[ ] 00 30 ( x s, x + s) 5% x s = 3, x + s = 7 s = 3 5 + 0 = 70 5 = 0 a = 3, b = 5 75.5 73.0 80.0 75.5 73.0 75.5 0 = 5 75.5 80.0 8 75.5 70.5 80.5 75.5 = 6. 70. 30% 5 A 50 + 5 0 8 0 = 56.5, 50 + 56 50 6 0 = 60 6 x = 70.7, s = 5. 7 :.5 8.5 5.0 7 0.058 7 0.058 8.5 5.5.0 0.09 8 0.50 5.5.5 9.0 3 0.08 3 0.58.5 9.5 6.0 0.75 5 0.33 9.5 36.5 33.0 0 0.67 7 0.600 36.5 3.5 0.0 7 0. 89 0.7 3.5 50.5 7.0 3 0.08 0 0.850 50.5 57.5 5.0 0 0.083 0.933 57.5 6.5 6.0 6 0.050 8 0.983 6.5 7.5 68.0 0.07 0.000 0.000
[ ] =. =3.5 3 =.3 =. =0.30 : (f i ) u i u i f i u i f i.5 8.5 5.0 7 8 8.5 5.5.0 3 33 99 5.5.5 9.0 3 6 5.5 9.5 6.0 9.5 36.5 33.0 0 0 0 0 36.5 3.5 0.0 7 7 7 3.5 50.5 7.0 3 6 5 50.5 57.5 5.0 0 3 30 90 57.5 6.5 6.0 6 96 6.5 7.5 68.0 5 0 50 0 589 ū = = 0.0 x = 33.0 0.0 7 = 3.93 0 { } 6.0 s u = 589 ( ) =. 0 0 s x = 7. = 5.5 0.7 3.5 < < ( x s, x + s) = (7.39, 8.7) 78 65.0% ( x s, x + s) = (.85, 6.0) 8 98.3% ( x 3s, x + 3s) = ( 3.69, 79.55) 0 00%
[ ] 3 ( x s, x + s).. 3 ( x s, x + s) 3 3: ().5 8.5 5.0 5 8.5 5.5.0 9 5.5.5 9.0 6 7.5 9.5 6.0 0 9.5 36.5 33.0 9 36.5 3.5 0.0 9 8 3.5 50.5 7.0 8 5 50.5 57.5 5.0 7 3 57.5 6.5 6.0 5 6.5 7.5 68.0 0 60 60 0 8 6 ( ) 0 0 8 6 ( ) 0 5 9 6 33 0 7 5 6 68 5 9 6 33 0 7 5 6 68 ( ) 0 = =3 3 == =0.3 x = 3.95 s = 5.56 =0.7 ( x s, x + s) = (7.39, 8.5) 78 ( x s, x + s) = (.83, 6.07) 8 ( x 3s, x + 3s) = ( 3.73, 79.63) 0 37.7 5.5 8.3.3
[ ] 8 :.5 5.5 0.0 9 0.090 9 0.090 5.5 36.5 3.0 9 0.90 38 0.380 36.5 7.5.0 3 0.30 6 0.60 7.5 58.5 53.0 9 0.90 80 0.800 58.5 69.5 6.0 0 0.00 90 0.900 69.5 80.5 75.0 0.00 9 0.90 80.5 9.5 86.0 3 0.030 97 0.970 9.5 0.5 97.0 0.00 98 0.980 0.5 3.5 08.0 0 0.000 98 0.980 3.5.5 9.0 0.00 99 0.990.5 35.5 30.0 0.00 00.000 00.000 3 3 =3.8 =.5 3 =55.9 =. =0.8 5 ū = 36 = 0.36 x =.0+0.36 = 5.96 00 ( ) 3.0 s u = 36 36 =.83 00 00 s x =.83 = 0.3 0.38.5 < < 5 ū = = 0. x =.0 + 0. =.3 98 ( ) 3.0 s u = 33 98 98 =.53 s x =.53 = 6.830.380
