2 2 GDP( ) 1 () 143,694 47,186 48,997 38,371 36,559 44,519 28,565 44,550 26,526 43,237 23,031 38,455 15,945 34,971 14,996 44,950 10,852 10,183 10,337

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1 2 1 R C C 1 5 ( mmhg) (1) (2) 2 () C (1) saikou<-c(116,128,120,116,118) median(saikou), mean(saikou), sd(saikou) =118, =119.6, = , 70, 1.58 (2) 2 () [110, 130], [67, 73] 2.2 OECD() 2008 GDP() 1 GDP ( ( ) 2010/ ) (1) GDP 1 GDP (2) GDP 1 GDP (3) GDP (1) GDP<-c(143694,48997,,168), GDP1<-c(47186,38371,,52568) 1 GDP rev(sort(gdp1)) 19 rank(gdp1) 1 GDP 12 (2) hist(gdp), hist(gdp1) summary(gdp), summary(gdp1) (3)

2 2 2 GDP( ) 1 () 143,694 47,186 48,997 38,371 36,559 44,519 28,565 44,550 26,526 43,237 23,031 38,455 15,945 34,971 14,996 44,950 10,852 10,183 10,337 48,049 9,291 19,115 8,729 53,094 7,300 10,270 5,283 13,861 5,049 47,151 5,003 64,885 4,790 51,954 4,518 94,763 4,129 49,527 3,503 31,174 3,408 62,054 2,706 50,931 2,663 59,944 2,436 22,929 2,161 20,719 1,542 15,363 1,278 29, , , ,568 kensu daisu plot(daisu,kensu) cor(daisu,kensu) ( 22 )1 ( ) ( ) (1) (2)

3 2 3 30,780 4, ,271 14,526 1,348 14,220 1, , ,748 1, , , ,031 1,093 14, ,693 2, , , , ,118 16, , , ,875 2,224 1, , (1) kyakusu rev(sort(kyakusu)) (2) hist(kyakusu) hist(log10(kyakusu)) ( ) ( C ) 2 (1 29.5, 31.0, 28.8, 31.6, 30.3, 30.6, 27.8, 30.1, 27.8, 31.1, 28.9, 29.1, 27.2, 31.3, 31.4, 31.9, 32.1, 31.7, 34.5, 34.5, 36.3, 36.1, 35.7, 35.8, 34.4, 33.3, 33.8, 34.2, 27.9, 29.2, 32.2 ) ( kw) (1) (2) 2

4 , 4916, 4049, 4066, 4984, 5022, 4716, 4916, 4640, 4146, 3725, 4689, 4367, 4679, 4899, 5249, 4492, 4228, 4793, 5726, 5918, 5965, 5999, 5213, 4715, 5550, 5666, 5596, 4659, 4953, 4712 (1) kion, denryoku plot(kion,denryoku) cor(kion,denryoku) 0.68 (2) % 40% 50% ( ) (1) (%) 2 (2) (1) cumsum ( ) (2) (2.2), (2.6) , ( ) (1) 65 (2) (1) 0.269, 0.173

5 (2) akita, aichi cumsum(akita)/sum(akita) () GNI() ( ) ( GNI GDP ) 1 1 GNI GNI 39, , , , , , , , , , , , , , , , , , , , (1) (2) ( ) 2009 (1) (1) 0.92 (2)

6 2 6 3, 6, 2, 7, 10, 19, 18, 1, 9, 8, 25, 39, 17, 4, 11, 5, 36, 45, 13, 16, 27, 44, 33, 48, 50, 41, 15, 28, 20, 38, 12, 29, 47, 14, 23, 49, 22, 46, 26, 21, 40, 32, 37, 43, 24, 30, 35, 31, 42, 34 nyushi<-1:50 nyugakugo<-c(3,6,2,,42,34) cor(nyushi,nyugakugo) 0.58

