400Hw09ans

400Hw09ans - STAT 400 Spring 2011 Homework #9 (due Friday,...

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STAT 400 Spring 2011 Homework #9 (due Friday, April 1, by 3:00 p.m.) 1. a) Suppose that the actual amount of instant coffee a filling machine puts into "6-ounce" c a n s v a r i e s f r o m c a n t o c a n a n d t h a t t h e a c t u a l f i ll may be considered a random variable h a v i n g a n o r m a l d i s t r i b u t i o n , w i t h a m e a n o f 6 . 0 3 ounces and a standard deviation of 0.04 o u n c e s . S u p p o s e a c a s e c o n t a i n s 1 2 c a n s a n d c a n b e considered as a random sample of c a n s . W h a t i s t h e p r o b a b i l i t y t h a t t h e a v e r a g e f i ll in a case is below 6 ounces? μ = 6.03, σ = 0.04, n = 12. Want P( X < 6 ) = ? S i n c e t h e d i s t r i b u t i o n w e s a m p l e f r o m i s n o r m a l , . n Z X = - σ μ P ( ) = - < 12 04 . 0 03 . 6 6 Z P = P( Z - 2.60 ) = 0.0047 . b) Suppose that the actual amount of instant coffee a filling machine puts into "6-ounce" c a n s v a r i e s f r o m c a n t o c a n a n d t h a t t h e a c t u a l f i ll may be considered a random variable h a v i n g a n o r m a l d i s t r i b u t i o n , w i t h a s t a n d a r d d e v i ation of 0.04 ounces. If on average o n l y 3 o u t o f e v e r y 2 0 0 c a n s c o n t a i n l e s s t h a n 6 o unces of coffee, what must be the m e a n f i l l o f t h e s e c a n s ? N o r m a l d i s t r i b u t i o n , μ = ? Know that P( X < 6 ) = 200 3 = 0.015. Find z such that P( Z < z ) = 0.015. T h e a r e a t o t h e l e f t i s 0 . 0 1 5 ( l e s s t h a n 0 . 5 0 ) . U s i n g t h e s t a n d a r d n o r m a l t a b l e , z = 2.17 . x = μ + σ z . 6 = μ + 0.04 (– 2.17). μ = 6.0868 oz .
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2. S u p p o s e t h a t 6 0 % o f a l l v o t e r s i n a c e r t a i n c o u n t y favor a tax increase to pay for a b e t t e r s c h o o l s y s t e m . A r a n d o m s a m p l e o f 4 0 0 v o ters is selected. Use Normal a p p r o x i m a t i o n t o f i n d t h e f o l l o w i n g : B i n o m i a l ( 4 0 0 , 0 . 6 0 ) . μ = 240, σ = 9.798. a) What is the probability that more than 250 of the sample members favor the tax i n c r e a s e ? P ( X > 2 5 0 ) = P ( X > 2 5 0 . 5 ) = P ( Z > 1 . 0 7 ) = 0.1423 . b) What is the probability that at least 260 of the sample members favor the tax i n c r e a s e ? P ( X 260 ) = P( X > 259.5 ) = P( Z > 1.99 ) = 0.0233 . c) What is the probability that at most 235 of the sample members favor the tax i n c r e a s e ? P ( X 235 ) = P( X < 235.5 ) = P( Z < – 0.46 ) = 0.3228 . d) What is the probability that between 225 and 245 (both inclusive) of the sample m e m b e r s f a v o r t h e t a x i n c r e a s e ? P ( 2 2 5 X 245 ) = P( 224.5 < X < 245.5 ) = P( – 1.58 < Z < 0.56 ) = 0.6552 . e) What is the probability that exactly 234 of the sample members favor the tax i n c r e a s e ? C o m p a r e t h e a n s w e r w i t h t h e e x a c t p r o b a bility. P ( X = 2 3 4 ) = P ( 2 3 3 . 5 < X < 2 3 4 . 5 ) = P ( 0.66 < Z < – 0.56 ) = 0.0331 . B i n o m i a l : P ( X = 2 3 4 ) = () 166 234 40 . 0 60 . 0 234 400 = 0.033565 .
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3. S u p p o s e t h e n u m b e r o f e - m a i l m e s s a g e s r e c e i v e d b y Mike follows Poisson d i s t r i b u t i o n w i t h t h e a v e r a g e r a g e o f 4 2 m e s s a g e s per week. Use Normal a p p r o x i m a t i o n t o f i n d t h e f o l l o w i n g : P o i s s o n , λ = 42.
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400Hw09ans - STAT 400 Spring 2011 Homework #9 (due Friday,...

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