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UIUC STAT 400 - 400Ex3_3

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STAT 400 Lecture AL1 Examples for 3.3 Spring 2015 Dalpiaz Normal (Gaussian) Distribution. µ – mean σ – standard deviation  2 ,σμ N ()( )2 2 σμ2 2 1σπ−−=xexf, − ∞ < x < ∞. Standard Normal Distribution. mean 0 standard deviation 1 N ( 0 , 1 )Example: For the standard normal distribution, find the area to the left of z = 1.24 Area to the left of z = 1.24 is 0.8925. Z ~ N ( 0 , 1 ) X ~ N ( µ , σ 2 ) σμX Z−= X = µ + σ Z1. At Initech , the salaries of the employees are normally distributed with mean µ = $36,000 and standard deviation σ = $5,000. a) Mr. Smith is paid $42,000. What proportion of the employees of Initech are paid less that Mr. Smith? b) What proportion of the employees have their salaries over $40,000? c) Suppose 10 Initech employees are randomly and independently selected. What is the probability that 3 of them have their salaries over $40,000? d) What proportion of the employees have their salaries between $30,000 and $40,000?e) Mrs. Jones claims that her salary is high enough to just put her among the highest paid 15% of all employees working at Initech . Find her salary. f) Ms. Green claims that her salary is so low that 90% of the employees make more than she does. Find her salary. 2. Suppose that the lifetime of Outlast batteries is normally distributed with mean µ = 240 hours and unknown standard deviation. Suppose also that 20% of the batteries last less than 219 hours. Find the standard deviation of the distribution of the lifetimes.Let X be normally distributed with mean µ and standard deviation σ. Then the moment-generating function of X is M X ( t ) = 2 σ μ 22 tte+. Let Y = a X + b. Then M Y ( t ) = e b t M X ( a t ). Therefore, Y is normally distributed with mean a µ + b and variance a 2 σ 2 ( standard deviation | a | σ ). 1. (continued) g) All Initech employees receive a memo instructing them to put away 4% of their salaries plus $100 per month ( $1,200 per year ) in a special savings account to supplement Social Security. What proportion of the employees would put away more than $3,000 per


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UIUC STAT 400 - 400Ex3_3

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