[ ] 5 5: (f i ) u i u i f i u i f i.5 5.5 0.0 9 8 36 5.5 36.5 3.0 9 9 9 36.5 7.5.0 3 0 0 0 7.5 58.5 53.0 9 9 9 58.5 69.5 6.0 0 0 0 69.5 80.5 75.0 3 36 80.5 9.5 86.0 3 8 9.5 0.5 97.0 5 5 5 0.5 3.5 08.0 0 6 0 0 3.5.5 9.0 7 7 9.5 35.5 30.0 8 8 6 00 36 36 ( x s, x + s) = (7.8, 6.) 66 67.3 ( x s, x + s) = (0.65, 77.97) 9 95.9 ( x 3s, x + 3s) = ( 6.8, 9.80) 98 00 3.. 00 98 00 98 3 3 3.8 3.6 0.5.5.0 3 55.5 55 55.9 55.0 3.5 3. 3. 0.87 0.8 0.8 0.79 3.0 3.0 x 5.79.5 5.96.3 s 0. 6.69 0.3 6.83 0.0 0.378 0.38 0.380 3.5% 6.5%
[ ] 6 00 98 ( x s, x + s) = (5.65, 65.93) : 79 (7.6, 60.8) : 66 ( x s, x + s) = (5.5, 86.07) : 96 (0.77, 77.53) : 9 ( x 3s, x + 3s) = (.63, 06.) : 98 ( 5.9, 9.) : 98 3 0.06 3 B A B A 3 5 5 00 0 50 30 6 a: 9% b: 8% c: 78% d: % e: 83% f: 7% ψ = 3. Q = 0.5
分 ) [ ] 7 7 i 0 5(.8) 0(0.) 35 30 0(0.) 8(7.8) 8 35 8 83 iii ψ =.05, Q = 0.0 0 30 35(35.6) 8(7.) 83 0 3(.) 6(6.6) 9 8 6 ψ = 0.897, Q = 0.05 ii 0 35(35.8) 5(.) 50 30 8(7.) 8(8.8) 66 83 33 6 iv ψ = 0.875, Q = 0.067 0 30 83(76.) 33(39.6) 6 0 9(35.6) 5(8.) 5 58 70 ψ =.7, Q = 0.369 i ii iii iv 0 30 0 0 8 x = y = 7 (, ), (, ), (3, 5) 9 5 r = 0. 待ち時間 ( 70 60 50 0 30 0 0 0 0 0 0 0 0 30 0 予約時刻 - 到着時刻 ( 分 ) 5 0.5. I ū = ( x a)c, v = (ȳ b)d s uv = n n (u i ū)(v i v) i=
[ ] 8 = = n cd n n {(x i a)c ( x a)c}{(y i b)d (ȳ b)d} i= n (x i x)(y i ȳ) = cds xy i= u v r uv x y r xy.5. I s u = cs x, s v = ds y r uv = s uv s u s v = cds xy cds x s y = r xy Ω = { AAAA, AAAD, AADA, AADD, ADAA, ADAD, ADDA, ADDD, DAAA, DAAD, DADA, DADD, DDAA, DDAD, DDDA, DDDD } { AAAA, AAAD, AADA, AADD }. { DDAA, DADA, DAAD, ADDA, ADAD, AADD } Ω = { ABCD, ABDC, ACBD, ACDB, ADBC, ADCB, BACD, BADC, BCAD, BCDA, BDAC, BDCA, CABD, CADB, CBAD, CBDA, CDAB, CDBA, DABC, DACB, DBAC, DBCA, DCAB, DCBA }. A B C D i ii iii 5 5 0 0 0 0 3 iv 0 0 0 3 8 9 8 5 5 8 5 5 7 5 6 5 8 5 7 5 6 5 3 5 0.7 5 = 0.05 + 0.0 + 0.5 + 0.5 + 0.05 = 0.5=(0.5 + 0.5 + 0.05) + (0.5 + 0.0 + 0.05) 0.5 = 0.5 0.5 = 0.85 6 0.6 0.75 7 3 + 3 0 + 3 = P (A G) = P (G) P (A G) 8 0.0099 0.097 3 9 3!!! 5! = 3,560,!!!! =,30 6 = 3
