t 2 Armitage t t t χ 2 F χ 2 F 2 µ, N(µ, ) f(x µ, ) = ( ) exp (x µ)2 2πσ 2 2 0, N(0, ) (00 α) z(α) t * 2. t (i)x N(µ, ) x µ σ N(0, ) 2 (ii)x,, x N(µ, ) x = x + +x ( N µ, σ2 ) (iii) (i),(ii) x,, x N(µ, ) z = x µ N(0, ) N(0, ) ( 95 97.5 N(0, 2), N(9, 4) N(0, 2) N(9, 4) N(0, ) *2 * 733 de Moivre 00 *2 (i) (iii) N(0, 2) 95 0 + 2z(0.95) N(9, 4) 9 + 4z(0.95)
(iii) x, x N(µ, ) z = x µ () N(0, ) () 0 (y,, y 0 ) 20 5 6 (σ = 6) *3 0 y y 2 y 3 y 4 y 5 y 6 y 7 y 8 y 9 y 0 2 0 05 8 02 34 30 94 0 7 ( ) *4 H 0 : µ = 20 H : µ 20 (iii) y,, y 0 z = ȳ µ 0 (2) N(0, ) ȳ = y + +y 0 0 = 09.5, µ = 20 σ = 6 ( ) z = 09.5 20 6 2 0 = 2.08 N(0, ) 5 z(0.975) =.96, z(0.025) =.96 ( 2.08 =) z < z(0.025) H 0 : µ = 20 3 t 3. t t t Studet( Gosset:876-937) 2 *3 *4 2
2 0 (y,, y 0 ) 20 5 2 0 ( ) y y 2 y 3 y 4 y 5 y 6 y 7 y 8 y 9 y 0 2 0 05 8 02 34 30 94 0 7 ( ) *5 H 0 : µ = 20 H : µ 20 ȳ = 09.5 2. (ii) ( ) ȳ N 20, σ2 0 2. (iii) z = ȳ 20 0 z N(0, ) (3) (3) z z z(0.975) =.96, z(0.025) =.96 z = 0 z z z = ȳ µ 0 = ȳ 20 0 = 0 i= (y i ȳ) 2 245.8 09.5 20 2.2 (4) 245.8 0 z N(0, ) *6 z z *5 t *6 z z(0.975) =.96 z(0.025) =.96.96.96 3
*7 (4) z Gosset z t y,, y 0 z 00 00 z z z t t z t t z t z t t t t ( ) *8 (0 ) = 9 t *9 t N(0, ) ((4) z t t ) t y,, y N(µ, ) t = ȳ µ ( ȳ = i= y i, = ) (y i ȳ) 2 i= ( ) t t( ) t( ) (00 α) t (α) () t t 9 t 5 t < t 9 (0.025) t 9 (0.975) < t *7 *8 χ 2 2 ( ) 2( ) *9 4
t 9 (0.025) = 2.26, t 9 (0.975) = 2.26 (4) z (= t) = 2.2 t 9 (0.025) t t 9 (0.975) H 0 : µ = 20 ( ) 3.2 t 2 N(5, 6) 0 ( ) t t 3.3 2 t t t 2 t t 3.3. 2 y, y 2,, y N(µ, ) y 2, y 22,, y 22 N(µ 2, ) H 0 µ = µ 2 H µ µ 2 3.3.2 µ = y j (= ȳ ), µ 2 = 2 y 2j (= ȳ 2 ) 2 j= µ, µ 2 ( µ N µ, d d = µ µ 2 N j= ) (, µ 2 N µ 2, ) 2 ( ( µ µ 2, + ) ) 2 5
* 0 H 0 : µ = µ 2 ( ( d N 0, + ) ) 2 d z = ( ) N(0, ) (5) + 2 t 3.3.3 = j= (y j ȳ ) 2, 2 = 2 (y 2j ȳ 2 ) 2 (6) 2 2 2 = (y j ȳ ) 2 + (y 2j ȳ 2 ) 2 (7) ( ) + ( 2 ) j= j= j= * *2 (7) (6), 2 { } = ( ) σ ( ) + ( 2 ) 2 + ( 2 ) 2 (8), 2 ( ) ( 2 ), 2 = j= (y j ȳ ) 2, 2 = 2 (y 2j ȳ 2 ) 2 2 t 2 2 3 * 3 j= *0 V [ d] = V [ µ µ 2 ] = V [ µ ] V [ µ 2 ] = + ( = + ) 2 2 * ( ) + ( 2 ) = ( + 2 2) χ 2 µ, µ 2 2 2 *2 H 0 : µ = µ 2 ȳ = + 2 = + 2 2 i (y ij ȳ) 2 i= j= 2 y ij i= j= t *3 Duett Tukey t F 6
