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# Mathematical Statistics with Applications, Seventh Edition

www.downloadslide.com Answers 887 c Not independent since ρ ≠ 0. d This is the bivariate normal distribution with µ 1 = µ 2 = 0, σ1 2 = 1, σ 2 2 1 = 2, and ρ = √ 2 6.69 a f (y 1 , y 2 ) = 1 , y y1 2 1 > 1, y2 2 y 2 > 1 e No 6.73 a g (2) (u) = 2u,0≤ u ≤ 1 b E(U 2 ) = 2/3, V (U 2 ) = 1/18 6.75 (10/15) 5 n! 6.77 a ( j − 1)!(k − 1 − j)!(n − k)! y j−1 j [y k − y j ] k−1− j [θ − y k ] n−k θ n , 0 ≤ y j < y k ≤ θ (n − k + 1) j b (n + 1) 2 (n + 2) θ 2 (n − k + j + 1)(k − j) c (n + 1) 2 (n + 2) 6.81 b 1 − e −9 6.83 1 − (.5) n 6.85 .5 6.87 a g (1) (y) = e −(y−4) , y ≥ 4 b E(Y (1) ) = 5 θ 2 6.89 f R (r) = n(n − 1)r n−2 (1 − r), 0 ≤ r ≤ 1 6.93 f (w) = 2 ( ) 1 √ − w ,0≤ w ≤ 1 3 w ⎧ 1 ⎪⎨ 0 ≤ u ≤ 1 6.95 a f U1 (u) = 2 ⎪⎩ 1 2u u > 1 2 b f U2 (u) = ue −u ,0≤ u c Same as Ex. 6.35. 6.97 p(W = 0) = p(0) = .0512, p(1) = .2048, p(2) = .3264, p(3) = .2656, p(4) = .1186, p(5) = .0294, p(6) = .0038, p(7) = .0002 6.101 f U (u) = 1, 0 ≤ u ≤ 1 Therefore, U has a uniform distribution on (0, 1) 1 6.103 π(1 + u 2 1 ) , ∞ < u 1 < ∞ 1 6.105 B(α, β) uβ−1 (1 − u) α−1 ,0< u < 1 ⎧ 1 ⎪⎨ 4 √ 0 ≤ u < 1 u 6.107 f U (u) = 1 ⎪⎩ 8 √ 1 ≤ u ≤ 9 u 6.109 P(U = C 1 − C 3 ) = .4156; P(U = C 2 − C 3 ) = .5844 Chapter 7 7.9 a .7698 b For n = 25, 36, 69, and 64, the probabilities are (respectively) .8664, .9284, .9642, .9836. c The probabilities increase with n. d Yes 7.11 .8664 7.13 .9876 7.15 a E( ¯X − Ȳ ) = µ 1 − µ 2 b V ( ¯X − Ȳ ) = σ1 2/m + σ 2 2/n c The two sample sizes should be at ( least ∑6 18. ) 7.17 P i=1 Z i 2 ≤ 6 = .57681 7.19 P(S 2 ≥ .065) = .10 7.21 a b = 2.42 b a = .656 c .95 7.27 a .17271 b .23041 d .40312 7.31 a 5.99, 4.89, 4.02, 3.65, 3.48, 3.32 c 13.2767 d 13.2767/3.32 ≈ 4 7.35 a E(F) = 1.029 b V (F) = .076 c 3 is 7.15 standard deviations above this mean; unlikely value. 7.39 a normal, E(ˆθ) = θ = c 1 µ 1 + c ( 2 µ 2 +···+c k µ k ) c 2 V (ˆθ) = 1 + c2 2 +···+ c2 k σ 2 n 1 n 2 n k b χ 2 with n 1 + n 2 +···+n k − k df c t with n 1 + n 2 +···+n k − k df 7.43 .9544 7.45 .0548 7.47 153 7.49 .0217 7.51 664 7.53 b Ȳ is approximately normal: .0132. 7.55 a random sample; approximately 1. b .1271

www.downloadslide.com 888 Answers 7.57 .0062 7.59 .0062 7.61 n = 51 7.63 56 customers 7.65 a Exact: .91854; normal approximation: .86396. 7.67 a n = 5 (exact: .99968; approximate: .95319); n = 10 (exact: .99363; approximate: .97312); n = 15 (exact: .98194; approximate: .97613); n = 20 (exact: .96786; approximate: .96886) 7.71 a n > 9 b n > 14, n > 14, n > 36, n > 36, n > 891, n > 8991 7.73 .8980 7.75 .7698 7.77 61 customers 7.79 a Using the normal approximation: .7486. b Using the exact binomial probability: .729. 7.81 a .5948 b With p = .2 and .3, the probabilities are .0559 and .0017 respectively. 7.83 a .36897 b .48679 7.85 .8414 7.87 .0041 7.89 µ = 10.15 7.91 Since X, Y , and W are normally distributed, so are ¯X, Ȳ , and ¯W . µ U = E(U) = .4µ 1 +.2µ 2 +.4µ ( ) 3 σ σ 2 2 U = V (U) = .16 1 n 1 ( ) ( ) σ 2 + .04 2 σ 2 + .16 3 n 2 n 3 7.95 a F with num. df = 1, denom. df = 9 b F with num. df = 9, denom. df = 1 c c = 49.04 7.97 b .1587 7.101 .8413 7.103 .1587 7.105 .264 Chapter 8 8.3 a B(ˆθ) = aθ + b − θ = (a − 1)θ + b b Let ˆθ ∗ = (ˆθ − b)/a 8.5 a MSE(ˆθ ∗ ) = V (ˆθ ∗ ) = V (ˆθ)/a 2 8.7 a = σ 2 2 − c σ 2 1 + σ 2 2 − 2c 8.9 Ȳ − 1 8.11 ˆθ 3 − 9ˆθ 2 + 54 8.13 b [n 2 /(n − 1)](Y/n)[1 − (Y/n)] 8.15 a ( 1 3n − 1 b MSE( ˆβ) = ) β 8.17 a (1 − 2p)/(n + 2) 2 (3n − 1)(3n − 2) β2 np(1 − p) + (1 − 2p)2 b (n + 2) 2 c p will be close to .5. 8.19 MSE(ˆθ) = β 2 8.21 11.5 ± .99 8.23 a 11.3 ± 1.54 b 1.3 ± 1.7 c .17 ± .08 8.25 a −.7 b .404 8.27 a .601 ± .031 8.29 a −.06 ± .045 8.31 a −.03 ± .041 8.33 .7 ± .205 8.35 a 20 ± 1.265 b −3 ± 1.855, yes 8.37 1020 ( ± 645.1 ) 2Y 8.39 9.48773 , 2Y .71072 8.41 a (Y 2 /5.02389, Y 2 /.0009821) b Y 2 /.0039321 c Y 2 /3.84146 8.43 b [Y (n) ](.95) −1/n 8.45 a Y /.05132 b 80% 8.47 c (2.557, 11.864) 8.49 c (3.108, 6.785) 8.57 .51 ± .04 8.59 a .78 ± .021 8.61 (15.46, 36.94) 8.63 a .78 ± .026 or (.754, .806) 8.65 a .06 ± .117 or (−.057, .177) 8.67 a 7.2 ± .751 b 2.5 ± .738 8.69 .22 ± .34 or (−.12, .56) 8.71 n = 100

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• Page 865 and 866: www.downloadslide.com 840 Appendix
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