# chapter7 - 1 Chapter 7 71(a P(0 Z 2 =(2(0 = 0.97725 0.5 =...

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1 Chapter 7 7–1. (a) P (0 Z 2) = Φ(2) - Φ(0) = 0 . 97725 - 0 . 5 = 0 . 47725 (b) P ( - 1 Z 1) = Φ(1) - Φ( - 1) = 2Φ(1) - 1 = 0 . 68268 (c) P ( Z 1 . 65) = Φ(1 . 65) = 0 . 95053 (d) P ( Z ≥ - 1 . 96) = Φ(1 . 96) = 0 . 9750 (e) P ( | Z | ≥ 1 . 5) = 2[1 - Φ(1 . 5)] = 0 . 1336 (f) P ( - 1 . 9 Z 2) = Φ(2) - Φ( - 1 . 9) = Φ(2) - [1 - Φ(1 . 9)] = 0 . 94853 (g) P ( Z 1 . 37) = 0 . 91465 (h) P ( | Z | ≤ 2 . 57) = 2Φ(2 . 57) - 1 = 0 . 98984 7–2. X N (10 , 9). (a) P ( X 8) = Φ ± 8 - 10 3 = Φ ± - 2 3 = 0 . 2525 (b) P ( X 12) = 1 - Φ ± 12 - 10 3 = 1 - Φ ± 2 3 = 0 . 2525 (c) P (2 X 10) = Φ ± 10 - 10 3 - Φ ± 2 - 10 3 = 0 . 5 - Φ( - 2 . 67) = 0 . 496 7–3. From Table II of the Appendix (a) c = 1 . 56 (b) c = 1 . 96 (c) c = 2 . 57 (d) c = - 1 . 645 7–4. P ( Z Z α ) = α Φ( Z α ) = 1 - α . (a) Z 0 . 025 = 1 . 96 (b) Z 0 . 005 = 2 . 57 (c) Z 0 . 05 = 1 . 645 (d) Z 0 . 0014 = 2 . 99

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2 7–5. X N (80 , 100). (a) P ( X 100) = Φ ± 100 - 80 10 = Φ(2) = 0 . 97725 (b) P ( X 80) = Φ ± 80 - 80 10 = 0 . 5 (c) P (75 X 100) = Φ ± 100 - 80 10 - Φ ± 75 - 80 10 = Φ(2) - Φ( - 0 . 5) = 0 . 97725 - 0 . 30854 = 0 . 66869 (d) P ( X 75) = 1 - Φ ± 75 - 80 10 = 1 - Φ( - 0 . 5) = Φ(0 . 5) = 0 . 69146 (e) P ( | X - 80 | ≤ 19 . 6) = Φ(1 . 96) - Φ( - 1 . 96) = 0 . 95 7–6. (a) P ( X > 680) = 1 - Φ ± 680 - 600 60 = 1 - Φ(1 . 33) = 0 . 09176 (b) P ( X 550) = Φ ± 550 - 600 60 = Φ( - 5 / 6) = 1 - Φ(5 / 6) = 0 . 20327 7–7. P ( X > 500) = 1 - Φ ± 500 - 485 30 = 1 - Φ(0 . 5) = 0 . 30854, i.e., 30.854% 7–8. (a) P ( X 28 . 5) = 1 - Φ ± 28 . 5 - 30 1 . 1 = 1 - Φ( - 1 . 36) = Φ(1 . 36) = 0 . 91308 (b) P ( X 31) = Φ ± 31 - 30 1 . 1 = 0 . 819 (c) P ( | X - 30 | > 2) = 1 - Φ ± 2 1 . 1 - Φ ± - 2 1 . 1 ¶‚ = 1 - [0 . 96485 - 0 . 03515] = 0 . 0703 7–9. X N (2500 , 5625). Then P ( X < ‘ ) = 0 . 05 implies that P ± Z < - 2500 75 = 0 . 05 or - 2500 75 = - 1 . 645 . Thus, = 2376 . 63 is the lower speciﬁcation limit.
3 7–10. M X ( t ) = E ( e tX ) = 1 σ 2 π Z -∞ e tx e - ( x - μ ) 2 2 σ 2 dx = σ σ 2 π Z -∞ e t ( + μ ) e - y 2 / 2 dy (letting y = ( x - μ ) ) = e μt 2 π Z -∞ e - ( y 2 - 2 σty ) / 2 dy = e μt 2 π Z -∞ e - ( y 2 - 2 σty + σ 2 t 2 - σ 2 t 2 ) / 2 dy = e μt 2 π Z -∞ e - ( y - σt ) 2 / 2 e σ 2 t 2 / 2 dy = e μt +(1 / 2) σ 2 t 2 1 2 π Z -∞ e - w 2 / 2 dw (letting w = y - σt ) = e μt +(1 / 2) σ 2 t 2 , since the integral term equals 1. 7–11. F Y ( y ) = P ( aX + b y ) = P ± X y - b a = Φ ± y - b a - μ σ = Φ ± y - b - = Φ ± y - ( + b ) This implies that Y N ( + b,a 2 σ 2 ). 7–12. X N (12 , (0 . 02) 2 ). (a) P ( X > 12 . 05) = 1 - Φ ± 12 . 05 - 12 0 . 02 = 1 - Φ(2 . 5) = 0 . 00621