7 (1) sample 10 R (2) 100 R (3) R (4) (3) 0.5, 0.05 R sample(), mean(), sapply(), hist(), curve(), dnorm(), replace=t, freq=f, add=t # (1) sample(c(" ",""), 10, replace=t) # (2) coin < - sample(c(" ",""), 100, replace=t) head < - which(coin == " ") length(head) / 100 # (3) jikken < - sapply(rep(100,100), function(n) { coin < - sample(c(" ",""), n, replace=t) head < - which(coin == " ") length(head) / 100 }) hist(jikken) # (4) (3) hist(jikken, freq=f) curve(dnorm(x, 0.5, 0.05), add=t) % sample mean sapply population < - c(rep("",20000), rep(" ",80000)) # 20% 10 z < - sapply(rep(1000,10000), function(x) { ss = sample(population, x) mean(ss == "") }) hist(z)

8 3 8 ±4% R (600 ) % c(rep(1,100000),rep(0,900000)) sample mean sapply ## population < - c(rep(1,100000),rep(0,900000)) # 10% 100 mean(sample(population, 600)) ## ww < - sapply(rep(600,1000), function(n) mean(sample(population, n))) hist(ww, freq=f) curve(dnorm(x,mean(ww),sd(ww)), add=t, col=2) ## ( 5% population population < - c(rep(1,50000),rep(0,950000)) # 10% 100 col=2 10% 3% 4% 5% 2% 3% , R m, s n rnorm(n,m,s) [a,b] curve(dnorm(x,m,s)), add=t ( freq=f ) ## n < # n = 1000 n < w < - rnorm(n, 10, 2) c(mean(w), sd(w)) hist(w, freq=f)

9 curve(dnorm(x,mean(w),sd(w)), add=t, col=2) 3 9

10 n, p 2 n ( n E(X) = i i i=0 n = np E(X(X 1)) = i=1 ( n 1 i 1 n ( n i(i 1) i i=0 = n(n 1))p 2 n ) p i (1 p) n i = n i=1 ) p i 1 (1 p) n i = np i=2 ) p i (1 p) n i = ( n 2 i 2 n! (i 1)!(n i)! pi (1 p) n i n i=2 n! (i 2)!(n i)! pi (1 p) n i ) p i 2 (1 p) n i = n(n 1)p 2 V (X) = E(X(X 1)) + E(X) (E(X)) 2 = np(1 p) 4.2 p k E(X) = k p(1 p) k 1 = p (1 p) k 1 = p (1 p) k 1 = (1 p) i 1 = 1 p k=1 k=2 k=1 i=1 k=2 i=1 i=1 k=i k 1 E(X(X 1)) = p k(k 1)(1 p) k 1 = 2p i (1 p) k 1 = 2p i 4.3 = 2 i=1 i(1 p) i = 2 1 p p 2 = 2 p 2 2 p V (X) = E(X(X 1)) + E(X) (E(X)) 2 = 1 p p 2 λ i=1 i=1 k=i+1 (1 p) k 1 E(X) = k=0 k λk k! e λ = E(X(X 1)) = k=0 = λ 2 k=1 λ k (k 1)! e λ = λ k(k 1) λk k! e λ = k=2 λ k 2 (k 2)! e λ = λ 2 k=1 k=2 λ k 1 (k 1)! e λ = λ λ k (k 2)! e λ

11 4 11 V (X)E(X(X 1)) + E(X) (E(X)) 2 = λ 4.4 a, b M(θ) = E(e θx ) = e θx ba x a 1 b a x a 1 0 Γ(a) e bx dx = 0 Γ(a) e (b θ))x dx ( ) a b (b θ) a x a 1 = e (b θ))x dx b θ 0 Γ(a) ( ) a b = b θ a, b θ 4 d dθ M(θ) = a (b θ) a+1 b a d 2 b a M(θ) = a(a + 1) dθ2 (b θ) a+2 a/b 2 a(a + 1)/b 2 a/b X, Y 2 g(x), h(x) E(g(X)h(Y )) = E(g(X))E(h(Y )) X, Y f X (x), f Y (x) 4.6 E(g(X)h(Y )) = = g(x)h(y) f X (x)f Y (y)dxdy g(x)f X (x)dx h(y)f Y (y)dy = E(g(X))E(h(Y )) X i a i, b (i = 1, 2,..., n) X 1 + X X n n i=1 a i, b 2 ( ) b a i, b M i (θ) = ( b ) ai b θ

