Section_3 - CEE 304 Section 3 Problems 1 Suppose the...

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CEE 304 – Section 3 Problems 1. Suppose the diameters of ball bearings are normally distributed. You are told that the diameter of a ball bearing must be within 3.0 ± 0.01 cm. No bearing outside this range is acceptable. You know from process that diameters have μ = 3.0 cm and σ = 0.005 cm. a.) On average, how many will be scrapped? Answer : 4.56% b.) How many bearings fall within ± 1 standard deviation of the mean? Answer : 68.3% c.) What diameters delineate the 95 th and 5 th percentile? 95 th percentile = Z 0.05 , [ Φ (Z 0.05 ) = 0.95] ==> from table, Z 0.05 = 1.645 => X = μ + 1.645 σ = 3.0082 By symmetry, 5 th percentile = - Z 0.05 = -1.645 ==> X = μ − 1.645 σ = 2.9918 2. Suppose the weekly demand for propane gas (in thousands of gallons) is a random variable X with pdf: 2 1 ( ) 2 1 f x x = for 1 x 2, and 0 otherwise a.) Show that it is a pdf. First must check if f ( x ) is defined for all x , which it is! Second, check if f ( x ) 0 for all x , which it is! Third, check if 2 1 ( ) 2 1 1 f x dx dx x +∞ +∞ −∞ −∞ = = , which it does!
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