ECON 214 - The Normal Distribution.pdf

The proportion of measurements weight of hamburgers

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The proportion of measurements (weight of hamburgers) within a given range can be found by the area under the smooth curve over this range. This curve is important because we can use it to determine the probability that measurements (e.g. weight of a randomly selected hamburger) lie within a given range (such as between 0.20 and 0.30 kg.) Slide 9
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Continuous Random Variables Slide 10
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Continuous Random Variables The smooth curve so obtained is called a probability density function or probability curve. The total area under any probability density function must equal 1. The probability that the random variable will assume a value between any two points, from say, x 1 to x 2 , equals the area under the curve from x 1 to x 2. Thus for continuous random variables we are interested in calculating probabilities over a range of values. Note then that the probability that a continuous random variable is precisely equal to a particular value is zero. Slide 11
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The Normal Probability Distribution The most important continuous probability distribution is the normal distribution. The formula for the probability density function of the normal random variable is Where X is said to have a normal distribution with mean µ and variance σ 2 . Slide 12 2 1 [( )/ ] 2 2 1 ( ) 2 x f x e 
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The Normal Distribution Examples of normally distributed variables: IQ Men’s heights Women’s heights The sample mean Slide 13
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The Normal Distribution The normal distribution has the following characteristics: Bell shaped Symmetric about the mean Unimodal The area under the curve is 100% = 1 Its shape and location depends on the mean and standard deviation Extends from x = - to + (in theory). Slide 14
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The Normal Distribution The two parameters of the Normal distribution are the mean and the variance 2 . x ~ N( , 2 ) Men’s heights are Normally distributed with mean 174 cm and variance 92.16 cm. x M ~ N(174, 92.16) Women’s heights are Normally distributed with a mean of 166 cm and variance 40.32 cm. x W ~ N(166, 40.32) Slide 15
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