12 4 12 X 1 + X X n M(θ) ( ) a1 ( ) a2 ( ) an b b b M(θ) = b θ b θ b θ ( ) a1 +a b 2 + +a n = b θ a 1 + a a n, b 4.7 (4.75) E(X Y ) Y g(y ) V (E(X Y )) = E(g(Y ) 2 ) (E(g(Y ))) 2 = E(g(Y ) 2 ) (E(X)) 2 V (X Y ) = E(X 2 Y ) g(y ) 2 E(V (X Y )) = E(X 2 ) E(g(Y ) 2 ) V (E(X Y )) + E(V (X Y )) = E(X 2 ) (E(X)) 2 = V (X) 4.1 A, B P (A B) = P (A) + P (B) P (A B) () A B A B c A c B A B A, B 2 A = (A B) (A B c ), B = (A B) (A c B) A B 3 A B = (A B) (A c B) (A B c ) P (A B) = P (A B) + P (A c B) + P (A B c ) = P (A) + P (B) P (A B) P (B i A) = P (A B i )P (B i ) n j=1 P (A B j)p (B j ) 3 B 1, B 2, B 3 50%, 30%, 20% , 0.010, B 1, B 2, B 3

13 4 13 P (B i A) = P (A B i) P (A) P (A) = n P (A B j ) = j=1 P (A B i ) = P (A B i )P (B i ) n P (A B j )P (B j ) A 1 B 1, B 2, B 3 B 1, B 2, B 3 j=1 P (B 1 ) = 0.5, P (A B 1 ) = P (B 2 ) = 0.3, P (A B 2 ) = 0.01 P (B 3 ) = 0.2, P (A B 3 ) = P (A B 1 )P (B 1 ) : P (A B 1 )P (B 1 ) : P (A B 1 )P (B 1 ) = 7.5 : 3 : 4 P (B 1 A) = 15 29, P (B 2 A) = 6 29, P (B 3 A) = 8 29 X F (x) X E(X) x = x 0 du. E(X) = 0 E(X) = ( x 0 0 ) dt df (x) = xdf (x) = ( 0 0 t (1 F (x))dx ) df (x) dt = 0 (1 F (t))dt 1 F (t) y = 1 y y = F (x) ( 1 (1 F (t))dt = 0 0 du F ((t) ) dt = 1 0 F 1 (u)du F 1 (x) = u 0 < u 1 < < u n = F 1 (u)du i u i = F (x i ) F 1 (u i )(u i+1 u i ) = i i = i F 1 (u i )(u i+1 u i ) x i (F (x i+1 ) F (x i )) x i F (x i+1 ) F (x i ) x i+1 x i (x i+1 x i )

14 4 14 n F (x i+1) F (x i ) x i+1 x i f(x i ) i F (x i+1 ) F (x i ) x i (x i+1 x i ) x i+1 x i i xf(x i ) (x i+1 x i ) 0 xf(x)dx 4.4 X x, y P (X > x + y X > x) = P (X > y) λ 1 e λx 4.5 P (X > x + y X > x) = P (X > x + y) P (X > x) = e λ(x+y) e λx = e λy = P (X > y) (= 5 60) X 1, X 2,... 1/5[ ] S n = X 1 +X X n 1 n S n 60 < S n+1 S n = s( 60) X n+1 > 60 s s 0 s 60 ( ) X 1, X 2,... 1/5[ ] S n = X 1 + X X n S n n, b(= 5) 1 N N = n S n (= s) 60 S n+1 > 60 X n+1 > 60 s 0 < S n 60 S n f Sn (s) P (N = n) = = P (X n+1 > 60 s)f Sn (s)ds e b(60 s) bn s n 1 (n 1)! e bs ds = (60b)n n! e 60b N 60b = n, p 2 p n np p n R R 2 dbinom(k,n,p)dpois(k,np) (n, p) = (10, 0.1), (20, 0.05), (100, 0.01), (100, 0.05) n n = 100; p = 0.05; nn = min(n,20) plot(dbinom(0:nn,n,p), type="b") lines(dpois(0:nn, n*p),col=2)

