STAT 400 Lecture AL1 Spring 2015 Dalpiaz Examples for 3 3 Normal Gaussian Distribution mean standard deviation N 2 f x 1 2 e x 2 2 2 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 z 1 24 is 0 8925 of Z N 0 1 X N 2 Z X X Z 1 At Initech the salaries of the employees are normally distributed with mean 36 000 and standard deviation 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 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 and standard deviation Then the moment generating function of X is MX t e t 2 2 t 2 Let Y a X b Then M Y t e b t M X a t Therefore Y is normally distributed with mean a b and variance a 2 2 standard deviation a 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 year
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