24 6 I., X, x X. Radom Samplig with Replacemet ( ) 1,.,, 1 X 1, 2 X 2,..., X., X 1, X 2,..., X ( ).,.,,. Estimate of Populatio Parameters ( ),..,,.. 6
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1 23 第 6 章 母数の推定 I 二項母集団の母比率 6.1 Audiece Ratig Survey (視聴率調査) テレビ局では視聴率の獲得にしのぎを削っているようである. 果たして, コンマ以下の数字に 意味はあるのだろうか? 2016 年 4 月 25 日 (月) 5 月 1 日 (日) ドラマ (関東地区) 視聴率ベスト 10 番組名 放送局 連続テレビ小説 とと姉ちゃん 真田丸 日曜劇場 ９９ ９ 刑事専門弁護士 世界一難しい恋 警視庁捜査一課９係 土曜ワイド劇場 再捜査刑事 片岡悠介 横山秀夫サスペンス 刑事の勲章 トットてれび グッドパートナー無敵の弁護士 ラヴソング 連続テレビ小説 とと姉ちゃん 他 ＮＨＫ総合 ＮＨＫ総合 ＴＢＳ 日本テレビ テレビ朝日 テレビ朝日 ＴＢＳ ＮＨＫ総合 テレビ朝日 フジテレビ ＮＨＫ総合 放送日 放送開始時刻 分数 16/04/27(水) 8: /05/01(日) 20: /05/01(日) 21: /04/27(水) 22: /04/27(水) 21: /04/30(土) 21: /04/25(月) 21: /04/30(土) 20: /04/28(木) 21: /04/25(月) 21: /04/29(金) 12:45-15 視聴率 (%) ビデオリサーチ社による番組平均世帯視聴率 日本の放送エリアは全部で 32 ありますが, それぞれの放送エリアごとに視聴率調査が行な われています. ビデオリサーチでは, 関東地区をはじめ全国 27 地区の調査エリアで, PM シ ステムによる調査とオンラインメータシステムによる調査を実施しています. 日本全国を ひとつの調査エリアとした視聴率調査は実施していません また, 調査対象世帯数は, PM システムによる調査の関東地区 関西地区 名古屋地区で 600 世帯, それ以外のオンライン メータシステムによる調査地区は 200 世帯です. (ビデオリサーチ社のウェッブページから 現在) 参考: 藤平芳紀 視聴率の正しい使い方 (朝日新書) 6.2 Samplig (標本抽出) 調査対象の集団 (母集団) に対して, 全数調査が不可能である場合に, その一部分 (標本) を調 査して全体の性質を推定することが重要である. 標本を 1 個取り出せば, 観測値 x が 1 個得られる. 観測値は取り出された標本ごとに違った数 値となるが, 母集団をよくかき混ぜて無作為に標本を選ぶのなら, 観測値 x の現れ方に母集団
2 24 6 I., X, x X. Radom Samplig with Replacemet ( ) 1,.,, 1 X 1, 2 X 2,..., X., X 1, X 2,..., X ( ).,.,,. Estimate of Populatio Parameters ( ),..,, Estimate of Biomial Parameter E 2, E p.. E 1, 0. X 1, X 2,..., X. k, { 1, k E, X k = 0, k E,, P (X k = 1) = p, P (X k = 0) = 1 p., X 1, X 2,..., X., f(x 1, X 2,..., X ) (poit estimatio).,. : ˆp = 1 (i) E[ˆp] = p ( ) (ii) P lim ˆp = p = 1 [ ] X k
3 6.4. Distributio of ˆp 25, ˆp ( ) (!)., ˆp p., ˆp,. (iterval estimatio). 6.4 Distributio of ˆp (1) X k B(, p). (2), B(, p) N(p, p(1 p)). p 5, (1 p) 5. (3), ( ) p(1 p) ˆp N p, ˆp p p(1 p)/ N(0, 1) (4), 2 ( ), ˆp p ˆp(1 ˆp)/ N(0, 1). 6.5 Iterval Estimatio of Biomial Parameter α = α/2 α, Z N(0, 1) ( ) P ( z Z z) = 1 α z N(0, 1) α. z α α N(0,1) 1 α α/2 α/2 -z 0 z
4 26 6 I p 1 α [ ] ˆp(1 ˆp) ˆp(1 ˆp) ˆp(1 ˆp) ˆp z, ˆp + z ˆp ± z. 90% (α = 0.1, z = 1.64) 95% (α = 0.05, z = 1.96) 99% (α = 0.01, z = 2.58). α 1 0 (1 α) 0% 100% 0 ( ) ( ) ( ), x 1..., x (, x k = 0 = 1). ˆp,.,.., 1 α, α.,. 6.1 ( ) %. 95%, 0.141( ) ± ± , 95% 0.01,? [38416] HW , 51% (NHK ). 90%. [0.51 ± 0.024] HW 24, 90% 0.01,? [26896] 8 100, 12.. [ ] 90%, 0.12(1 0.12) 0.12 ± ± ,,.