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4 (b) P ( X > c ) = 0 . 9 1 - Φ ± c - 12 0 . 02 = 0 . 9 Φ ± c - 12 0 . 02 = 0 . 1 c - 12 0 . 02 = - 1 . 28 c = 12 - 0 . 0256 = 11 . 97 (c) P (11 . 95 X 12 . 05) = Φ ± 12 . 05 - 12 0 . 02 - Φ ± 11 . 95 - 12 0 . 02 = Φ(2 . 5) - Φ( - 2 . 5) = 0 . 9876 7–13. X N ( μ, (0 . 1) 2 ). (a) Take μ = 7 . 0. Then P ( X > 7 . 2) + P ( X < 6 . 8) = 1 - Φ ± 7 . 2 - 7 0 . 1 + Φ ± 6 . 8 - 7 0 . 1 = 1 - Φ(2) + Φ( - 2) = 1 - 0 . 97725 + 0 . 02275 = 0 . 0455 (b) 1 - Φ ± 7 . 2 - 7 . 05 0 . 1 + Φ ± 6 . 8 - 7 . 05 0 . 1 = 1 - Φ(1 . 5) + Φ( - 2 . 5) = 1 - 0 . 93319 + 0 . 00621 = 0 . 07302 (c) Φ ± 7 . 2 - 7 . 25 0 . 1 - Φ ± 6 . 8 - 7 . 25 0 . 1 = Φ( - 0 . 5) - Φ( - 4 . 5) . = 1 - Φ(0 . 5) = 0 . 3085
5 (d) Φ ± 7 . 2 - 6 . 75 0 . 1 - Φ ± 6 . 8 - 6 . 75 0 . 1 . = 1 - Φ(0 . 5) = 0 . 3085 7–14. X N (50 , 25), Y N (45 , 6 . 25). If Y X , i.e., if Y - X 0, a transaction will occur. Let W = Y - X N ( - 5 , 31 . 25). P ( W > 0) = P ± Z 0 + 5 5 . 59 = 1 - Φ(0 . 89) = 0 . 1867. 7–15. \$9.00 = revenue / capacitor, k = manufacturing cost for process A , 2 k = manufac- turing cost for process B . The proﬁts are P * A = 9 - k if 1000 X 5000 9 - k - 3 otherwise P * B = 9 - 2 k if 1000 X 5000 9 - 2 k - 3 otherwise Therefore, E ( P * A ) = (9 - k ) P (1000 X 5000) + (6 - k )[1 - P (1000 X 5000)] = (9 - k )0 . 9544 + (6 - k )0 . 0456 = 8 . 8632 - k E ( P * B ) = (9 - 2 k ) P (1000 X 5000) + (6 - 2 k )[1 - P (1000 X 5000)] = (9 - 2 k )(1) + (6 - k )(0) = 9 - 2 k Since E ( P * A ) < E ( P * B ) when k < 0 . 1368, use process B ; When k 0 . 1368, use process A .

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chapter7 - 1 Chapter 7 71(a P(0 Z 2 =(2(0 = 0.97725 0.5 =...

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