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HW9_08_soln

# HW9_08_soln - IE 111 Fall 2007 Solutions for Homework...

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IE 111 Fall 2007 Solutions for Homework Assignment 9 Note: I used Excel to evaluate Normal probabilities and percentiles instead of the Normal tables. Because the table’s accuracy is limited in significant digits, my answers may vary somewhat from those of people using the Tables. Students should not be penalized for such discrepancies. Question 1. The lower and upper specifications on the diameter of a certain manufactured part are [0.99 cm to 1.01 cm]. If the diameter of a part falls in this range then it is "good". Otherwise it is defective. The actual diameter of parts produced is a random variable distributed N(1.0 , 0.000025). a) What percentage of parts diameters fall within the specifications? σ =0.005 ( 29 ) 2 ( ) 2 ( 2 2 005 . 0 1 01 . 1 005 . 0 1 99 . 0 ) 01 . 1 99 . 0 ( - Φ - Φ = < < - = - < < - = < < X P Z P X P =0.977250-0.022750 = 0.9545 b) Suppose I change the mean from 1.0 to 1.005. Now what percentage of parts diamet- ers fall within the specifications? ( 29 ) 3 ( ) 1 ( 1 3 005 . 0 005 . 1 01 . 1 005 . 0 005 . 1 99 . 0 ) 01 . 1 99 . 0 ( - Φ - Φ = < < - = - < < - = < < X P Z P X P = 0.841345 - 0.00135 = 0.84 c) Suppose the mean is 1.0 as in part a). Suppose further that I can adjust the standard deviation σ . What value of σ will result in 1% of the parts being defective? 0.003882 2.575829 01 . 0 or tables) function Norminv Excel (From 2.575829 000 . 1 01 . 1 995 . 0 000 . 1 01 . 1 99 . 0 000 . 1 01 . 1 000 . 1 99 . 0 = = = - = - < = - < < - σ σ σ σ σ Z P Z P

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Question 2. The number of miles my car can go on a gallon of gasoline is Normally distributed with a mean of 25 miles and a standard deviation of 7 miles. a) I currently have exactly one gallon in my tank, and have to travel 30 miles to my des- tination. What is the probability I make it to my destination before I run out of gas? Let X = the number of miles I go. P(I make it) = P(X>30) 0.237525 0.762475 - 1 0.714286) ( 1 7 25 30 1 ) 30 ( 1 = = < - = - < - = < - = Z P Z P X P b) Find the distance D so that I have a 90% chance of traveling D miles before running out of gas (assuming again that I start with one gallon). 03 . 16 D tables) (from -1.28155 7 25 1 . 0 7 25 1 . 0 ) ( 9 . 0 ) ( 1 = = - = - < = < = < - = D D Z P D X P D X P c) Suppose I can change my mean mileage per gallon by adjusting the carburetor. What value should I set the mean to so that I have a 90% chance of making it 30 miles on one gallon of gas? 38.9709 tables) (from -1.28155 7 30 1 . 0 7 30 1 . 0 ) 30 ( 9 . 0 ) 30 ( 1 = = - = - < = < = < - = μ μ μ Z P X P X P Question 3. Steel balls are manufactured for use in ball bearings. The diameter of a randomly selected ball is Normally distributed with a mean of 3 millimeters and a standard deviation of 0.2 millimeters. a) If the specifications say that ball diameter must be in the range 3 ± 0.5 millimeters in order to meet quality requirements, what percentage of the balls will meet these re- quirements?
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HW9_08_soln - IE 111 Fall 2007 Solutions for Homework...

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