15 [0,1] R runif m n R m = 2, 4, 12 n = , 1/12m mean(runif(m)) sapply hist freq=f curve(dnorm(x,1/2,sqrt(1/12/m)),add=t) m = 2 m = 4 m = 12 ## n = m = 12 z = sapply(rep(m,n), function(m) mean(runif(m))) hist(z, freq=f) curve(dnorm(x, 0.5, sqrt(1/12/m)), add=t)

16 X 1, X 2,..., X n p 2 n X 1 + X X n n, p 2 X ( P X = k ) ( ) n = p k (1 p) k, k = 0, 1,..., n n k 5.2 X 1, X 2,..., X n a, b (4.6.8 ) n X M(θ) = ( b ) a b θ ( ) an b M X(θ) = = b θ/n ( bn ) an bn θ an, bn E ( an X) = bn = a b V ( an X) = (bn) 2 = 1 a n b X 1, X 2,..., X n µ, σ 2 X i X X i X 2 X X i X C( X, X i X) = C( X, X i ) V ( X) = 1 n n j=1 C(X j, X i ) σ2 n = 1 n V (X i) σ2 n = n i=1 (X i X) 2 /(n 1) 2σ 4 /(n 1) 5.2 W = n ( Xi X i=1 σ ) 2

17 5 17 n 1 2 W 2(n 1) V V = σ 2 W/(n 1) V 2σ 4 /(n 1) 5.5 n t X X 2 1, n F t 2 P (X 2 x) = P ( X x ) x = 2 = x 0 Γ((n + 1)/2) Γ(n/2) nπ 0 ( u n + 1 ) (n+1)/2 du u, 1, n F ( ) Γ((n + 1)/2) t 2 (n+1)/2 Γ(n/2) nπ n + 1 dt (t 2 = u ) Z n 2 W Z/ W/n n t Z ( F Z/ 2 W/n) = Z 2 /(W/n) 1, n F R R rnorm(n)^2 2 X g(x) = x 2 Y = X 2 P (Y x) = P (X 2 x) = P ( x X x ) = 2P ( 0 X x ) 2 x = e u2 /2 du π 0 u 2 = v u = v, du = dv 2 v F Y (x) = 2 x π 0 f Y (x) = d dx F Y (x) = π = Γ(1/2) 1 2 v e v/2 dv 2 π 1 2 x e x/2 f Y (x) = 1 Γ(a) ba x a 1 e bx, (a = b = 1/2) a, b 5.7 a, b X X/(2a)/((1 X)/(2b)) 2a, 2b F

18 5 18 a, b B a,b (x) Y = X/(2a) (1 X)/(2b) ( ) ( ) X/(2a) P Y = (1 X)/(2b) y y/(2b) = P X 1/(2a) + y/(2b) ( ) ay = B a,b b + ay Y ( Γ(a + b) ay Γ(a)Γ(b) = Γ(a + b) Γ(a)Γ(b) ) a 1 ( ) b 1 b ab b + ay b + ay (b + ay) 2 ( ) b ( b y a 1 y + b ) (a+b) a a 2a, 2b F 5.8 m, n F 100α% F α (m, n) F α (m, n)f 1 α (n, m) = 1 F 2 2 m 2 X, n 2 Y (X/m)/(Y/n) m, n F (Y/n)/(X/m) n, m F ( ) X/m α = P Y/n F α(m, n) = P ( ) Y/n = 1 P X/m 1 F α (m, n) ( ) Y/n X/m 1 F α (m, n) n, m F 100(1 α)% F 1 α (n, m) 1/F α (m, n) (5.1) (5.2)

19 p = n = 100 ˆp = % 2 95% 2 R pbinom(k,n,p) p = 0.4 p(1 p)/n /100 = = % qnorm(0.975) % 0.35 ± [0.257, 0.443] pbinom(kb,100,0.35)-pbinom(ka,100,0.35)> =0.95 ka,kb kb-ka 2.5% ( 2.5% ) ka 2.5% ( 2.5% ) kb pbinom(44,100,0.35)= pbinom(25,100,0.35)= % [0.25, 0.44] pbinom(44,100,0.35)-pbinom(24,100,0.35)= % 96% 6.1 µ, σ 2 n T 1 (X) = X i, T 2 (X) = n i=1 i=1 T 1 (X), T 2 (X) µ, σ 2 X 2 i µ, σ 2 n x 1, x 2,..., x n l(µ, σ 2 ; x) = n 2 log 2πσ2 1 2σ 2 n (x i µ) 2 i=1 = n 2 log 2πσ2 1 2σ 2 ( T2 (x) 2µT 1 (x) + nµ 2) n T 1 (x), T 2 (x) µ, σ 2 T 1 (X), T 2 (X) ( 6.2) 6.2 µ, σ , % 2 σ % 1.96σ/ n σ = n < 2