[ ] 9 0 3 9 C C = 55 9 C 3 3 C C = 8 55 5 x 0 60 80 00 p(x) 0.0 0.7 0.3 0.9 8 3 x 3 p(x) 3 = 6 3 x 0 p(x) 3 0 5 3 5 0 E(x) = 0 3, E(x) = 5, V (x) = 3 5 x 3 3 p(x) 3 = 3 3 = 6 µ =, σ = V (x) = 5 9 ( ) 3 x p(x) =, 7 90 x =,, 3,...,000 8 + 0 +,000 + 8,000 8 = 3,50 00,500 0,750 00 6 35 8 37 56 8 8 5 6. 3 B(, 0.9) 0.659 0m 5m 8 8 x ( B 8, ) x = 35 8
[ ] 0 5 7 8 8 7 8 6 P o() p(x) = x x! e p(0) = e, 0.35 p() = e, 0.7 p() = e, (e + e + e ) = 5e 0.33 7 P o() p(x) = x x! e p(0) + p() = 3e, 0.06 5e 0.33 8 p(x) = 3x x! e 3 p(0) = e 3, 0.050 p() p(0) = 3e 3 0.57 e 3 ( ) x ( ) 00 x ( 00 9 B(00, 00 ) p(x) = 99 00C x 00 00, p(0) = 99 00) = 0.3660, p() = ( 99 00 ) 99 = 0.3637, p() = 00 99 ( 00 ) ( 99 00) 98 = 0.89 P o() : p(x) = x! e, p(0) = e = 0.3679, p() = e = 0.3679, p() = e = 0.839 0 p(x) = x x! e 0 p(0) + 7,000 p() +,000 p() +,000 [ {p(0) + p() + p()}] 8,000 =,7,000,003 =.5, Q(.5) = 0.0668 6.7 % 998,003,00,003 =.5, Q(.5) = 0.006 (0.006 + 0.3085) = 0.6853 69% = 0.5, Q(0.5) = 0.3085.985 5.000 0.0.985 5.005 0.0 =.5, =.0, 5.05 5.000 0.0 5.05 5.005 0.0 µ u(0.) = 0.706 5.05 µ 0.0 3 6 65 5 0.70 7% u(0.) =.8, = 0.8, Q(0.8) = 0.9, a 65 5 =.5, Q(.5) = 0.866 87% =.0, {Q(.0) + Q(.0)} = 0.885 8% 63.5 65 5 = 0.706 µ = 5.0079 = 0.3, Q(0.3) = 0.38, 0.38 0.9 = =.8 a = 7. 7. cm
[ ] B(50, 0.8) N(0, ( 8) ).5 0 8 = 0.5303, Q(0.53) = 0.98, 0.98 = 0.709 70%.5 0 8 = 0.8839, Q(0.88) = 0.89, 9% 5 00,000 00 00 B(00, 0.) N(0, ).5 0 =.875, Q(.88) = 0.030 3% 6 B (,600, 899.5 86 8.8 u(0.0) =.36, 7 ) N(86, 8.8 ) 5 =.33, Q(.3) = 0.093 % a 86 8.8 =.36 a = 93 P {x = 0, y = 0} = 0., P {x = 0} P {y = 0} = 0.5 0.3 = 0.5 E(x) = 0.5, V (x) = 0.5, E(y) =, V (y) = 0.6. x + y 0 3 0. 0. 0. 0. E(x + y) =.5, V (x + y) = 0.65 E(x + y) = E(x) + E(y) V (x + y) = V (x) + V (y) P {x = y} = 0.5, P {y > x} = 0.30 E(x) = 0.88, V (x) = 0.6056, E(y) = 0.95, V (y) = 0.6075 E(xy) = 0.90 Cov(x, y) = 0.90 0.88 0.95 = 0.06, ρ xy = 0.06 0.6056 0.6075 = 0.055. 3 N(360, (0 ) ) 365 365 360 0 = 0.3536, Q(0.35) = 0.6368 6%