3.3.4 (5) d t = ( ) + 2 H 0 : µ = µ 2 ( ) + ( 2 ) = ( + 2 2) t 5 t < t + 2 2(0.025) t + 2 2(0.975) < t 3.3.5 3 3 ( = 6, 2 = 5) 8 9 6 4 8 8 9 5 7 4 µ, µ 2 5 H 0 : µ = µ 2 H : µ µ 2 ( ) µ = ȳ = 6 (8 + 9 + 6 + 4 + + 8) =.0, µ 2 = ȳ 2 = (8 + 9 + 5 + 7 + 4) = 0.6 5 d = µ µ 2 =.0 0.6 = 0.4 = [{ (8 ) 2 + (9 ) 2 + (6 ) 2 + (4 ) 2 + ( ) 2 + (8 ) 2} 5 + 4 + { (8 0.6) 2 + (9 0.6) 2 + (5 0.6) 2 + (7 0.6) 2 + (4 0.6) 2}] = 25.5 t d t = ( ) = + 2 ( 6 + 5 0.4 ) 0.3 25.5 ( ) ( ) 7
H 0 : µ = µ 2 t ( ) + ( 2 ) = (6 ) + (5 ) = 9 t t t 9 (0.025) = 2.26, t 9 (0.975) = 2.26 t 9 (0.025) t (= 0.3) t 9 (0.975) H 0 : µ = µ 2 3.3.6 4 4 5 4 ( = 5, 2 = 4) * 4 5 5 3 4 0 2 µ, µ 2 H 0 : µ = µ 2 H : µ µ 2 5 ( ) t µ = 5, µ 2 = 0, σ = 3, = 2 d d = µ µ 2 = 5 0 = 5 σ = 3, = 2 = 5 2 = 4 (8) = t { } ( ) σ ( ) + ( 2 ) 2 + ( 2 ) 2 = 7 (4 32 + 3 2 2 ) 6.9 d t = ( ) = + 2 5 ( 5 + 4 ) 6.9 2.84 7 t t 7 (0.025) = 2.36, t 7 (0.975) = 2.36 H 0 : µ = µ 2 t 7 (0.975) < t (= 2.84) *4 y,, y σ x = x i i= σ = (x i x) 2 i= 8
3.4 2 t 2 t 3.4. t 2 t t µ µ 2 p,, p * 5 ( ) y N(µ + p, ) y 2 N(µ 2 + p, ) (2 ) y 2 N(µ + p 2, ) y 22 N(µ 2 + p 2, ) (3 ) y 3 N(µ + p 3, ) y 32 N(µ 2 + p 3, )... ( ) y N(µ + p, ) y 2 N(µ 2 + p, ) µ, µ 2 p + p 2 + p 3 + + p = 0 * 6 H 0 µ = µ 2 H µ µ 2 3.4.2 p,, p d i = y i2 y i (i =,, ) d i * 7 E[d i ] = E[y i2 y i ] = (µ 2 + p i ) (µ + p i ) = µ 2 µ V [d i ] = V [y i2 y i ] = V [y i2 y i ] = V [y i2 ] + V [y i ] = + = 2 *5 *6 ȳ = y i i= [ ] E[ȳ ] = E y i = E[y i ] = (µ + p i ) = µ + (p + p 2 + + p ) = µ i= i= i= µ *7 9
d i N(µ 2 µ, 2 ) d i H 0 : µ = µ 2 d i N(0, 2 ) = 2 d,, d N(0, ) d,, d t * 8 d = i= = t = d i (d i d) 2 (9) i= d t 5 t < t (0.025) t (0.975) < t 3.4.3 5 5 (mg/dl) 2 3 4 5 6 30 35 52 33 40 50 20 30 40 38 42 40 t * 9 ( ) t 2 3 4 5 6 30 35 52 33 40 50 20 30 40 38 42 40 ( ) ( ) 0 5 2 5 2 0 *8 H 0 : µ 2 µ = 0 d,, d 0 *9 t 0
0 t d = (0 + 5 + 2 + ( 5) + ( 2) + 0) = 5.0 6 = 5 {(0 5)2 + (5 5) 2 + (2 5) 2 + ( 5 5) 2 + ( 2 5) 2 + (0 5) 2 } 49.6 t = 5.0 0 49.6 6 =.739 6 t 5 t 2.5 t 5 (0.025) = 2.57, t 5 (0.975) = 2.57 t 5 (0.025) < (.739 =) t < t 5 (0.975) H 0 : µ = µ 2 3.4.4 SAS t SAS data d; iput before after @@; cards; 30 20 35 30 52 40 33 38 40 42 50 40 ; ru; proc ttest data=d; paired after*before; ru;