20 6 20 n n > σ 2 = 20 = X p(0 < p < 1) P (X [r, s]) = p [r, s] s r r + s = 0, s = r + g(r) r λ s r λ(φ(s) Φ(r) p) r, s λϕ(r) = 0, 1 λϕ(s) = 0 ϕ(r) = ϕ(s) r + s = 0 r, s 6.4 θ > 0 X 1, X 2,..., X n U(0, θ) T (X) = c max {X 1, X 2,..., X n } θ c T (X) P (T (X) < x) = P ( x ) n = cθ E(T (X)) = E(T (X) 2 ) = cθ 0 cθ 0 V (T (X)) = ( X 1 < x c, X 2 < x c,..., X n < x ) c x nxn 1 (cθ) n dx = n n + 1 cθ = θ c = n + 1 n x 2 nxn 1 (cθ) n dx = n n + 2 (cθ)2 = (n + 1)2 n(n + 2) θ2 θ 2 1 = n(n + 2) θ2 (n + 1)2 n(n + 2) θ (1) p (2) a (1) log f(x; p) = x log p + (1 x) log(1 p) p log f(x; p) = x p 1 x 1 p 2 p 2 log f(x; p) = x p 2 1 x (1 p) 2

21 6 21 (2) ) I 1 (p) = E ( 2 log f(x; p) = 1 p2 p p = 1 p(1 p) 6.6 log f(x; a) = log a x a a log f(x; a) = 1 a + x a 2 2 a 2 log f(x; a) = 1 a 2 2x a 3 ) I 1 (p) = E ( 2 log f(x; a) = 1 a2 a 2 θ (1) n θ ˆθ n (2) n X med π 2 /(4n) f(x; θ) = 1 π(1 + (x θ) 2 ) (1) ) E ( 2 l(θ; X) θ2 l(θ; x) = log(1 + (x θ) 2 ) log π 2(x θ) l(θ; x) = θ 1 + (x θ) 2 2 θ 2 l(θ; x) = (x θ) 2 + 4(x θ) 2 (1 + (x θ) 2 ) 2 = = (x θ) 2 4 (1 + (x θ) 2 ) 2 J k = 2 π(1 + (x θ) 2 ) 2 dx π(1 + x 2 ) k dx 4 π(1 + (x θ) 2 ) 3 dx x 2 J k = 2k 0 π(1 + x 2 ) k+1 dx = 2kJ k 2kJ k+1 J k+1 = 2k 1 2k J 1 = 0.5 J k ) E ( 2 l(θ; X) = 2(4J θ2 3 2J 2 ) = 1 2 ˆθ n 2/n (2) V (X med ) π 2 /(4n) V (ˆθ n ) V (X med ) 8 π

