Lecture6U3 - ECON1203/ECON2292 Business and Economic...

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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 309-314, 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 & bell-shaped 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. B8-B9 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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Lecture6U3 - ECON1203/ECON2292 Business and Economic...

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