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**Unformatted text preview: **The Normal distributions BPS chapter 3 © 2006 W.H. Freeman and Company 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 Density curves come in any imaginable shape. Some are well-known mathematically and others aren’t. Apply your knowledge Go to page 67 and work on the following problem: 3.1 Sketch density curves • The median of a density curve is the equal-areas point, the point that divides the area under the curve in half. •The mean of a density curve is the balance point, at which the curve would balance if made of solid material. The median and mean are the same for a symmetric density curve. The mean of a skewed curve is pulled away from the median in the direction of the long tail. Median and mean of a density curve Apply your knowledge Go to page 69 and work on the following problems: 3.2 A uniform distribution 3.3 Mean and median 3.4 Mean and median 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 ) ( ! " # $ % & ’ ’ = ( μ ) x e x f x x 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 A family of density curves 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 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 ( μ = 15) while the standard deviations are different ( σ = 2, 4, and 6). mean µ = 64.5 standard deviation σ = 2.5 N ( µ , σ ) = N (64.5, 2.5) All Normal curves N ( μ , σ ) share the same properties Reminder : µ (mu) is the mean of the idealized curve, while is the mean of a sample....

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