22 () µ i, σi 2 (i = 1, 2) m, n X 1, X 2,..., X m ; Y 1, Y 2,..., Y n S 2 X, S2 Y 2 σ1/σ (1 α)% F α (m, n) m, n F 100α% [ ] S 2 X 1 F α/2 (m 1, n 1), S2 X F α/2 (n 1, m 1) S 2 Y S 2 Y (S X /σ 1 ) 2 m 1 2 F α (m, n)f 1 α (n, m) = 1 (m 1)SX 2 /σ2 1 m 1 2 (n 1)S2 Y /σ2 2 n 1 2 ( S 2 X /σ2 1) / ( S 2 Y /σ 2 2) m 1, n 1 F α = P ( F 1 α/2 (m 1, n 1) < S2 X /σ2 1 SY 2 /σ2 2 ( S 2 = P X S 2 Y 1 F α/2 (m 1, n 1) < σ2 1 σ 2 2 ) < F α/2 (m 1, n 1) < S2 X S 2 Y ) 1 F 1 α/2 (m 1, n 1) F F α (m, n)f 1 α (n, m) = 1 100(1 α)% 6.8 X 1, X 2,..., X n N(µ, σ 2 ) n T 1 (X) = X i, T 2 (X) = n i=1 i=1 T 1 (X), T 2 (X) µ, σ 2 X 2 i n x 1, x 2,..., x n X med n 1 n n x i θ i=1 θ θ = X med n θ = X med n x 1, x 2,..., x n x (1) x (2) x (n) x (k) < θ < θ < x (k+1) n x i θ i=1 n x i θ = (2k n)(θ θ) i=1 k < n/2 θ k > n/2 θ θ = x (k), θ = x (k+1) n x i θ i=1 n x i θ = 2x (k+1) + (2k + 2 n)θ (2k n)θ i=1 = (2k n)(x (k+1) x (k) ) k < n/2 θ k > n/2 θ n

23 6 23 k = n/2 0 x (n/2) < θ < x (n/2+1) θ = X med

24 exp(θa(x) + b(θ) + c(x)) H 0 : θ = θ 0, H 1 : θ = θ 1 (> θ 0 ) θ a(xi ) > d 1 1 ( exp( (x µ) 2 /2) = exp µx µ 2 /2 x 2 /2 log ) 2π 2π log L(θ 0 ; x) = (θ 0 a(x i ) + b(θ 0 ) + c(x i )) log L(θ 1 ; x) = (θ 1 a(x i ) + b(θ 1 ) + c(x i )) L(θ 1 ; x) L(θ 0 ; x) > c log L(θ 1; x) log L(θ 0 ; x) > log c log L(θ 1 ; x) log L(θ 0 ; x) = (θ 1 θ 0 ) a(x i ) + b(θ 1 ) b(θ 0 ) ( ) log c (b(θ1 ) b(θ 0 )) d = exp θ 1 θ p H 0 : p = 1 6, H 1 : ṗ > i=30 ( 100 i ) ( ) i ( ) 100 i % 7.3 () 40 A B A 39.0, 39.6, 39.9, 40.4, 39.8, 39.7, 40.0, 40.4, 40.0, 39.4 B 39.5, 40.7, 40.6, 39.3, 38.9, 40.4, 41.6, 41.6, 42.3, 39.1

25 7 25 A σ 2 A B σ2 B H 0 : σ 2 A = σ2 B, H 1 : σa 2 < σ2 B A B T 5 (X, Y ) 1.398/0.184 = 7.6 9, 9 F 1% % p m = 20 n = 10 x = 1500 ȳ = 1000 µ X, µ Y (1) X, Ȳ ( ) (2) F F = ( X/µ X )/(Ȳ /µ Y ) (3) µ X /µ Y 100(1 α)% (4) µ X = µ Y µ X = 1.5µ Y (5) (1) µ X X 2X/µ X m X m µ 1 X 2m X/µ X 2m 2 ( 1/2 ) 2m 1/2 θ X θ θµ X /(2m) ( ) m ( ) m 1/2 m/µx = 1/2 θµ X /(2m) m/µ X θ m, m/µ X X 20, 20/µ X Ȳ 10, 10/µ Y (2) (1) 2nȲ /µ Y 2n 2 2m X/µ X 2m 2 F = 2m X/µ X /(2m) 2nȲ /µ Y /(2n) 2m, 2n(= 40, 20) F ( 5.6) (3) (2) ( X/µX P Ȳ /µ Y = X/µ X Ȳ /µ Y ) ( 1 X x = P x Ȳ µ ) X = F 2n,2m (x) µ Y µ X /µ Y 100(1 α)% [ 1 F α/2 (2m, 2n) X/Ȳ X Ȳ, 1 F 1 α/2 (2m, 2n) ] X Ȳ = 1.5 α α = 0.1 [0.752, 2.758] α = 0.05 [0.656, 3.102] α = 0.01 [0.496, 3.898]

