1 21 10 5 1 E-mail: qliu@res.otaru-uc.ac.jp
1 1 ( ) ( 1.1 1.1.1 60% mm 100 100 60 60% 1.1.2 A B A B A 1
B 1.1.3 boy W ID 1 2 3 DI DII DIII OL OL 1.1.4 2
1.1.5 1.1.6 1.1.7 1.1.8 1.2 1.2.1 1. 2. 3
1.2.2 1. 2. 50 1.2.3 1. 2. 2 Excel 3 Excel 3.1 sum,mean,max,min,round,if... Σ [ ] 4
SUM =sum( =sum(a1:a5) AVERAGE sum MAX MIN 1 ROUND =round( ) =round(123.246,2) =round(b1,3) 1 5
IF =if( =if(a1>2,1,0) =if(a1=1,, ) 3.2 > > > > > > > 2 http://www.geocities.jp/qfliuwind/ DATA01 2 = ( 2 /10000) 22 x ( 5 <= x <= 5 abs(x) <= 5) (x < 5 x > 5) 3 Word 2 3 6
3.3 DATA02 > > ( D)=Sheet1!A1:E7 =a1:e7) > > > 7
( D)=Sheet1!A1:E7 =a1:e7) 8
専 攻 35 30 25 人 数 20 15 クラス1 クラス2 クラス3 クラス4 10 5 0 中 国 語 韓 国 語 英 語 フランス 語 ドイツ 語 イタリア 語 言 語 3 4 DATA01 4 50 DATA01 9
階 級 度 数 累 積 度 数 相 対 度 数 累 積 相 対 度 数 143 152 9 9 18% 18% 152 161 10 19 20% 38% 161 170 14 33 28% 66% 170 179 12 45 24% 90% 179 188 2 47 4% 94% 188 198 3 50 6% 100% 5 / 6 / 25 ヒストグラム 20 度 数 15 10 5 0 143 155 167 179 191 203 階 級 の 下 限 7 DATA01 10 10
5 8 ( ) 9 () 10 () 07 5.1 1. x = 1 n n x i = x 1 + x 2 + + x n n n (x i x) = 0. 2. 5 x 3 3. {x 1, x 2,, x t } t t 11
3 4 x t = x t 1 + x t + x t 3 x t = x t 1 + x t + x t+1 + x t+2 4 { x 2, x 3,, x n 1 } { x 3, x 4,, x n 2 } TOPIX 4 日 付 終 値 3 項 移 動 平 均 5 項 移 動 平 均 7 項 移 動 平 均 18/04/2006 1741.75 17/04/2006 1719.05 1734.96 14/04/2006 1744.07 1735.63 1738.31 13/04/2006 1743.77 1743.58 1743.99 1748.44 12/04/2006 1742.89 1752.28 1755.65 1754.43 11/04/2006 1770.18 1763.47 1763.58 1762.52 10/04/2006 1777.34 1777.08 1769.96 1762.80 07/04/2006 1783.72 1778.91 1770.59 1763.64 06/04/2006 1775.67 1768.48 1766.49 1765.32 05/04/2006 1746.05 1757.12 1761.95 1759.32 04/04/2006 1749.65 1750.11 1750.83 1752.08 03/04/2006 1754.64 1744.15 1741.04 1741.77 31/03/2006 1728.16 1736.49 1734.13 30/03/2006 1726.68 1722.13 29/03/2006 1711.54 TOPIX 4 Yahoo Japan Finance TOPIX 12
1,800.00 1,750.00 1,700.00 終 値 3 項 移 動 平 均 半 月 移 動 平 均 一 ヶ 月 移 動 平 均 1,650.00 1,600.00 1,550.00 06/2/7 06/2/22 06/3/9 06/3/24 06/4/8 TOPIX 4. x = n x 1 x 2 x n 1 x n 3 r 1 = 23%, r 2 = 27%, r 3 = 28% 10 R = (1 + r 1 )(1 + r 2 )(1 + r 3 ) 1 1 r = 3 (1 + r 1 )(1 + r 2 )(1 + r 3 ) 1 26% 13
3 2 200% 10 r r = 3 1 + R 1 = 3 2 1 26% 25% 3 1 + r 3 (1 + r) 3 = (1 + 26%) (1 + 26%) (1 + 26%) 2. 2 5.2 1. http://www.geocities.jp/qfliuwind/ DATA01 10 14
