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Unformatted text preview: ECON1203/ECON2292 Business and Economic Statistics Unit 3 Lecture 6 2 Lecture 6 topics Normal distribution Calculating areas under the normal curve Normal approximation to the Binomial Concept of an estimator Properties of estimators Key references Keller 8.2, 9.2 pp 309314, 10.1 Learning Outcomes: You should be able to Find probabilities using normal distribution Find normal distribution percentiles Approximate binomial with normal Know and check properties of estimators Normal distribution Plays a pivotal role in statistical theory & practice In practice heights of men & women are well approximated by normal distributions Will play a prominent role in upcoming discussion of statistical inference It is a continuous rv P ( X = x ) = 0 P(a < X < b) = area under curve between a & b Completely characterized by its mean m & variance s 2 3 Normal distribution Graphically, the probability density function (pdf) is symmetric, unimodal & bellshaped mean=median=mode Basic features include Range of support is unlimited, so  x Despite unlimited range little area in tails of a normal distribution 4.6% outside m 2 s ; 0.3% outside m 3 s (confirm from tables) Relevant tables in Keller,Table 3, p. B8B9 Also Part II of tutorial program 4 5 Normal distribution 6 Normal distribution If X is normally distributed with mean m and variance s 2 then we write X ~ N ( m , s 2 ) Algebraic formula for pdf of a normal rv x e x f x 2 2 1 2 1 ) ( s m s 7 Normal distribution No rules for when normal distribution is appropriate Determined empirically & from past experience Theoretical reasons why the normal is useful in problems involving samples from other populations Can also use statistical methods to assess whether a normal model might be reasonable Refer Week 12 Normal distribution as a model How can heights be normally distributed when negative heights arent possible? Let X be heights of 2 year olds in cms & assume X ~ N (80, 16) How far is zero from mean of X ? Recall (Week 2, slide 14) the (population) coefficient of variation is CV = / = 4/80 = 0.05 in this case Thus the model would suggest a negative height for a person 20 standard deviations less than the mean! So, normality may very well be an excellent approximation 8 9 Standard normal Recall Z scores Z = ( X m )/ s In general this standardization yields a rv with zero mean & standard deviation of one How are Z & X related?...
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This note was uploaded on 03/29/2012 for the course ECON 1102 taught by Professor Henry during the Three '08 term at University of New South Wales.
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