26 7 26 (4) F = X/Ȳ 40, 20 F µ X = 1.5µ Y F > F α (2m, 2n) F = 1.5 F 0.1 (40, 20) = % R na = 20; ma = 1/1000; nb = 10; mb = 1/1000 nn = 1000 z = sapply(1:nn, function(x) mean(rexp(na,ma))/mean(rexp(nb,mb)) ) hist(z) c(length(which(z> 1.5))/nn, 1-pf(1.5,40,20)) (5) H 1 : µ X > µ Y

27 ( 8.1) Q Q P P Q Q 151.3,157.2,156.0,152.2,151.4,154.1,158.9,158.9,155.4,156.6, 151.9,153.4,167.8,168.7,157.2,158.2,155.6,152.5,161.2,152.1, 151.5,159.2,161.1,162.6,151.5 Q Q 1, P P 2 P P Q Q R data = c(151.3,157.2,156.0,152.2,151.4,154.1,158.9,158.9,155.4,156.6, 151.9,153.4,167.8,168.7,157.2,158.2,155.6,152.5,161.2,152.1, 151.5,159.2,161.1,162.6,151.5) hist(data) qqnorm(data, xlim=c(-2.5,2.5)) qqline(data) n = length(data) qqplot((1:n-0.5)/n, pnorm(data,mean(data),sd(data)), xlim=c(0,1), ylim=c(0,1)) abline(0,1) 8.2 1

28 ( 8.1) P P λ = m = 4 (m 1) 1 = % χ (2) = chisq.test p pchisq 1.09 p 0.78 data = c(226,104,28,6,1) n = sum(data) # m = sum(data * (0:4)) / n # obs = c(data[1:3],7) # 2 exp = c(dpois(0:2,m), 1-ppois(2,m)) chisq.test(obs, p=exp) ##

29 ( 8.2) Z (ρ = 0.5) chi2 = sum((obs - n*exp)^2 / (n*exp)) 1 - pchisq(chi2, 3) 8.3 µ = ν = 0, σ 2 = τ 2 = 1 ρ = n = 20 (8.33) Z N = ρ 2 log 1 ρ, 1/(n 3) Z ρ = 0.9 rr N hist((log(1+rr)-log(1-rr))/2, freq=f)z freq=f 8.4 n = 20 2 N sapply ρ = 0.5 3, ρ = hist(rr) ( ) n = 20 r = 0.5 N = rr = sapply(rep(n,n), function(n) { z = rnorm(n) w = r*z+sqrt(1-r^2)*rnorm(n) cor(z,w)}) hist((log(1+rr)-log(1-rr))/2, freq=f) curve(dnorm(x,(log(1+r)-log(1-r))/2, 1/sqrt(n-3)), add=t)

30 ( 8.2) Z (ρ = 0.9) ( C) ( ) () Excel R R lm data= attach(temp) temp < - read.table("clipboard", header=t) plot(temp, type="l") model < - lm( ~, data=temp) abline(model) summary(model)

31 ( 9.1) t ( mmhg, C. 1mmHg Pa) P, T T = log 10 P () 680 P 775 ( 6) R boil < - read.table("clipboard", header=t) plot(boil) model < - lm( ~, data=boil) abline(model) curve(1730/(8.07-log10(x))-233.4, add=t, col=2)

32 ( 9.2) 9.3 ( C) ( C ) (Ω) , 0.06 R regist < - read.table("clipboard", header=t) plot(regist) model < - lm( ~, data=regist) abline(model) 9.4 ( ) () ( )

33 ( 9.3) , , , , , , , , , , , , , 0.94 R bank < - read.table("clipboard", header=t) plot(bank) n < - length(bank[,1]) x < - bank[4:n,1] y < - bank[4:n,2]/10000 plot(x, y) model < - lm(y ~x) abline(model) 9.5 (CO 2 ) () ( ppm) ( 2010 )

34 ( 9.4) , 1.78 R carbon < - read.table("clipboard", header=t) plot(carbon) model < - lm(carbon$ ~carbon$ ) abline(model) () ( ) (1) () (2) (1) (3) (2)

35 ( 9.5) ( ppm) Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec (1) (1 132 )

