Statistics & Probability MS2

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Mathematics HL STATISTICS & PROBABILITY Worksheet 2 MARKSCHEME 11. EITHER let y i = x i - 12 M1A1 σ x = σ y = 3 A1 M1A1 = 10(9 + 4) = 130 A1 OR M1A1 A1 σ x = 3, (M1) = 10(9 + 100) A1 = 1090 - 2400 + 1440 = 130 A1 [6] 12. (a) M1A1 M1A1 - e - a + 1 = 1 - A1 Note: Accept e 0 instead of 1. e - a = e a = a = A1 a = ln 2 AG 2 10 = = y x 9 10 2 10 1 2 = = y y i i = 10 1 2 i i y = = = = + = 10 1 10 1 10 1 2 10 1 2 1 144 24 ) 12 ( i i i i i i i x x x 100 10 10 1 1 = = = i x x 9 10 2 10 1 2 = = x x i i = 10 1 2 i i x = 10 1 2 ) 12 ( i i x 2 1 1 d e 1 0 = x a ax 2 1 1 e 1 0 = ax 2 1 2 1 2 = 2 1 2 1 2 ln accept 2 ln a 2 1
(b) M1A1 - e - Ma + 1 = e - Ma = A1 Ma = ln 2 M = = 2 A1 (c) P(1 < X < 3) = M1A1 = - e - 3 a + e - a A1 P( X < 3| X > 1) = M1A1 = A1 = A1 = ( - e - 3 a + e - a ) = A1 = A1 Note: Award full marks for P ( X < 3│ X > 1) = P ( X < 2) = or quoting properties of exponential distribution. [20] 13. recognition of X ~ B (M1) P( X = 3) = A1 P( X = 2) = A1 A1 [4] 2 1 d e 0 = x a M ax 2 1 e 0 = M ax 2 1 2 1 a 2 ln x a ax d e 3 1 ) 1 P( ) 3 P(1 X X ) 1 P( 1 e e 3 + X a a 2 1 e e 3 a a + 2 + 2 1 2 3 2 2 2 2 1 2 1 7 4 , 6 = 6 3 3 3 3 7 3 4 20 7 3 7 4 3 6 = 6 4 2 4 2 7 3 4 15 7 3 7 4 2 6 = = = = 9 16 45 80 ) 2 P( ) 3 P( X X
14. weight of glass = X X ~ N(160, σ 2 ) P( X < 160 + 14) = P( X < 174) = 0.75 (M1)(A1) Note: P( X < 160 - 14) = P( X < 146) = 0.25 can also be used. P = 0.75 (M1) = 0.6745... (M1)(A1) σ = 20.8 A1 [6] 15. (a) p + q = 0.44 A1 2.5 p + 3.5 q = 1.25 (M1)A1 p = 0.29, q = 0.15 A1 (b) use of Var( X ) = E( X 2 ) - E( X ) 2 (M1) Var( X ) = 2.10 A1 [6] 16. (a) required to solve P = 0.8 (M1) = 0.842... (or equivalent) (M1) σ = 7.13 (days) A1 N1 (b) P (survival after 21 days) = 0.337 (M1)A1 [5] 17. (A1) 14 Z 14 15 21 Z 6
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