正規分布と簡単な統計理論／ t 分布と信頼区間 / Normal distribution, simple statistical theory, t-distribution and confidence intervals

早稲田大学大学院経営管理研究科「企業データ分析」2023 冬の第5-6回で使用したスライドです。

Transcript

1. generated by Stable Diffusion XL v1.0 2023 5-6 t (WBS) 2023 5-6 t — 2023-12-14 – p.1/28
2. https://speakerdeck.com/ks91/collections/corporate-data-analysis-2023-winter 2023 5-6 t — 2023-12-14 – p.2/28
3. ( ) 1 11 30 • 2 11 30 (B A ) • 3 12 7 • 4 12 7 • 5 12 14 • 6 12 14 t • 7 12 21 2 ( ) t 8 12 21 2 ( ) t 9 1 11 P 10 1 11 11 1 18 12 1 18 13 1 25 14 1 25 W-IOI 2023 5-6 t — 2023-12-14 – p.3/28
4. ( 20 ) 1 • 2 R • 3 • 4 • 5 • 6 ( ) • 7 (1) • 8 (2) • 9 R ( ) (1) 10 R ( ) (2) 11 R ( ) (1) 12 R ( ) (2) 13 GPT-4 14 GPT-4 15 ( ) LaTeX Overleaf 8 (12/21 ) / (2 ) OK / 2023 5-6 t — 2023-12-14 – p.4/28
5. 3 1 ( ) 2 ( ) 1 2 4 σ2 σ s2 s df 2023 5-6 t — 2023-12-14 – p.5/28
6. 5 1 : 2 : 3 : 4 : 2 6 t µ 95% Student t σ 95% 95% 95% 2023 5-6 t — 2023-12-14 – p.6/28
7. 2023 5-6 t — 2023-12-14 – p.7/28
8. 2. 1 2 (1) 1 2 (2) 2023 12 10 ( ) 23:59 JST ( ) Waseda Moodle (Q & A ) (1) Discord 2023 5-6 t — 2023-12-14 – p.8/28
9. . . . . . . 10 9 (12/13( ) ) ( ) 1 2 2023 5-6 t — 2023-12-14 – p.9/28
10. W iPhone Android iPhone Android Android 1 Android iPhone 2 iPhone Android ⇒ ( 120g ) iPhone Android 1 2 2023 5-6 t — 2023-12-14 – p.10/28
11. S UFO 30% 10 7 UFO 70% 10 4 ⇒ ( ) “ .R” ( ^^;) 30% 70% 2 1 1% 2 4.7% 2023 5-6 t — 2023-12-14 – p.11/28
12. S 1 2 1 20 1 8 → 8 → 1 2 2 1 2 20 1 5 → 1 → 2 ⇒ “ .R” 1% 2 30% ( 2 48% ) 2023 5-6 t — 2023-12-14 – p.12/28
13. S 1 α 2 β α α β β α β ⇒ 2023 5-6 t — 2023-12-14 – p.13/28
14. S 2 n bidistg(n, p0 , p) n = 100, p0 = 0.6, p = 0.8 n = 20 2 p0 p 0.6 vs 0.8 n 100 100 ⇒ n n = 100 100 0.6 vs. 0.8 3 3 0.5 vs. 0.55 1 n = 100 p ( ) 6 . . . n ( p.136–137) t n = 100 ( p.140) ← 2023 5-6 t — 2023-12-14 – p.14/28
15. ⇒ . . . 2023 5-6 t — 2023-12-14 – p.15/28
16. 5 1 : 2 : 3 : 4 : 2 2023 5-6 t — 2023-12-14 – p.16/28
17. (normal distribution) N(µ, σ2) ( µ σ2 ) (probability density) x 1 x N(µ, σ2) x µ − σ µ + σ 68.3% µ − 2σ µ + 2σ 95.4% µ − 3σ µ + 3σ 99.7% N(0, 12) (standard normal distribution) ( z) 2.5% z0.05 1.96 N(µ, σ2) x z z = x − µ σ z N(0, 12) 2023 5-6 t — 2023-12-14 – p.17/28
18. 1 : x ( ) x N(µ, σ2) x x N(µ, σ2 n ) (n ) ( ) µ 1 n 1 √n n N(µ, σ2 n ) n → ∞ x µ ( (law of large numbers)) . . . σ √n (standard error) 2023 5-6 t — 2023-12-14 – p.18/28
19. 2 : (central limit theorem) x 1 √n x N(µ, σ2 n ) (n ) n ( ) µ 1 n 1 √n 1 2023 5-6 t — 2023-12-14 – p.19/28
20. 3 : A B N(µA , σ2 A ) N(µB , σ2 B ) ( ) xA xB (xA + xB ) (xA − xB ) ( (reproductive property)) (xA + xB ) ( ) (µa + µb ) (xA − xB ) ( ) (µa − µb ) (xA + xB ) (xA − xB ) (σ2 A + σ2 B ) (xA + xB ) (xA − xB ) σ2 A + σ2 B 2023 5-6 t — 2023-12-14 – p.20/28
21. 4 : (xA − xB ) 1 3 A B N(µA , σ2 A ) N(µB , σ2 B ) xA xB xA xB (xA − xB ) (xA − xB ) ( ) (µA − µB ) (xA − xB ) xA xB σ2 A nA + σ2 B nB σ2 A nA + σ2 B nB 2023 5-6 t — 2023-12-14 – p.21/28
22. J ( p.128) “ J.R” z pnorm (( ) ) 5 2023 5-6 t — 2023-12-14 – p.22/28
23. 6 t µ 95% Student t σ 95% 95% 95% 2023 5-6 t — 2023-12-14 – p.23/28
24. µ 95% (95% confidence interval) µ 95% 95% ( µ ) 20 19 µ σ . . . ( ) x 95% −z0.05 +z0.05 µ ( 5 ) σ s σ s z N(0, 12) ( n ) → z t → t (t distribution) t n df ( : t(df)) t0.05 (df) ( ) 95% [x − t0.05 (df) × s √ n , x + t0.05 (df) × s √ n ] 2023 5-6 t — 2023-12-14 – p.24/28
25. L ( p.147) µ “ L.R” p.146 t source 95% 2023 5-6 t — 2023-12-14 – p.25/28
26. 2023 5-6 t — 2023-12-14 – p.26/28
27. 3. µ 95% (1) (t ) µ 95% (2) 2023 12 17 ( ) 23:59 JST ( ) Waseda Moodle (Q & A ) (1) Discord 2023 5-6 t — 2023-12-14 – p.27/28
28. 2023 5-6 t — 2023-12-14 – p.28/28