[ ] 30 = 0 5 = 0 7 6.5 0 0 = 0.65, Q(0.65) = 0.578 0.6 5 N(60, ) 00 N(60, ( 5) ) 56 56 60 5 =.789, Q(.79) = 0.0367 = 0.9633 96.3% σ u(0.0) =.36 + σ ( ) σ = + σ.0 g.36 =.36 8 5 a 5 a 3 {( 5, 5), ( 5, 5), ( 5, 5), ( 5, 5), ( 5, 5), ( 5, 5), (5, 5), (5, 5), (5, 5)}. x \x 5 5 5 5 0.09 0.8 0.03 5 0.8 0.36 0.06 5 0.03 0.06 0.0 x x 0 0 0 0 0 0.03 0. 0.6 0. 0.03 E(x x ) = 0 V (x x ) = 7 V (x ) = V (x ) = 36 3 N(., (0.0 ) ).06. 0.0 =., Q(.) = 0.0793 8% 30 00 = 3,000 00 = 0 N(3,000, 0 ) 3,00 3,000 0 =.0, Q(.0) = 0.587. 0.6 5.5 36 = 6 36 8.5 = 97 N(6,97) 80 6 3 33 =.0, Q(.0) = 0.9 0.5
[ ] 3 6 00 50 5 0.5 0.05,500,50 5 0.5 0.0 0,000 5,000 50 0.5 0.005,000,000 500,000 500 0.5 0.0005 9 µ µ (0.,.8) 95% 95% 00 95 µ,050 ± t(5, 0.05) 00 = (997,,03) n =.96 00 0 = 38.6 385 ( ( ) 3 6 x N µ, ) P { 0.3 < x µ < 0.3} = {0.5 Q(.)} = 0.7698 0.77.65.0 n 0. n 70.6 n 7. ±.96 3.0 = (9.50,.90) s = 3.0, t(, 0.05) =.0. ±.0 3.0 = (9.9, 3.) σ 3.0 s 3.0 5 iii,000,000 B(, 000, p) p 6 n =.96 pq 0.0.96 0.0 =,0 ˆpˆq 95%.96 500 3.9 50 = 0.039
[ ] 7 n \ p 0. 0. 0.3 0. 0.5 5,000 0.8..3..,500..6.8.9.0,000.9.5.8 3.0 3. 500.6 3.5.0.3. 00 5.9 7.8 9.0 9.6 9.8 e \ p 0. 0. 0.3 0. 0.5 0.05 39 6 33 369 385 0.0 865,537,07,305,0 0.0 3,58 6,7 8,068 9,0 9,60 % (,050,050,500 ±.96 8 p : p : 990,500 ±.96,500 50,500 ( 990,500 50,500 ) = (0.677, 0.73),500 ) = (0.636, 0.68),500,050 990,500 ±.96,050,500 50,500,500 + 990,500 50,500 = (0.007, 0.073),500 95% 95% 0 0 N(0, ).65 9 =.355 Q(.36) = 0.0869 9% u(0.0) =.36.65 n.36 n n 6 3 µ 0 t(n, 0.05) s x µ 0 + t(n, 0.05) s µ 0 n n x t(n, 0.05) s n µ 0 x + t(n, 0.05) s n [ x t(n, 0.05) s, x + t(n, 0.05) s ] n n x ± t(n, 0.05) s n µ 95% µ 0