5 27 7 II 7.1 Poit Estimatio, X 1, X 2,..., X X = 1 X k ( ) ( ) E[ X] = m.,. 7.2 ( ) X, ( ) P X = m = 1. lim 7.3 (Strog law of large umbers ( )) X 1, X 2,..., m., ( P lim 1 ) X k = m = ( ), 1, 0., x 1, x 2,... t = 1 x k. t,.
6 28 7 II Cetral Limit Theorem ( ) 7.5 ( ) X 1, X 2,..., m = 0, σ 2 = 1., ( ) lim P 1 X k x = 1 x e t2 /2 dt. 2π,, 1 X k N(0, 1). 7.6 ( ) m, σ 2 X 1, X 2,..., X, X., : ) X N (m, σ2 X m σ/ N(0, 1), X k m σ 1 X k m σ N(0, 1),,., 1 X k m σ = 1 σ (X k m) = 1 σ ( X m ) = X m σ/, X m σ/ N(0, 1) X N ) (m, σ2.
7 7.3. ( ) ( ) m ( ), σ 2 ( ), X 1, X 2,..., X. m 1 α, X ± z σ z N(0, 1) α (= α/2 ), N(0, 1) α, Z N(0, 1) P ( z Z z) = 1 α z. z α α N(0,1) 1 α α/2 α/2 -z 0 z HW 25, 200, 2.2 g., 1.5 g., g?. [95% 2.2 ± 0.208] 7.4 ( ) m ( ), σ 2 ( ), X 1, X 2,..., X. U 2 = 1 1 (X i X) 2, S 2 = 1 i=1 (X i X) 2,. (,, ) 7.7 U 2 : E(U 2 ) = σ 2. i=1
8 30 7 II,.,, S 2 U N(m, σ 2 ) X 1,..., X, T = X m U/ t 1 ( 1) t-,. t- 1 B ( 2, 1 2) ( ) t2 2 = Γ( +1 2 ) Γ( 2 )Γ( 1 2 ) ( ) t2 2 (1) Γ. Γ(x) = 0 t x 1 e t dt, x > 0. (2) B. B(x, y) = 1 (3) N(0, 1),. 0 t x 1 (1 t) y 1 dt = Γ(x)Γ(y), x > 0, y > 0. Γ(x + y) (4) = t- N(0, 1). (5), 30 N(0, 1). m 1 α, X ± t U t t 1 α
9 7.4. ( ) ,. 90% [ x = , u 2 = = , t 7 = ± 0.375] 10,. 95% [33 ± 4.17] g., 8g. 1. [95% 156 ± 2.48] 12 11, 95% 1g? [984] 13 ( ) m, σ, ( ) = x m σ,., 20 80,.
![95%. 23 42 33 29 34 41 30 36 34 28 [33 ± 4.17] 11 1. 40 156g., 8g. 1. [95% 156 ± 2.](/docs-images/48/23792178/images/page_9.jpg "48] 12 11, 95% 1g? [984] 13 ( ) m, σ, ( ) = 50 + 10 x m σ,., 20 80,.")