6 1. 2. 3. 4. Excel 5. 7 7.1 median 11 15
12 7 X = {4, 7, 2, 30, 9, 7, 1} X = {1, 2, 4, 7, 7, 9, 30} 13 100, 90, 10, 110, 110 10 84 100 7.2 mode 14 15 X = {2, 3, 2, 4, 6, 4, 6, 6, 7} 6 3 6 7.3 range 16 R = max(x) min(x) 17 X = {2, 3, 2, 4, 6, 4, 6, 6, 7} X 2 7 = 7 2 = 5 16
7.4 mean deviation 18 X = {x 1, x 2,, x n } d = n x i x n = x 1 x + x 2 x + + x n x n x 19 X = {x 1, x 2,, x 10 } = {2, 3, 2, 4, 6, 4, 6, 6, 3, 4} x = 2 + 3 + 2 + 4 + 6 + 4 + 6 + 6 + 7 10 = 4 2 4 + 3 4 + 2 4 + + 7 4 d = 10 = 2 + 1 + 2 + 0 + 2 + 0 + 2 + 2 + 3 = 7 10 5 x 7/5 7.5 Excel max( -min( 20 max(a1:a10)-min(a1:a10) MEDIAN( ) 21 =MEDIAN(a1:a10) MODE( ) 22 =MODE(a1:a10) 17
8 8.1 Variance) 23 S 2 = = n (x i x) 2 n n x2 i n x 2 = (x 1 x) 2 + (x 2 x) 2 + + (x n x) 2 n x P n x2 i n x 2 s 2 = n (x i x) 2 n 1 n 8.2 Standard Deviation 24 S = S 2 = n (x i x) 2 n S = s 2 = n (x i x) 2 n 1 18
8.3 5 Yahoo! 2006 3 1 4 28 0.002, 0.0001), (0.003, 0.0001) (-0.003, 0.0002) 株 価 の 収 益 率 の 例 0.0400 0.0300 0.0200 0.0100 収 益 率 0.0000 0.0100 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 収 益 率 トヨダ 自 動 車 収 益 率 日 産 自 動 車 収 益 率 新 日 本 製 鐵 0.0200 0.0300 0.0400 0.0500 T 8.4 C.V. = S x 5 19
8.5 25 z i = x i S x S x x T i = 50 + 10zi 8.6 Excel S 2 =VARP( ) s 2 =VAR( ) S =STDEVP( ) s: =STDEV( ) 26 Yahoo 3 3 20
8.7 27 X 1 5 3 Y 13 1 20 Z 15 50 120 S 2 = = n (x i x) 2 n n x2 i n x 2 = (x 1 x) 2 + (x 2 x) 2 + + (x n x) 2 n x = 1 + 5 + 3 = 3 3 n Sx 2 = (x i x) 2 n = ( 2)2 + 2 2 + 0 2 3 S 2 x = n x2 i n S 2 y = 132 + 1 2 + 20 2 3 S 2 z = 152 + 50 2 + 120 2 3 = (1 3)2 + (5 3) 2 + (3 3) 2 3 = 4 + 4 3 = 2.67 x 2 = 12 + 5 2 + 3 2 3 3 2 = 2.67 ( ) 2 13 + 1 + 20 = 61.56 3 ( ) 2 15 + 50 + 120 = 1905.6 3 S 2 z > S 2 y > S 2 x. Z X 1, 5, 3 Z 15, 50, 120 21
25 20 15 10 5 0 5 10 15 20 25 0 50 100 150 200 250 300 X Y Z 28 X = {x 1, x 2,, x 100 }, Y = {y 1, y 2,, y 100 }, Z{y 1, y 2,, y 100 } 0 1, 3, 10 8.8 DATA01 9 22
9.1 DATA01 体 重 図 1 身 長 と 体 重 の 散 布 図 100 90 80 70 60 50 40 30 20 100 150 200 250 身 長 2 1980 2000 GDP( GDP 9.2 155cm 155 175cm 175cm 50kg 1 13 10 24 50kg 6 11 4 21 7 24 14 45 23
図 2 一 人 当 たりGDPと 乳 児 死 亡 率 8.5 7.5 乳 児 死 亡 率 6.5 5.5 4.5 3.5 2.5 2.5 3 3.5 4 4.5 一 人 当 たりGDP 1: GDP SNA 9.3 X Y S xy X Y ρ xy X Y 9.3.1 29 S xy = 1 n = 1 n n (x i x) (y i ȳ) n x i y i xȳ 30 24