36 (a) banka < - read.table("clipboard", header=t) attach(banka) model < - lm( ~ ) plot(banka[,3]) ## abline(model) ## ahat < - model$coefficients[1] bhat < - model$coefficients[2] res < - ahat + bhat*banka[,1] plot(banka[,3]-res) abline(h=0) (2) log 9.1 ahat + bhat * R predict(model) logbank < - log( ) logkaiki < - lm(logbank ~ ) summary(logkaiki) plot(logbank-predict(logkaiki)) abline(h=0) (3) logkaiki$coefficients[2] i y i y i+12 y i y i

37 (b) () z i = log y i y i+12 y i y i = ezi+12 e zi 1 = e zi+12 zi 1 = e = %

38 H I T O p ( ) data < - read.table("clipboard", header=t) data1 < - c(data[,2], data[,3], data[,4], data[,5]) level < - c(rep(1,7), rep(2,7), rep(3,7), rep(4,7)) level < - factor(level) summary(aov(data1 ~ level)) ## Df Sum Sq Mean Sq F value Pr(> F) level e-13 *** Residuals ## 5 plot(level,data1,xaxp=c(1,4,3),xlab="") points(1:4,c(mean(data[,2]),mean(data[,3]),mean(data[,4]),mean(data[,5])),pch=16) lines(1:4,c(mean(data[,2]),mean(data[,3]),mean(data[,4]),mean(data[,5]))) lines(1:4,c(mean(za),mean(zb),mean(zc),mean(zd))) abline(h=mean(data1), lty=2) ( ( )

39 10 39 ) 10 ( / ) (1) ( 2 ) (2) (3) () (1) ( 10.1) 9 data < - read.table("clipboard", header=t) brand < - factor(floor((10:99)/10))

40 10 40 count <- c(); for(i in 1:9) count = c(count, data[,i]) ## () apply(data, 2, mean) apply(data, 2, sd) apply(data, 2, sd) / apply(data, 2, mean) ## plot(level,count) (2) (3) (summary(aov(count ~brand))) > summary(aov(count ~brand)) Df Sum Sq Mean Sq F value Pr( > F) brand e-14 *** Residuals ( / ) , 3 2 p p 0.87 >time < - factor(rep(1:10,5)) > summary(aov(count ~point + time)) Df Sum Sq Mean Sq F value Pr( > F) point < 2e-16 *** time

41 ( 10.2) Residuals A, B, C, D 4 3 () ( kg) A B C D (1) (2) (1) 14 A B 4 p % 5% (2) 2 p 0.77 ## (1) data < - read.table("clipboard", header=t) person < - factor(c(rep("a",3),rep("b",3),rep("c",3),rep("d",3))) count < - c(data[,1],data[,2],data[,3],data[,4]) summary(aov(count ~person)) plot(person, count) ## (2)

42 ( 10.4) brand < - factor(c(rep("",4),rep(" ",4),rep(" ",4))) count < - c(t(data[1,]),t(data[2,]),t(data[3,])) summary(aov(count ~brand)) 10.5 (A1, A2, A3, A4 4 ) (B1, B2, B3 3 ) 3 ( %) A1 A2 A3 A4 B1 55.6, 56.6, , 60.8, , 60.0, , 55.7, 48.4 B2 66.3, 65.6, , 71.1, , 55.1, , 59.0, 59.9 B3 52.0, 50.9, , 58.3, , 52.6, , 47.2, p % p % aov ~ * data < - read.table("clipboard", header=t) < - rep(data[,1],3) < - c(rep("b1",12),rep("b2",12),rep("b3",12)) < - c(data[,3],data[,4],data[,5]) summary(aov( ~))

43 ( 10.5) 16 ( 10.5) summary(aov( ~)) summary(aov( ~ + )) summary(aov( ~ * ))

201711grade1ouyou.pdf

201711grade1ouyou.pdf 2017 11 26 1 2 52 3 12 13 22 23 32 33 42 3 5 3 4 90 5 6 A 1 2 Web Web 3 4 1 2... 5 6 7 7 44 8 9 1 2 3 1 p p >2 2 A 1 2 0.6 0.4 0.52... (a) 0.6 0.4...... B 1 2 0.8-0.2 0.52..... (b) 0.6 0.52.... 1 A B 2