10 32 7 II t P ( T t (α)) = α \α α 0 t ( α)
11 33 8 Testig Hypotheses 8.1 Sir Roald Aylmer Fisher ( ) 1. (ull hypothesis) H 0 (alterative hypothesis) H T ( ), H 0,. 3. (sigificace level) 0 < α < 1 (critical regio)., H 0., 10%, 5%, 1%., T, T α (P (T W ) = α). ( H 1. ),. 4. T t, W (t W ). t W. T, H 0. α, H 0 (reject), H 1 (accept). t W. T, α, H 0 (, ) , 223.? 1. p. H 0 : p = 1 2 H 1 : p X. H 0, X B(400, 1/2) N(200, 10 2 )., Z = X 200 N(0, 1) 10.
12 34 8 Testig Hypotheses 3. α = 0.05., 5% ( ). 5% (= 2.5% ) 1.96, W : z x = 223 Z z = = 2.3., H 0., 5% H 0.,. 5. 1%, 1% 2.58, z = % H 0. α α α W W W W N(0, 1) α α z α ( ) m, σ 2, X = 1 ) X k N (m, σ2 X m σ/ N(0, 1),, (. N(m, σ 2 ) ).
13 (Two Types of Error) ( ) 25 mm.,.,, 0.8 mm mm.? [ 5% H 0 : m = 25 ( )] 8.3 ( ). 120,., , [ m. H 0 : m = 120 H 1 : m > 120] HW 26 ( ), HW 27 ( ), m = 60 (g).,, m 50 70, σ = 3 ( )., 25,, m = 60? (Two Types of Error) H 0, 4. \ H 0 H 0 H 0 2 H 0 1 α: 1 (Type I error) = β: 2 (Type II error) 1 = = 2 = = , 215.? 2., α β.
14 36 8 Testig Hypotheses θ θ β α c c, H 0,. H 0, ( 2 β). H ( )1 3, 7 22 ( ) ,. 4. ( ),. 5.,.,.
15 37 9 William Sealy Gosset ( ) 9.1 ( ) m, σ 2, X = 1 ) X k N (m, σ2, ( ). X m σ/ N(0, 1) 9.2 ( ) N(m, σ 2 ) X 1,..., X, U 2 = 1 (X i 1 X) 2,. X, i=1 T = X m U/ t 1 ( 1) t (g) 9 494, 8 2.,? [ α = 0.05, t = 2.25 > H 0., N(0, 1), 2.25 < 1.96 H 0.] 9.2 ( ),. 50kg, 50kg. 12 (kg), x = 48.6, u 2 = [ 5% H 0 : m = 50 ( )]
16 38 9 HW (kg), kg, HW A , A. A. [ 5% ] 9.3 P (P-value), α H 0.,, H 0. t, H 0, P = t, t P.,,. HW 30 A P. HW 31 ( ) 250.,, ? P N(m 1, σ1), 2 N(m 2, σ2) 2 1, 2 X 1, X2, ( ) X 1 X 2 N m 1 m 2, σ2 1 + σ
17 9.5. ( ) ( ). A B A 0.7, B [H 0 : m 1 = m 2, H 1 : m 1 m 2. z = % H 0.] HW 32 A 36, B 40, A x A = 64.5, B x B = A B., N(m 1, σ 2 ), N(m 2, σ 2 ) 1, 2 X 1, X2, U 2 1, U 2 2. U 2 = ( 1 1)U ( 2 1)U , T = X 1 X 2 ( ) U t. 9.6 ( ) 2 A, B. A 6, B 8. A : B : A,B. [ x A = , u 2 A = , x B = , u 2 B = , u 2 = , t = , t % %.] 9.5 ( ) 14, 45, 55.? , 38, 62. [ 5% ]
18 , 23.5 ( ) ? cm cm, 4.63 cm. [ 1% ] 18, 100g 2g., 2g. 200, 2.2g.,, 1.5g.. [ 5% ] 19 8%., 175, 25..
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23(2011) (1 C104) 5 11 (2 C206) 5 12 http://www.math.is.tohoku.ac.jp/~obata,.,,,.. 1. 2. 3. 4. 5. 6. 7.,,. 1., 2007 ( ). 2. P. G. Hoel, 1995. 3... 1... 2.,,. ii 3.,. 4. F. (),.. 5.. 6.. 7.,,. 8.,. 1. (75%
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