3 0 4 0 2 0 3 0 2 0 1 0 1 0 0 0 1 0 1 0 2 0 2 0 3 0 3 0 4 0 2 5 2 0 1 5 1 0 5 0 5 1 0 1 5 2 0 2 5 4 0 2 5 2 0 1 5 1 0 5 0 5 1 0 1 5 2 0 2 5 3 9.3.2 31 ρ xy = S xy S x S y = 1 n n 1 n n (x i x) (y i ȳ) n (x i x) 2 1 n (y i ȳ) 2 n (x i x) (y i ȳ) (x i x) 2 n (y i ȳ) 2 n 1 ρ 1 4 25
3 2 1 0 1 2 3 4 2 0 2 4 3 2 1 0 1 2 3 ρ xy =0 4 4 2 0 2 4 ρ xy = 0.7 4 2 0 2 4 4 2 0 2 4 3 2 1 0 1 ρ xy =0.7 2 2 1 0 1 2 3 ρ xy =1 32 33 ID a b c d e f g h i j (g) 10 15 20 25 30 35 40 45 50 55 (kg) 6 15 19 18 7 19 24 22 34 35 26
(kg) 4 0 3 5 3 0 2 5 2 0 1 5 1 0 5 0 1 0 2 0 3 0 4 0 5 0 6 0 (g) 5 ρ xy = 0.84 34 x = {4, 3, 5, 1, 5}, y = {1, 3, 3, 0, 1} x y 9.3.3 Excel 1. =COVAR( 1 2 ) 2. =CORREL( 1 2 ) 9.3.4 DATA01 10 27
10.1 x, y y = a + bx x y a = 3, b = 2 y = 3 + 2x x = 0 y = 3 + 2 0 = 3, x = 2 y = 3 + 2 2 = 7..., 4 x 0 2 5 7 y 3 7 13 17 4 20 y 15 10 5 0 10 5 0 5 10 5 x 10 15 20 10.2 28
学 籍 番 号 身 長 (cm) 体 重 (kg) 161111 172 59 161112 166 58 161113 164 65 161114 175 73 161115 149 41 161116 144 48 161117 161 63 161118 159 56 161119 172 63 161120 167 57 161121 166 52 161122 187 80 90 80 70 体 重 60 50 40 30 130 140 150 160 170 180 190 身 長 29
Ìd W e i g h t 9 0 8 5 8 0 7 5 7 0 6 5 6 0 5 5 5 0 4 5 4 0 1 4 0 1 5 0 1 6 0 1 7 0 1 8 0 1 9 0 2 0 0 H e i g h t 9 0 8 5 8 0 7 5 7 0 6 5 6 0 5 5 5 0 4 5 4 0 1 4 0 1 5 0 1 6 0 1 7 0 1 8 0 1 9 0 2 0 0 g x y x i y i, i = 1, 2, 3,..., 12. ŷ = a + bx d i = y i ŷ = y i (a + bx i ) 30
S 12 S = (y i (a + bx i )) 2 S = n (y i (a + bx i )) 2. (1) 35 ( ) S b = n x iy i n xȳ n x2 i n x2 (2) a = ȳ b x (3) 11 a b S a b 1 ( n n ) ( n ) S = yi 2 + na 2 + b 2 + x i 2ab x 2 i ( n ) ( n ) y i 2a x i y i 2b (4) x i y i 3 a a ( ( n ) ( n ) S = na 2 y i 2a + x i 2ab ( nȳ 2 2nb xȳ + nb 2 x 2)) + ( nȳ 2 2nb xȳ + nb 2 x 2) + ( n n yi 2 + x 2 i ) ( n ) b 2 x i y i 2b S = n(a (ȳ b x)) 2 + 31
S a = ȳ b x (5) 3 b b S = n x 2 i ( b S ( n x iy i a n x )) 2 i n + x2 i n x2 i b = ( n x iy i ) a n x i (6) 5 a 6 b = ( n x iy i ) n xȳ ( n x2 i ) n x2 (7) a = ȳ b x (8) 1 { S = 0 a S = 0 b 11.1 32
