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# ch 9 - Chapter 9 9.2-THE NORMAL DISTRIBUTION The most...

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Chapter 9 9.2--THE NORMAL DISTRIBUTION - The most important probability distribution in statistics because: o Many variables in science/nature have probability distributions that closely resemble it o It’s a convenient model for estimating probabilities for other theoretical distributions o Provides excellent approximation to binomial distribution when number of trials is large - Serendipity produced normal distribution o Abraham de Moivre searching for shortcut method for computing probabilities for binomial random variables, and came up with normal distribution Derived function rule for determining height of normal distribution, f(X), for any value of random variable, X. - Characteristics : o A random variable X is said to be normally distributed it its probability distribution is given by the function rule for the normal distribution o Function rule for normal distribution: --blue box pg. 232-- e = the base of the system of natural logarithms = mean of particular norm. distr. = standard deviation of particular norm. distr. o Shaped like bell o Inflection points = (mean +/- standard deviation) Defines place where curve changes from being concave to convex or vice versa. o Tails of curve extend indefinitely in both directions, never quite touching horizontal axis o Total area under curve = 1 Converting Scores to Standard Scores - Standard normal distribution o The normal distribution that’s the standard o Has mean= 0 and standard deviation= 1 o Random variable values for this distribution called standard scores and denoted by z o If case doesn’t have mean=0 and standard deviation=1, random variable must be transformed into standard score to use standard normal distribution table. Formula for this is : pg. 233 Applying z-score transformation to each X in distribution will result in mean of 0, z (put bar over z!!!!) = 0, a standard deviation of 1, and S z =1. o A “z” score” transformation alters the mean and standard deviation of the transformed random variable but not the relative location of scores in the distribution

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o Graph of X scores and graph of “z” scores identical except for the central tendencie dispersions. o Transforming scores to standard scores DOES NOT change shape of distribution or the relative position of scores---only changes the MEAN and STANDARD DEVIATION o For normaldistributions, most “z” scores between+ /- 3 standard deviations (99.73%) Finding Areas under the Normal Distribution (graphs on pg. 234) Finding Scores When the Area Is Known - if you have a percentile rank in mind or know the size of the area above or below a point in a distribution and you want to determine the untransformedscore corresponding to that rank or point, use formula: --pg. 235-- - EX: --pg. 236 2
Normal Approximation to the Binomial Distribution - Although normal distribution function rule was originally created for large number of trials, can approximate excellently even when “n” is small - When using the continuous normal distribution to estimate a discrete random variable, must think of four heads as occupying the interval from 3.5 to 4.5

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