QZ 6-B - and variance Var X – Y = Var X Var Y = 0.20 2...

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STAT 400 Fall 2011 Version B Name ANSWERS . Quiz 6 (10 points) Be sure to show all your work; your partial credit might depend on it. No credit will be given without supporting work. 1. In Someville, the price of a gallon of milk ( X ) varies from day to day according to normal distribution with mean \$3.00 and standard deviation \$0.20. The price of a package of Oreo cookies ( Y ) also varies from day to day according to normal distribution with mean \$2.70 and standard deviation \$0.15. Assume the prices of a gallon of milk and a package of Oreo cookies are independent. a) (5) Find the probability that on a given day, the price of a package of Oreo cookies is higher than the the price of a gallon of milk. That is, find P ( Y > X ). P ( Y > X ) = P ( X – Y < 0 ). X – Y has Normal distribution with mean E ( X – Y ) = 3.00 – 2.70 = \$0.30
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Unformatted text preview: and variance Var ( X – Y ) = Var ( X ) + Var ( Y ) = 0.20 2 + 0.15 2 = 0.0625 ( standard deviation = \$0.25 ). P ( X – Y < 0 ) = -< 25 . 30 . Z P = P ( Z < – 1.20 ) = Φ ( – 1.20 ) = 0.1151 . b) (5) Alex is planning a Milk-and-Oreos party for his imaginary friend. He buys 4 gallons of milk and 7 packages of Oreo cookies. Find the probability that he paid less than \$30. That is, find P ( 4 X + 7 Y < 30 ). 4 X + 7 Y has Normal distribution with mean E ( 4 X + 7 Y ) = 4 ⋅ 3.00 + 7 ⋅ 2.70 = \$30.90 and variance Var ( 4 X + 7 Y ) = 16 ⋅ Var ( X ) + 49 ⋅ Var ( Y ) = 16 ⋅ 0.20 2 + 49 ⋅ 0.15 2 = 1.7425 ( standard deviation ≈ \$1.32 ). P ( 4 X + 7 Y < 30 ) = -< 32 . 1 90 . 30 30 Z P = P ( Z < – 0.68 ) = Φ ( – 0.68 ) = 0.2483 ....
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