i x i y i x 2 i x i y i 1 172 59 29584 10148 2 166 58 27556 9628 3 164 65 26896 10660 4 175 73 30625 12775 5 149 41 22201 6109 6 144 48 20736 6912 7 161 63 25921 10143 8 159 56 25281 8904 9 172 63 29584 10836 10 167 57 27889 9519 11 166 52 27556 8632 12 187 80 34969 14960 12 (x iy i ) 12 x i 12 y i 12 x2 i 1982 715 328798 119226 x ȳ 165.17 59.58 b = ( n x iy i ) n xȳ ( 119226 12 165.17 59.58 n x2 i ) = = 0.80 n x2 328798 12 165.17 2 a = ȳ b x = 60 0.80 165.17 = 72.14 ŷ = 72.14 + 0.80x 1cm 0.8kg 36 x = {2, 5, 6, 9}, y = {4, 6, 8, 9} 33
12 12.1 x, y y = a + bx x y a = 3, b = 2 y = 3 + 2x x = 0 y = 3 + 2 0 = 3, x = 2 y = 3 + 2 2 = 7..., 4 x 0 2 5 7 y 3 7 13 17 4 20 y 15 10 5 0 10 5 0 5 10 5 x 10 15 20 34
12.2 学 籍 番 号 身 長 (cm) 体 重 (kg) 161111 172 59 161112 166 58 161113 164 65 161114 175 73 161115 149 41 161116 144 48 161117 161 63 161118 159 56 161119 172 63 161120 167 57 161121 166 52 161122 187 80 90 80 70 体 重 60 50 40 30 130 140 150 160 170 180 190 身 長 35
Ìd W e i g h t 9 0 8 5 8 0 7 5 7 0 6 5 6 0 5 5 5 0 4 5 4 0 1 4 0 1 5 0 1 6 0 1 7 0 1 8 0 1 9 0 2 0 0 H e i g h t 9 0 8 5 8 0 7 5 7 0 6 5 6 0 5 5 5 0 4 5 4 0 1 4 0 1 5 0 1 6 0 1 7 0 1 8 0 1 9 0 2 0 0 g x y x i y i, i = 1, 2, 3,..., 12. ŷ = a + bx d i = y i ŷ = y i (a + bx i ) 36
S 12 S = (y i (a + bx i )) 2 S = n (y i (a + bx i )) 2. (9) 37 ( ) S b = n x iy i n xȳ n x2 i n x2 (10) a = ȳ b x (11) 13 a b S a b 1 ( n n ) ( n ) S = yi 2 + na 2 + b 2 + x i 2ab x 2 i ( n ) ( n ) y i 2a x i y i 2b (12) x i y i 3 a a ( ( n ) ( n ) S = na 2 y i 2a + x i 2ab ( nȳ 2 2nb xȳ + nb 2 x 2)) + ( nȳ 2 2nb xȳ + nb 2 x 2) + ( n n yi 2 + x 2 i ) ( n ) b 2 x i y i 2b S = n(a (ȳ b x)) 2 + 37
S a = ȳ b x (13) 3 b b S = n x 2 i ( b S ( n x iy i a n x )) 2 i n + x2 i n x2 i b = ( n x iy i ) a n x i (14) 5 a 6 b = ( n x iy i ) n xȳ ( n x2 i ) n x2 (15) a = ȳ b x (16) 1 { S = 0 a S = 0 b 13.1 38
i x i y i x 2 i x i y i 1 172 59 29584 10148 2 166 58 27556 9628 3 164 65 26896 10660 4 175 73 30625 12775 5 149 41 22201 6109 6 144 48 20736 6912 7 161 63 25921 10143 8 159 56 25281 8904 9 172 63 29584 10836 10 167 57 27889 9519 11 166 52 27556 8632 12 187 80 34969 14960 12 (x iy i ) 12 x i 12 y i 12 x2 i 1982 715 328798 119226 x ȳ 165.17 59.58 b = ( n x iy i ) n xȳ ( 119226 12 165.17 59.58 n x2 i ) = = 0.80 n x2 328798 12 165.17 2 a = ȳ b x = 60 0.80 165.17 = 72.14 ŷ = 72.14 + 0.80x 1cm 0.8kg 38 x = {2, 5, 6, 9}, y = {4, 6, 8, 9} 39
