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# chapter+3 - The Normal distributions BPS chapter 3 2006 W.H...

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The Normal distributions BPS chapter 3 © 2006 W.H. Freeman and Company

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Objectives (BPS 3) The Normal distributions Density curves Normal distributions The 68-95-99.7 rule The standard Normal distribution Finding Normal proportions Using the standard Normal table Finding a value given a proportion
Density curves A density curve is a mathematical model of a distribution. It is always on or above the horizontal axis. The total area under the curve, by definition, is equal to 1, or 100%. The area under the curve for a range of values is the proportion of all observations for that range. Histogram of a sample with the smoothed density curve theoretically describing the population

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Density curves come in any imaginable shape. Some are well-known mathematically and others aren’t.
Normal distributions e = 2.71828… The base of the natural logarithm π = pi = 3.14159… Normal—or Gaussian—distributions are a family of symmetrical, bell- shaped density curves defined by a mean μ ( mu ) and a standard deviation σ ( sigma ): N ( μ,σ ). 2 2 1 2 1 ) ( - - = π x e x f x x

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0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 A family of density curves Here the means are different ( μ = 10, 15, and 20) while the standard deviations are the same ( σ = 3). Here the means are the same ( = 15) while the standard deviations are different ( = 2, 4, and 6).
µ = 64.5 standard deviation σ = 2.5 N ( µ , ) = N (64.5, 2.5) All Normal curves  (μ , ) share the same  properties Reminder : µ (mu) is the mean of the idealized curve, while is the mean of a sample. σ (sigma) is the standard deviation of the idealized curve, while s is the s.d. of a sample. About 68% of all observations are within 1 standard deviation ( σ29 of the mean ( μ ). About 95% of all observations are within 2 of the mean . are within 3 of the mean. Inflection point x

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## This note was uploaded on 06/24/2008 for the course MATH 1070 taught by Professor Akbas during the Summer '08 term at Georgia State.

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chapter+3 - The Normal distributions BPS chapter 3 2006 W.H...

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