µ B *9 *5 10 0.1 ) H 0 : µ = 0 10,000 0.1 H 0 : µ = 0 10 10,000 *6 *7 *8 *9 3
A H 0 : µ A = 120 H 1 : µ A > 120 A B H 0 : µ B = 120 H 1 : µ B > 120 B (a) A 200 x 1,, x 200 (b) B 600 y 1,, y 600 x 1,, x 200 N(120, 20 2 ) y 1,, y 600 N(120, 20 2 ) (a) A 200 x 120, 202 200 = 2 N(120, 2) (b) B 600 ȳ 120, 202 600 = 1 3 N(120, 1 3 ) 1.2 1 0.8 0.6 0.4 0.2 1.2 1 0.8 0.6 0.4 0.2 115 120 125 130 1 A 200 N(120, 2) 115 120 125 130 2 B 600 N(120, 1 ) 3 200 600 4
2.5 0.2 0.15 0.1 0.05 1.2 1 0.8 0.6 0.4 0.2 123.8 3 A 200 N(120, 2) 2.5 120.7 4 B 600 N(120, 1 ) 2.5 3 2 2.5 120 200 123.8 600 120.7 * 10 200 122 = 120 122 600 122 = 120 4 2.5 4.1 2.3 3, 4 2.5 3, 4 0.025 *10 5
4.2 H 0 : µ = 120 H 1 : µ > 120 120 20 2 N(120, 20 2 ) N(120, 2) N(120, 1 3 ) 120 H 1 : µ = 122 ( ) H 1 H 1 N(122, 202 ) N(122, 20 2 ) N(120, 20 2 ) N(122, 20 2 ) A 200 N(122, 2) B 600 N(120, 1 3 ) 2.5 3 4 123.8 120.7 * 11 0.4 0.3 0.2 0.1 1.2 1 0.8 0.6 0.4 0.2 123.8 5 A 200 N(122, 2) H 1 H 0 120.7 6 B 600 N(122, 1 ) 3 H 1 H 0 H 1 H 0 H 1 H 1 H 0 200 5 3 600 6 (α) *11 6
4.3 5 6 600 120 2 1 1,000,000 120 0.1 1 0.1 120 80 0.8 H 1 : µ = 120 + 5 5.1 (i) α (ii) 1 β 2 4.3 (iii) (iv) σ 2 5.2 1 A 400 20 A 10 ( = 10) 2.5 80 : 1 1 X 1,, X Y 1,, Y 400 20 7
N(µ y, 400), N(µ x, 400) H 0 : µ y = µ x H 1 : µ y < µ x = 10 H 1 : µ x = µ y + 10 H 0 : µ x µ y = 0 H 1 : µ x µ y > 0 H 1 : µ x µ y = 10 Y = 1 Y i, X = 1 X i i=1 i=1 Y N ( ) ( ) µ y, 400, X N µx, 400 d = X Y * 12 ( d N µ x µ y, ) d H 0 : µ x = µ y H 1 : µ x = µ y + 10 H 0 µ x µ y = 0 H 1 µ x µ y = 10 d N d N ( 0, ) ( 10, ) 7 *12 X N(µ x, σ 2 x), Y N(µ y, σ 2 y) X Y N(µ x µ y, σ 2 x + σ 2 y) 8
0.4 0.3 0.2 0.1 z 7 2.5 80 H 0 H 1 7 z 2 (a) H 0 H 0 7 z 0.025 d N ( ) 0, z N ( ) 0, 2.5 97.5 z 0 = z 0.975 (1) * 13 z 0.975 = 2.5 1.96 z z = 1.96 (2) (b) H 1 H 1 7 0.8 z 80 H 1 d N ( ) ( ) 10, z N 10, 20 (a) (z 0.20 = 0.84) z 10 = z 0.20 z 10 = 0.84 z = 10 0.84 (3) *13 N(µ, σ 2 ) (100 α) z z N(0, 1) (100 α) z α z µ σ 2 = zα z = µ + zα σ 2 9
(2) (3) z z 1.96 = 10 0.84 1.96 + 0.84 = 10 2.8 = 10 2 2.8 = 10 = 2.8 10 = 62.72 1 63 * 14 5.3 ( ) 1 A σ 2 ) A α β : 1 X 1,, X Y 1, Y X 1,, X N(µ A, σ 2 ) Y 1, Y N(µ P, σ 2 ) H 0 : µ P = µ A H 1 : µ P < µ A A H 1 : µ A = µ P + H 0 : µ A µ P = 0 H 1 : µ A µ P > 0 H 1 : µ A µ P = *14 0 10
Y = 1 Y i, X = 1 i=1 ( d N ( X i Y N i=1 µ A µ P, 2σ2 µ P, σ2 ) (, X N µ A, σ2 d = X Y ) H 0 : µ A µ P = 0 d N ) (0, 2σ2 H 1 : µ A µ P = ) d N (, 2σ2 ) (a) H 0 ( z N (b) H 1 H 1 0, 2σ2 ) (100 α) 100 (1 α) z 0 = z 1 α 2σ 2 2σ 2 z = z 1 α 2σ 2 z = z 1 α 2σ2 z N(, ) (100 β) z 2σ 2 = z β z = z β 2σ 2 z = + z β 2σ 2 (4) (5) (4) (5) z z 1 α 2σ 2 = + z β 2σ 2 z 1 α 2σ 2 = + z β 2σ 2 = z 1 α 2σ 2 z β 2σ 2 = z 1 α 2σ 2 z β 2σ 2 = 2σ2 (z 1 α z β ) 11
2 = 2σ2 (z 1 α z β ) 2 z 1 α = z α = 2σ2 ( z α z β ) 2 2 (6) = 2σ2 (z α + z β ) 2 2 (7) 5.4 α α 2 (7) α α 2 * 15 = 2σ2 (z α 2 + z β) 2 2 *15 12
6 (α) (1 β) ( ) (σ 2 ) (i) (ii) d (iii) d (iv) α, 1 β (v) z (vi) z (z 1 α, z β ) (vii) (vi) 2 z = z = ( ) ( ) = 2σ2 (z α + z β ) 2 2 ( ) = 2σ2 (z α 2 + z β) 2 2 13
7 SAS 7.1 SAS proc power 1 : = 10 : 10, 15, 20 3 : 5 : 80 1 t * 16 proc power; twosamplemeas test = diff meadiff = 10 stddev = 10, 15, 20 alpha = 0.05 power = 0.8 total =. ; ru; twosamplemeas total (;) Two-sample t Test for Mea Differece Computed N Total Idex Std Dev Actual Power N Total 1 10 0.807 34 2 15 0.808 74 3 20 0.801 128 (Std Dev) 10, 15, 20 3 Actual Power 1 80 80 N Total 1 *16 14
7.2 1 20 1 63 2 63 2 = 126 SAS 128 2 1 2.5 5 1 t 15