14 15 39 x = {4, 6, 9}, y = {6, 6, 9} x y 15.1 3 DATA03 Excel 2007 5 2007 7 37 2007 Excel 12 6 16 Excel OK Y X OK 40
概 要 回 帰 統 計 重 相 関 R 0.641785 重 決 定 R2 0.411887 補 正 R2 0.390883 標 準 誤 差 20.70401 観 測 数 30 分 散 分 析 表 自 由 度 変 動 分 散 観 測 された 分 散 比 有 意 F 回 帰 1 8405.914 8405.914 19.60993 0.000132 残 差 28 12002.37 428.656 合 計 29 20408.28 係 数 標 準 誤 差 t P 値 下 限 95% 上 限 95% 下 限 95.0% 上 限 95.0% 切 片 94.39179 171.995 0.548805 0.587489 257.924 446.7081 257.924 446.7081 X 値 1 20.47776 4.624284 4.42831 0.000132 11.00534 29.95019 11.00534 29.95019 y x 7 y = a + b x + u u a 94.4 b X 1 20.5 y = 94.4 + 20.5x 17, 40 A 1 A = {... } 41 A = {2, 1, 5}, B = {3, 6, 5} A B A B A B = {2, 1, 5, 3, 6} A B A B A B = {5} 41
18 18.1 42 100 1000 1/2 43 1 1/6 44 100 18.2 90 n n m n m/n p p p 42
18.3 1. A P (A) 0 P (A) 1 2. Ω P (Ω) = 1 3. A B A B P (A B) = P (A) + P (B) A B 18.4 45 2 2 46 1 0.5 2 { { 4 1 1/4 47 A = { 2 }, B = { } P {A B}? 48 {A B} = { 2 } P {A B} = 1 49 C = { 2 } D = { 2 } 50 P (C D) C = { } D = { } C D C D = φ φ P {φ} = 0 2 2 51 P {B} P {A B} = P {A} + P {B} 52 3 2?{ 1 2 }?{ 1 2 }? 43
18.5 10 50 18.6 Excel Excel 1. 2. 1 10 P = 0.5 OK 3. 1 0 50 1000 19 1 0 X X = 0 0.5, X = 1 0.5 x = 1 1 6, x = 2 1 6,... 19.1 X 0 1 0, 1, 2, 3, 4, 5, 6 0 1 X x 1 x 2 x 3 x 4 P (x) P (x 1 ) P (x 2 ) P (x 3 ) P (x 4 ) x i P (x) P (x 1 ), P (x 2 )... x 44
F (x) {X x} F (x) = P ({X x}) F (x) = P ({X x}) = x i <x P (x i ) x x i F (x 3 ) = P (X x 3 ) = P (x 1 ) + P (x 2 ) + P (x 3 ) x i X E (X) E (X) = n x i P (x i ). X V (X) V (X) = E [ (X E (X)) 2] = n [ (xi E (X)) 2 P (x i ) ]. 53 () X P (x) X P (x) 19.1.1 1. { P (x) = p P (x) = 1 p x = 1 x = 0 E (X) = 1 p + 0 (1 p) = p. V (X) = (0 p) 2 (1 p) + (1 p) 2 p = p 2 p 3 + p 2p 2 + p 3 = p p 2 45
F(x) 1 0.75 0.5 0.25 0 0 0.25 0.5 0.75 1 p = 0.7 0 F (x) = 1 p 1 x < 0 0 x < 1 x 1 46
F(x) 1 0.75 0.5 0.25 0 4 3 2 1 0 1 2 3 4 5 p = 0.5 p = 0.5 2. P (x) = C x np x (1 p) n x x = 0, 1, 2,..., n. Y i 0 p X = n ( n ) E (X) = E Y i = np V (X) = V ( n ) Y i = n(p p 2 ) C x n Cn x = P n x n!/ (n x)! = Px x x! n (n 1) (n 2) (n x + 1) = x (x 1) (x 2) 3 2 Y i 47
F (x) = x P (x i ) n x p = 0.5 3. P (x) = e λ λ x p = λ/n. p = λ/n x! C x np x (1 p) n x = (n 1) (n 2) n n = n!/ (n x)! x! n n (n 1) n (n 2) n ( m ) x ( 1 m ) n x n n (n (x 1)) n x! ( m x 1 m ) n ( 1 m ) x n n ) x 1 ( 1 m n ) n e m n (n (x 1)) 1, ( 1 m n n n P (x) = e λ λ x x! E (X) = λ. V (X) = λ. F (x) = x P (x i ) 19.1.2 n p 48