[ ] 5 N(µ, 3.5 ) 0 x = 5.9 H 0 : µ = 60. H : µ 60. 0.05 x = 5.9 < 56.0 = 60..96 3.5 0 A 5 00 x x H 0 : µ = 0 6 x N(50, (/5) ) 9.5 50.5 9.5 50 /5 =.5, 50.5 50 /5 =.5, Q(.5) = 0.0 0.0 µ = 9. N(9., (/5) ) [9.5, 50.5] 9.5 9. /5 =.5, 50.5 9. /5 = 6.5, Q(6.5) 0 Q(.5) Q(6.5) =. Q(.5) = 0.0668, 0.07 µ = 9. 0.3 µ = 9.6 0.69 7 Z = (5 ) + (39 0) 0 + (6 6) 6 = 9.65 χ (, 0.05) = 5.99, χ (, 0.0) = 9., χ (, 0.005) = 0.60 0.0 0.005 8 n = n = 600, ˆp = 30 600 = 8 5, ˆp = 80 600 = 7 30 + 80, ˆp = = 0.5 5 600 + 600 ( ˆp ˆp = 0.0667 > 0.0566 =.96 0.5 600 + ) 600 p 95% 8 ( 8 5 ±.96 5 7 ) = (0.93, 0.573) 5 600 p 95% 7 ( 7 5 ±.96 5 8 ) = (0.7, 0.507) 5 600
[ ] 6 H 0 : p = p 0.05 9 α 0. 0.05 0.0 0.0 c 0.058 0.0693 0.08 0.09 α 0. 0.05 0.0 0.0 x = 3 x = 7 α 0.0768 x 7 x 6.5 ˆp 0.585 ˆp 0.585 0.585 0.5000 /(0 ) =.333, Q(.33) = 0.0099, 0.0099 = 0.098 0.098 x x y y nˆα + x i ˆβ = yi () xi ˆα + x ˆβ i = x i y i () () () xi n { x i ( x i ) } n ˆβ = x i y i xi yi n (xi x) ˆβ = (xi x)(y i ȳ) ˆβ = (xi x)(y i ȳ) (xi x) = s xy s x () ˆα = ȳ ˆβ x.
[ ] 7 3 6 50 製作時間 0 30 0 3 5 製作個数 6 x i y i x i x (x i x) y i ȳ (y i ȳ) (x i x)(y i ȳ) x i x i y i 0 6 56 3 0 30 6 36 6 60 3 35 0 0 0 9 05 5 9 8 9 6 80 5 50 96 8 5 50 5 80 0 570 75 55 65 x = 3, ȳ = 36 5ˆα + 5 ˆβ = 80, 5ˆα + 55 ˆβ = 65 ˆα = 3.5, ˆβ = 7.5 ˆβ = s xy s x = 75 0 = 7.5, ˆα = ȳ ˆβ x = 36 7.5 3 = 3.5. ŷ = 3.5 + 7.5x x = 6 ŷ = 3.5 + 7.5 6 = 58.5 58.5 y i ŷ i y i ŷ i (y i ŷ i ) 0 30 8.5.5.5 35 36 5 3.5.5.5 50 5 0 7.5 S e = 7.5 S T = 570 r = S e = 7.5 S T 570 = 75 = 0.9868 76 σ s = 7.5 =.5 5.5 7.5 ± t(3, 0.05) 0 = (5.9, 9.09)
[ ] 8 ˆβ = 7.5 >.59 = t(3, 0.05) 7.5 0 読書 音楽鑑賞の時間 0 9 8 7 6 5 3 0 5 0 5 0 5 テレビを視る時間 7 ŷ = 9.758 0.35x S e = 0.9, r = 0.738 ( 0.9, 0.5) ˆβ = 0.35 > 0.77 = t(7, 0.05) 0.076 5 8 3300 300 民間最終消費支出 ( 千億円 ) 300 3000 900 800 700 600 700 800 900 5000 500 500 5300 500 5500 5600 国内総生産 ( 千億円 ) 8 ŷ =. + 0.80x GDP r = 0.9695 s = 553.3 0.80 ± t(, 0.05) 0.067 = (0.8, 0.536) ˆα =. > 73.8 = t(, 0.05) 5.6