10 5 10 5 2 λ = 10 P (X 2) = F (2) = = e λ λ 0 0! 2 e λ i λ x i i=0 + e λ λ 1 1! x i! + e λ λ 2 2! = e 10 10 0 0! = 0.0028. + e 10 10 1 1! + e 10 10 2 2! 0.0028 p = 0.3, n = 10 P (6) 20 20.1 Probability Density Funtion PDF 20.1.1 20 x 1000 1000 x x 1000 P (c) 6 = C c np c (1 p) n c x = 0, 1, 2,..., n. (17) 6 P (c), f (c), F (c) P (x), f (x), F (x) 49
20.1.2 f (c) = 1 σ (c µ) 2 2π e 2σ 2 (18) 20.1.3 f (x) X c c 1 10 2 0 0 50
21 (Cumulative Distribution Function CDF) 21.0.4 F (c) = P (X c) = c P (x i ) = c C x i n p x i (1 p) n x i x = 0, 1, 2,..., n. (19) 51
21.0.5 F (c) = P (X c) = c 1 σ (x µ) 2 2π e 2σ 2 dx (20) 21.0.6 k k k F (x) c c c 21.0.7 P (c) f (c) F (c) a b a < b X a b a b X 52
X c d c d X F (c) = P (X c) = c P (x i ) (21) F (c) = P (X c) = c f (x) dx (22) 0.4 0.35 0.3 0.25 0.2 The area of this part 0.15 0.1 0.05 0 5 4 3 2 1 0 1 2 3 4 5 F ( 1) 53
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 This value 0.1 0 5 4 3 2 1 0 1 2 3 4 5 F ( 1) 21.0.8 E (X) = n x i P (x i ). X V (X) V (X) = E [ (X E (X)) 2] = n [ (xi E (X)) 2 P (x i ) ]. E (X) = V (X) = E [ (X E (X)) 2] = xf (x) dx. (x E (X)) 2 f (x) dx. 54
22 1. P (x) = C x np x (1 p) n x x = 0, 1, 2,..., n. (23) x F (x) = P (x i ) (24) ( n ) E (X) = E Y i = np (25) ( n ) V (X) = V Y i = n(p p 2 ). (26) 2. f (c) = 1 σ (c µ) 2 2π e 2σ 2 (27) F (c) = 1 c σ e (x µ)2 2σ 2 dx (28) 2π 55
µ σ 2 E (X) = V (X) = = 1 σ 2π = µ = 1 σ 2π = σ 2 xf (x) dx c xe (x µ)2 2σ 2 dx (x µ) 2 f (x) dx c (x µ) 2 e (x µ)2 2σ 2 dx 22.1 54 10 Excel Excel 10 p = 0.5 100 1 x i, i = 1, 2,..., 100 x i 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 55 Excel 1000 4.2 4.2 0.4 56 p = 0.3, n = 6 P (3) 23 57 C 2 5 58 10 5 7 Excel 56
0.4 0.35 0.3 0.25 0.2 0.15 3 0.1 0.05 2 1 0 5 4 3 2 1 0 1 2 3 4 5 2: 1 59 Excel p = 0.5, n = 10 x = 0, 1, 2,..., 10 60 1 X 0 2 (1) 2 1 (2) 1 0 (3) 0.0228 0.1359 0.3413 P (X 1) P (X 0) f(0), f(1) 61 X 2 F ( 1) F (0) P (X 1) P (X 0) 57
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 5 4 3 2 1 0 1 2 3 4 5 3: 58
62 10 1 5 23.1 Excel Cn x nc x = combin (n, x) C6 3 = combin (6, 3) 24 1. P (x) = Cnp x x (1 p) n x x = 0, 1, 2,..., n. (29) x F (x) = P (x i ) (30) ( n ) E (X) = E Y i = np (31) ( n ) V (X) = V Y i = n(p p 2 ). (32) 2. µ σ 2 f (c) = 1 σ (c µ) 2 2π e 2σ 2 (33) F (c) = 1 c σ e (x µ)2 2σ 2 dx (34) 2π 59
3. P (x) = e λ λ x x! (35) p = λ/n E (X) = λ. (36) V (X) = λ. (37) 25 p n x P (x) = C x np x (1 p) n x x = 0, 1, 2,..., n. (38) P (x) = e λ λ x 1 5 10 λ x x! (39) 26 1 X 0 2 (1) 2 1 (2) 1 0 (3) 0.0228 0.1359 0.3413 P (X 1) 60
P (X 0) f(0), f(1) 0.4 0.35 0.3 0.25 0.2 0.15 3 0.1 0.05 2 1 0 5 4 3 2 1 0 1 2 3 4 5 1 27 X P (X c) P (X c) = F (c) = 1 σ 2π c e (x µ)2 2σ 2 dx (40) c P (X c) µ = 0 σ 2 = 1 0 1 61
27.1 E (X) = µ V (X) = σ 2 X X Z = X µ σ E (Z) = 0 V (Z) = 1 63 X µ = 0 σ 2 = 1 X 1.64 F (1.64) P (X 1.64) P (0 < X 1.64) 1.6 0.04 1.6 0.04 X 1.64 P (X 1.64) = 0.95 P (0 < X 1.64) = P (X 1.64) P (X 0) P (X 0) P (X 1.64) P (0 < X 1.64) = P (X 1.64) P (X 0) = 0.95 0.5 = 0.45 64 X µ = 1 σ 2 = 4 X 4.28 µ = 1 σ 2 = 4 X µ = 0 σ 2 = 1 Z Z = X µ σ 2 = X 1 2 Z x = 4.38 x X z = x 1 = 4.28 1 = 1.64 2 2 P (X 4.28) = P (Z 1.64) = 0.95 28 62
28.1 X {X 1, X 2, X 3,..., X n } X E (X) µ V ar (X) = σ 2 65 (at random ) µ n X = 1 n X i σ 2 S 2 = 1 n 1 n (X i X) 2 X: 100 {X 1, X 2, X 3,..., X 100 } E (X) µ µ X = 1 100 100 X i σ 2 S 2 = 1 100 1 100 (X i X) 2 66 ( ) Γ γ E (γ) = Γ γ Γ 63
X S 2 µ σ 2 X E ( ( ) 1 n X) = E X i n = 1 n E (X i ) n X i X E (X i ) = µ = 1 n µ = µ. n 29 67 ( ) X i µ σ 2 lim n X = µ. 68 ( ) Γ γ lim n γ = Γ γ Γ X µ 30 1000 2 4 1 2 3... 1000 1000 2 64
31 31.1 H 0 H 1 100 1000 H 0 H 1 : 31.2 69 ( ) X i µ σ 2 n n X n( X µ) d N(0, 1). σ 31.3 σ σ σ 31.4 65
70 X 100 X = 175 σ = 10 µ 170 H 0 µ = 170 H 1 µ > 170 X µ = 170 σ = 10 N (170, 100) X 175 α 1% 5% µ = 170 n X Z 0 Z 0 1. H 0 µ = 170 H 1 µ > 170 2. α α = 5% 3. α Z P (Z > Z ) = α Z P (Z > Z ) = 5% Z P (Z Z ) = 1 5% Z Z = 1.65 4. Z 0 = n ( X µ ) /σ Z 0 = 100 (175 170) /10 = 5 66
5. Z 0 Z Z 0 > Z Z 0 Z Z 0 = 5, Z = 1.65 Z 0 > Z µ = 170 µ > 170 Z 0 Z µ = 170 0.4 0.35 0.3 0.25 0.2 0.15 0.1 5% 0.05 0 5 4 3 2 1 0 1 2 1.65 3 4 5 31.5 10 σ 2 = 9 25 X = 12 H 0 µ = 10 H 1 µ > 10 67
31.6 7 5 DATA01 µ 160 cm σ = 6 32 32.1 σ 100 X = 3.2 cm σ 2 = 4 µ = 3 µ σ 2 X µ = 3 1. H 0 µ = 3 cm H 1 µ > 3 cm µ < 3 cm µ 3cm 2. α 1/2 α α = 5% 1/2 α = 2.5% 3. 1/2 α Z P (Z > Z ) = 1/2 α Z P (Z > Z ) = 2.5% Z P (Z Z ) = 1 2.5% z Z = 1.96 4. Z 0 = n ( X µ ) /σ Z 0 = 100 (3.2 3) /2 = 1 68
5. Z 0 Z Z 0 > Z Z 0 < Z Z Z 0 Z Z 0 = 1, Z = 1.96 Z 0 < Z µ = 3 Z 0 Z Z µ = 3 0.4 0.35 0.3 0.25 0.2 0.15 0.1 2.5% 2.5% 0.05 0 5 4 3 2 1 0 1 2 3 4 5 1.96 1.96 69
33 7 22 34 0 35 36 36.1 9.1 9 8.4, 9.3, 8, 7, 8.9, 9.8, 9.3, 9.2, 9, 8.9 α = 1% σ 2 = 4 37 σ n 37.1 σ σ n( X µ) σ d N(0, 1). σ σ σ s n( X µ) s n 1( X µ) s d N(0, 1) d t (n 1). 70
t (n 1) n 1 t n 1( X µ) t t n 1 t s 37.2 t t t t 3 t 37.3 σ σ t t σ 71 X 25 71
µ 170 1. H 0 µ = 170 H 1 µ > 170 2. α α = 5% 3. υ = n 1 υ = 25 1 = 24 4. α n 1 t t t P (t > t ) = α t 24 t P (t > t ) = 5% t t = 1.71 5. X s s 2 = n (x i x) 2 n 1 X = 175, s 2 = 100 6. t 0 = n 1 ( X µ ) /s t 0 = 24 (175 170) /10 2.5 7. 6 t 0 4 t t 0 > t t 0 t t 0 = 2.5, t = 1.71 t 0 > t µ = 170 µ > 170 170 cm 72
37.4 1. H 0 µ = µ 0 H 1 µ µ 0 2. α 1/2 α 3. υ = n 1 υ = 25 1 = 24 4. 1/2 α n 1 t t t P (t > t ) = α t 5. X s s 2 = n (x i x) 2 n 1 6. t 0 = n 1 ( X µ ) /s 7. t 0 t t 0 > t t 0 < t t 0 t t 0 t 72 DATA01 9 µ > 160 38 38.1 σ 73
100 X = 3.8 cm σ 2 = 4 µ = 3 µ σ 2 X µ = 3 1. H 0 µ = 3 cm H 1 µ > 3 cm µ < 3 cm µ 3cm 2. α 1/2 α α = 5% 1/2 α = 2.5% 3. 1/2 α Z P (Z > Z ) = 1/2 α Z P (Z > Z ) = 2.5% Z P (Z Z ) = 1 2.5% z Z = 1.96 4. Z 0 = n ( X µ ) /σ Z 0 = 100 (3.2 3) /2 = 1 5. Z 0 z Z 0 > Z Z 0 < Z Z 0 Z Z 0 Z Z 0 = 1, Z = 1.96 Z 0 < Z µ = 3 Z 0 Z Z 74
µ = 3 0.4 0.35 0.3 0.25 0.2 0.15 0.1 2.5% 2.5% 0.05 0 5 4 3 2 1 0 1 2 3 4 5 1.96 1.96 39 σ 4 10 cm, 12 cm, 15 cm, 9 cm 13 cm 40 73 X = {3, 6, 9} Y = {2, 3, 8} 74 75
75 10 76 100 23 100 21 4 1% 77 7 10 10 7 31 7 1 7 29 41 41 [1] ˆb Y i = a + b X i (41) / n t ˆb (x i x) 2 t n 2 t Excel t 41.1 DATA03 Excel 2007 5 2007 7 37 2007 Excel 76
概 要 回 帰 統 計 重 相 関 R 0.641785 重 決 定 R2 0.411887 補 正 R2 0.390883 標 準 誤 差 20.70401 観 測 数 30 分 散 分 析 表 自 由 度 変 動 分 散 観 測 された 分 散 比 有 意 F 回 帰 1 8405.914 8405.914 19.60993 0.000132 残 差 28 12002.37 428.656 合 計 29 20408.28 係 数 標 準 誤 差 t P 値 下 限 95% 上 限 95% 下 限 95.0% 上 限 95.0% 切 片 94.39179 171.995 0.548805 0.587489 257.924 446.7081 257.924 446.7081 X 値 1 20.47776 4.624284 4.42831 0.000132 11.00534 29.95019 11.00534 29.95019 y x 7 y = a + b x + u u a 94.4 b X 1 20.5 y = 94.4 + 20.5x x 37 852.9 t t p t p p [1] 2002) 77